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A strategy for ambulance system designs : an investigation of the ambulance system in the greater Vancouver… Tan, Eddy 1974

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A STRATEGY FOR AMBULANCE SYSTEM DESIGNS: An investigation of the Ambulance System in the Greater Vancouver Regional D i s t r i c t by Eddy Tan BSc., University of V i c t o r i a , 1972. A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN BUSINESS ADMINISTRATION IN THE MANAGEMENT SCIENCE DEPARTMENT OF THE FACULTY OF COMMERCE We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September 1974 A. D. In p resent ing t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the requirements fo r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree t h a t permiss ion for e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that copy ing or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l ga in s h a l l not be a l lowed without my w r i t t e n p e r m i s s i o n . Department of C OA1 »7 C- r~t f The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada Date 4-€£/ / 0 - / c/ 7 </ ( i i ) ABSTRACT OF THE DISSERTATION T h i s d i s s e r t a t i o n i s a study o f the e f f e c t s o f a l t e r n a t i v e d e s i g n parameters and c o n t r o l p o l i c i e s i n the p l a n n i n g and o p e r a t i o n o f emergency ambulance systems. The ambulance system i s c h a r a c t e r i z e d by a sequence o f events b e g i n n i n g w i t h the n o t i f i c a t i o n o f an i n c i d e n t and endin g w i t h the d e l i v e r y o f the p a t i e n t t o an emergency m e d i c a l f a c i l i t y . A computer s i m u l a t i o n model was s u b s e q u e n t l y d e v e l o p e d and v a l i d a t e d f o r the G.V.R.D. Experiments are conducted u s i n g t h i s S i m u l a t i o n Model and a L o c a l Search A l g o r i t h m which d e t e r m i n e d o p t i m a l l o c a t i o n s t o e v a l u a t e the e f f e c t o f the f o l l o w i n g on system p e r f o r m a n c e : * number o f ambulances * d i s p a t c h i n g p o l i c i e s * demand r a t e R e s u l t s from the above experiments l e a d t o the f o l l o w i n g recommendations: * Ambulance s e r v i c e s h o u l d be p r o v i d e d on a r e g i o n a l b a s i s . * Ambulances s h o u l d be l o c a t e d s t r a t e g i c a l l y to m i n i m i z e response t i m e . * For a g i v e n d e s i r e d l e v e l of response t i m e , the number and l o c a t i o n o f ambulances s h o u l d be d i f f e r e n t i n d i f f e r e n t p e r i o d s o f the day. * P e r i o d i c r e v i e w o f the l e v e l and s p a t i a l d i s t r i b u t i o n o f demand s h o u l d be made and ad j u s t m e n t s s h o u l d be made i n the number and l o c a t i o n s o f ambulances. * The d e t e r m i n a t i o n o f the p r o p e r r e s p o n s e time l e v e l s h o u l d be based i n p a r t on an a n a l y s i s o f the t r a d e o f f between c o s t and response t i m e . ( i i i ) TABLE OF CONTENTS Page ABSTRACT OF THE DISSERTATION ( i i ) LIST OF TABLES ( v) LIST OF FIGURES ( v i ) LIST OF MAPS ( v i i ) CHAPTER 1 INTRODUCTION 1 1.1. I n t r o d u c t i o n 1 1.1.1. Scope and O b j e c t i v e s o f Research 1 1.1.2. Research O u t l i n e 2 1.1.3. F u n c t i o n o f the Ambulance System 3 1.2. Ambulance System Models 6 1.2.1. Queuing Model 6 1.2.2. S i m u l a t i o n Model 8 1.2.3. O p t i m i z a t i o n Method t o Locate Ambulances 8 CHAPTER 2 EXISTING AMBULANCE SERVICES IN G.V.R.D. 10 2.1. E x i s t i n g Ambulance S e r v i c e 10 2.2. Demand For Ambulance S e r v i c e s 13 2.2.1. Data C o l l e c t i o n 13 2.2.2. D e f i n i t i o n s 15 2.2.3. C u r r e n t Demand 16 a. Temporal V a r i a t i o n s ( i ) Seasonal V a r i a t i o n s ( i i ) Day o f Week V a r i a t i o n s ( i i i ) Time o f Day V a r i a t i o n s 2.2.4. Demand Growth P a t t e r n 20 2.3. P r o d u c t i o n and E v a l u a t i o n 24 2.3.1. P r o d u c t i o n Components 24 2.3.2. A n a l y s i s o f Components 26 2.3.3. Ambulance U t i l i z a t i o n 34 CHAPTER 3 COMPUTER SIMULATION MODEL 36 3.1. Use o f S i m u l a t i o n i n System A n a l y s i s 36 ( i v ) Page 3.2. General D e s c r i p t i o n of Computer S i m u l a t i o n Models 3 7 3.2.1. Ambulance C a l l G e n e r a t o r 37 3.2.2. Main S i m u l a t o r 37 3.3. Model V a l i d a t i o n f o r the Vancouver Ambulance System 40 3.3.1. General V a l i d i t y o f the Model 41 3.3.2. S t a t i s t i c a l Test 41 3.3.3. D e t e r m i n i n g the Length o f the S i m u l a t i o n P e r i o d 42 3.3.4. Correspondence o f Real World and S i m u l a t e d Data 42 CHAPTER 4 THE AMBULANCE LOCATION PROBLEM 48 4.1. E s s e n t i a l F e a t u r e s i n the L o c a t i o n o f Ambulances 48 4.2. An A n a l y t i c a l A p p r o x i m a t i o n to Mean Response Time 48 4.3. A L o c a l Search A l g o r i t h m 50 4.4. E x p e r i m e n t a l E v a l u a t i o n o f the A l g o r i t h m 51 4.5. S e n s i t i v i t y o f A l g o r i t h m to Choice o f S t a r t i n g L o c a t i o n s 52 CHAPTER 5 DESIGN OF THE EXPERIMENT 54 5.1. I n t r o d u c t i o n 54 5.2. E x p e r i m e n t a l Design 55 5.3. The C o l l e c t i o n o f S i m u l a t i o n S t a t i s t i c s 57 CHAPTER 6 EXPERIMENT RESULTS AND THEIR IMPLICATIONS 58 6.1. Optimal Ambulance L o c a t i o n 58 6.2. V a r i a t i o n s i n Response Time by M u n i c i p a l i t y 71 6.3. D i s p a t c h Rules 76 6.4. S e n s i t i v i t y o f Response Times to Changes i n Demand 80 6.5. T r a d e - o f f s Between Annual O p e r a t i n g Cost and Response Time 81 6.6. Summary and C o n c l u s i o n s . 84 ( v ) LIST OF TABLES Page Table 1 Ambulance Companies S e r v i n g the G.V.R.D 11 2 Ambulance S e r v i c e by M u n i c i p a l i t y 12 3 D a i l y Ambulance Demand Rate by Time o f Day and M u n i c i p a l i t y 19 4 D i s t r i b u t i o n o f Ambulance D e s t i n a t i o n s by M u n i c i p a l i t y 21 5 D i s t r i b u t i o n o f C a l l O r i g i n by D e s t i n a t i o n . . . . 22 6 Average Response Time and S e r v i c e Time by M u n i c i p a l i t y 28 7 Average O p e r a t i o n Times by Type and Time of C a l l 33 8 Ambulance U t i l i z a t i o n by Time o f Day 35 9 CHI Square Goodness o f F i t Test o f Response Time P r o b a b i l i t y D i s t r i b u t i o n 46 10 Average and 90th F r a c t i l e Response Time by M u n i c i p a l i t y and by Number o f Ambulances (8 a.m. - 6 p.m., a l l c a l l s ) 72 11 Average and 90th F r a c t i l e Response Times by M u n c i p a l i t y and by Number o f Ambulances ( 6 . p . m . m i d n i g h t , a l l c a l l s ) 73 12 Average and 90th F r a c t i l e Response Times by M u n i c i p a l i t y and by Number o f Ambulances (midnight - 8 a.m., a l l c a l l s ) 74 13 Average Response Time by M u n i c i p a l i t y w i t h O ptimal Placement o f 21 Ambulances f o r True, Squared, and Cubed T r a v e l Times 77 14 Optimal Placement o f Ambulance Depots by Time of Day and Number o f Ambulances 86 ( v i ) LIST OF FIGURES Page F i g . 1 Average G.V.R.D. C a l l Rate ( c a l l s / h o u r ) by Time o f Day 18 2. D i s t r i b u t i o n o f Response Time, C u r r e n t Data, A l l C a l l s 31 3 D i s t r i b u t i o n o f Response Time, C u r r e n t Data, Emergency 32 4 D i s t r i b u t i o n o f Response Time f o r G.V.R.D. Ambulance System 45 5 Stan d a r d D e v i a t i o n v e r s u s number o f c a l l s 47 6 E f f e c t o f P r o b a b i l i t y o f an i n c i d e n t d i s p a t c h by K-th c l o s e s t Ambulance 53 7 D i s t r i b u t i o n o f Response Time w i t h Optimal Placement o f Ambulances (8 a.m. - 6 p.m.) 60 8 D i s t r i b u t i o n o f Response Time w i t h Optimal Placement o f Ambulances (7 p.m. - m i d n i g h t ) 61 9 D i s t r i b u t i o n o f Response Time w i t h O p t i m a l Placement o f Ambulances ( m i d n i g h t - 8 a.m.) 62 10 Average Response Time by Number o f Ambulances by Time of Day, f o r a l l c a l l s 66 11 Average Response Time by Number o f Ambulances by Time of Day, f o r emergency c a l l s o n l y 67 12 E f f e c t o f Number o f Ambulances on Response Time, S e r v i c e Time and Ambulance U t i l i z a t i o n (8 a.m. - 6 p.m.) , 68 13 E f f e c t o f Number o f Ambulances on Response Time, S e r v i c e Time and Ambulance U t i l i z a t i o n (6 p.m. - m i d n i g h t ) 69 14 E f f e c t o f Number o f Ambulances on Response Time, S e r v i c e Time and Ambulance U t i l i z a t i o n ( m i d n i g h t - 8 a.m.)_ 70 15 M u n i c i p a l V a r i a t i o n i n Average Response Time by Number o f Ambulances f o r a l l c a l l s (8 a.m.-6pcmB)75 16 Trade o f f Between Average Response Times and Annual O p e r a t i n g Cost 82 ( v i i ) LIST OF MAPS Page Map 1 C u r r e n t Ambulance L o c a t i o n s 63 2 Optimal Placement o f 21 Ambulances (8 a.m. - 6 p.m.) 64 3 Optimal Placement of 21 Ambulances w i t h Squared T r a v e l Time (8 a.m. - 6 p.m.) 78 4 Optimal Placement o f 21 Ambulances w i t h cubed T r a v e l Time (8 a.m. - 6 p.m.) 79 - 1 -CHAPTER 1 1 . 1 . I n t r o d u c t i o n An e f f e c t i v e ambulance s e r v i c e i s a v i t a l p a r t o f every community. I t p r o v i d e s l i f e - g i v i n g s u p port t o people i n need. Without i t , hundreds o f peop l e would d i e each y e a r - a c c i d e n t v i c t i m s unable t o get m e d i c a l a t t e n t i o n i n t i m e . Time i s the essence o f every ambulance s e r v i c e . The speed w i t h which an ambulance can respond t o a c a l l i s c r u c i a l . In the G r e a t e r Vancouver R e g i o n a l D i s t r i c t (G.V.R.D.), the ambulance system's a b i l i t y t o respond t o emergency c a l l s depends on both p l a n n i n g and management p o l i c i e s . T h i s s t u d y i s an attempt t o e x p l a i n why t h i s i s so. 1 . 1 . 1 . Scope and O b j e c t i v e s of Research The p r i m a r y o b j e c t i v e of t h i s r e s e a r c h i s to study the management and c o n t r o l of the ambulance system and, s p e c i f i -c a l l y determine how i t s p l a n n i n g and management p o l i c i e s a f f e c t the system's a b i l i t y t o respond t o emergency c a l l s . In o r d e r t o meet the r e s e a r c h o b j e c t i v e s , i t was n e c e s s a r y t o conduct s i m u l a t i o n s t u d i e s o f a l t e r n a t i v e ambulance systems to p r o v i d e knowledge and methodologies f o r the p l a n n i n g and o p e r a t i o n o f t h i s subsystem w i t h i n community m e d i c a l system. Three s u b - o b j e c t i v e s were e s t a b l i s h e d to det e r m i n e : * the e f f e c t s o f d e s i g n parameters on the ambulance system performance, i . e . , the number o f ambulances and t h e i r l o c a t i o n * the e f f e c t o f d i s p a t c h i n g p o l i c y on ambulance system performance - 2 * the s e n s i t i v i t y of ambulance system performance in response to change in the demands for service i . e . , the occurrence rate of ambulance c a l l s . 1.1.2. Research Outline Section 1.2. of this chapter relates the ambulance system to three theoretical frameworks: a queuing model, a simulation model, and an optimization model. A b r i e f review of three different studies on the subject of ambulance systems is also included. Chapter 2 contains an analysis of the ambulance service which is now being provided in the G.V.R.D. Included is a discussion about: * who provides the ambulance service * the temporal and s p a t i a l d i s t r i b u t i o n of c a l l o r i g i n * the d i s t r i b u t i o n of c a l l destinations * current "response time" levels ( i . e . , the time from when the c a l l for help is placed to the time the ambulance arrives on the scene) * ambulance u t i l i z a t i o n . Chapter 3 is devoted to the description of the simulation model used in this study. The chapter also discusses: * why a simulation i s used * a general description of the simulation model * a discussion of the assumptions made * a v a l i d a t i o n of the model Chapter 4 contains the "optimization algorithm" and includes: - 3 -* a n a l y t i c a l a p p r o x i m a t i o n o f mean response time from s i m u l a t i o n r e s u l t s * comparison between the v a l u e o f mean response time o b t a i n e d from the s i m u l a t i o n and the a n a l y t i c a l a p p r o x i m a t i o n * the " a l g o r i t h m s t e p s " t o l o c a t e ambulances w i t h the o b j e c t i v e o f m i n i m i z i n g mean response time * e x p e r i m e n t a l e v a l u a t i o n o f the " a l g o r i t h m " Chapter 5 c o n t a i n s e x p e r i m e n t s . T h i s c h a p t e r develops the p l a n o f the experiments u s i n g the s i m u l a t i o n and the o p t i m i z a t i o n a l g o r i t h m . The s t a t i s t i c a l output from the s i m u l a t i o n i s e x p l a i n e d . F i n a l l y , Chapter 6 c o n t a i n s e x p e r i m e n t a l r e s u l t s , p o l i c y i m p l i c a t i o n s , and c o n c l u d i n g remarks. T h i s c h a p t e r a l s o examines: * the average o v e r a l l response time i n each m u n i c i p a l i t y f o r each o f t h r e e l e v e l s o f s e r v i c e * the impact o f d i s p a t c h r u l e s on response times * the s e n s i t i v i t y of response time t o l e v e l o f demand (e . g . , by how much does average response time d e t e r -i o r a t e when c a l l r a t e s p e r hour are doubled. 1.1.3. F u n c t i o n o f the Ambulance System The ambulance system has f i v e p r i n c i p l e f u n c t i o n s : r e s c u e , l i f e s u p p o r t , p r e l i m i n a r y emergency c a r e , t r a n s p o r t t o emergency h e a l t h care f a c i l i t i e s , and t r e a t m e n t at emergency f a c i l i t i e s . U n t i l q u i t e r e c e n t l y r e s p o n s i b i l i t y f o r the s e f u n c t i o n s r e s i d e d among a number o f p r i v a t e and p u b l i c o r g a n i z a t i o n s w i t h i n the G.V.R.D.: p r i v a t e ambulance companies, p o l i c e and f i r e departments. - 4 -and h o s p i t a l emergency departments. The importance o f an e f f e c t i v e and e f f i c i e n t ambulance s e r v i c e can be i l l u s t r a t e d by a s i m p l e n u m e r i c a l c a l c u l a t i o n . C u r r e n t l y , t h e r e are r o u g h l y 45,000 ambulance c a l l s per year i n the G.V.R.D. Of t h e s e , a p p r o x i m a t e l y 10,000 are " h i g h p r i o r i t y " or "emergency" c a l l s - - c a l l s i n which the s i r e n i s r e q u i r e d t o speed the ambulance t o the scene o f the c a l l . Of these h i g h p r i o r i t y c a l l s t h e r e i s some pe r c e n t a g e i n which a r a p i d response by h i g h l y t r a i n e d ambulance p e r s o n n e l i s c r u c i a l t o the maintenance o f the p a t i e n t ' s l i f e . I f , f o r the sake o f argument, t h i s p e r c e n t a g e were o n l y 2%, t h a t would t r a n s l a t e i n t o 240 c a l l s per y e a r i n which the p a t i e n t may l i v e or d i e , depending on the q u a l i t y o f the ambulance s e r v i c e . F u r t h e r e vidence o f the importance of ambulance s e r v i c e i s c o n t a i n e d i n a study by Frey e t . a l . o f 150 deaths r e s u l t -i n g from a u t o m o b i l e a c c i d e n t s ( 5 ) . I t was found t h a t 18% o f the v i c t i m s would p r o b a b l y have s u r v i v e d had the q u a l i t y o f ambulance s e r v i c e been h i g h e r . The r e c e n t r e p o r t of the H e a l t h Systems Task F o r c e (the " F o u l k e s r e p o r t " ) has a l s o s t r e s s e d the importance of and need f o r improved ambulance s e r v i c e ( 4 ) . Let us now b r i e f l y c o n s i d e r each o f the f i v e f u n c t i o n s o f the ambulance system. 1. Rescue In some emergencies c e r t a i n s p e c i a l i z e d s k i l l s and e q u i p -ment are r e q u i r e d j u s t t o get a p a t i e n t t o a p o i n t of s a f e t r e a t m e n t . For example, a v i c t i m may have t o be e x t r i c a t e d from a wrecked automobile o r r e s c u e d from a c l i f f . S i n c e t h e s e - 5 -situations may require very rapid response and since the s k i l l s and equipment required are often the same as those used to fight f i r e s , many municipalities currently rely on the f i r e department to provide rescue service. 2. Life Support In the f i r s t c r i t i c a l moments after an emergency, immediate response i s often necessary to maintain l i f e . This i s part-i c u l a r l y true i f breathing has stopped or there is heavy bleeding. The primary consideration then is to reach the victim quickly and render f i r s t aid to restore breathing or stop bleeding. In such cases rapid response is essential but a sophisticated level of training i s generally not. Frequently f i r e rescue vehicles and f i r e department personnel perform this function. To combat the wide range of emergencies, however, broader and more sophisticated t r a i n i n g and more complex technological support are required. Since each s p e c i a l l y equipped and trained unit is quite expensive, the decision-maker must be constantly aware of the trade-offs between training and equipment costs and the r a p i d i t y of response. 3. Preliminary Emergency Medical Care This function consists not only of the provision of f i r s t aid, but also the decision on the appropriate preliminary treatment path: What services should be done at the scene to s t a b l i z e the patient? How quickly must the patient be moved to a treatment f a c i l i t y ? Must the treatment f a c i l i t y be forewarned of the patient's a r r i v a l ? 4. Transport to the Emergency Health Care F a c i l i t y Once the patient is in the ambulance and is being transported to the emergency f a c i l i t y , f i r s t aid may be continued and - 6 -additional information may be gathered. 5. Treatment at an Emergency Health Care F a c i l i t y It is not common to regard the treatment at the emergency f a c i l i t y as part of the emergency ambulance service system. However, there are instances in which the attendants have s u f f i c i e n t medical training and experience to be of continued value to the patient by assi s t i n g the emergency room s t a f f . 1.2. Ambulance System Models The following is a description of two models related to the ambulance system and a discussion of an optimization method. 1 . 2 . 1 . Queuing Model The ambulance system can be conceptualized as a queuing system in the following sense: (i) A patient (customer) "a r r i v e s " at the moment he places a c a l l for an ambulance; ( i i ) Service begins at the moment the ambulance is dispatched to the patient; ( i i i ) Service ends when the patient is delivered to the hospital and when the ambulance becomes available for another c a l l . In one study, Bell ( 1 ) developed a queuing model for comput-ing the size of an emergency ambulance f l e e t required to meet specified standards of service. He used the example of a system with a uniform incident pattern with a single ambulance location. By analyzing a queuing system with unlimited ambulance a v a i l -a b i l i t y , he provides approximate results to overcome the d i f f i -c u l t i e s of computing steady state results for a multi server queuing system. He presented e a s i l y computed guidelines to assist - 7 -the a d m i n i s t r a t o r o f an ambulance system i n d e t e r m i n i n g o p t i m a l f l e e t s i z e . U n f o r t u n a t e l y , h i s method can o n l y be a p p l i e d f o r u n i f o r m i n c i d e n t p a t t e r n s u s i n g a s i n g l e ambulance l o c a t i o n , which of c o u r s e i s not the case i n the G.V.R.D. Morse, has examined s i n g l e queue and m u l t i p l e queue systems and found t h a t the s i n g l e queue system e x h i b i t e d a s m a l l e r mean w a i t i n g time than the m u l t i p l e queue system. ( 6 ) T h i s r e s u l t s u p p o r t s the common p r a c t i c e o f f o r m i n g a s i n g l e d i s p a t c h queue when a l l ambulances are busy and a s s i g n i n g the f i r s t a v a i l a b l e ambulance t o the i n c i d e n t which has been w a i t i n g the l o n g e s t . T h i s p o l i c y i s o f t e n a l t e r e d s l i g h t l y by a s s i g n i n g a busy ambulance t o respond to a nearby t r a n s f e r c a l l a f t e r c o m p l e t i n g i t s c u r r e n t c a l l . The d i s t r i b u t i o n o f i n c i d e n t a r r i v a l r a t e s found i n the G.V.R.D. was found to f i t the f a m i l i a r P o i s s o n d i s t r i b u t i o n w i t h a time v a r i a b l e r a t e . The d i s t r i b u t i o n o f s e r v i c e time however was found not t o be e x p o n e n t i a l . Moreover, i t was h i g h l y dependent on the p a r t i c u l a r ambulances busy at the time each c a l l was r e c e i v e d . In a d d i t i o n , the s e r v i c e time d i s t r i b u t i o n i s not the same f o r every s e r v e r except f o r the s p e c i a l case when a l l the ambulances are l o c a t e d at the same s t a t i o n . T h i s c h a r a c t e r i s t i c o f s e r v e r s s u b s t a n t i a l l y c o m p l i c a t e s any a n a l y t i c a l e f f o r t t o model such a queuing system because the l o c a t i o n of the s e r v e r s and s p a t i a l d i s t r i b u t i o n o f i n c i d e n t l o c a t i o n s i n f l u e n c e the t r a v e l time which i s a s i g n i f i c a n t component o f the s e r v i c e t i m e . There i s i n i t i a l l y no t r e a t m e n t of queuing systems w i t h m obile s e r v e r s i n the l i t e r a t u r e . ( 3 ) . 1.2.2. S i m u l a t i o n Model Many r e s e a r c h e r s s t u d y i n g ambulance systems have used s i m u l a t i o n as a t o o l . Savas (9) r e p o r t s the r e s u l t s o f a s i m u l a t i o n s t u d y o f the ambulance system of King County H o s p i t a l D i s t r i c t i n New York C i t y . He u t i l i z e s a GPSS s i m u l a t i o n as the b a s i s f o r a d e c i s i o n t o a l l o c a t e ambulances w i t h a d i s t r i c t . T h i s model was based upon the assumption o f a P o i s s o n d i s t r i b u t e d r a t e of emergency o c c u r r e n c e and a s t o c h a s t i c s e r v i c e p r o c e s s w i t h a s e r v i c e r a t e t h a t i s a p i e c e - w i s e l i n e a r f u n c t i o n o f t r a n s i t d i s t a n c e . He d i s c o v e r e d t h a t the b e s t way to u t i l i z e ambulances i s to d i s p e r s e them. In a n other s t u d y , Swoveland e t . a l . (10) developed a ( F o r t r a n ) d i g i t a l s i m u l a t i o n which was used f o r e s t i m a t i n g i n f o r m a t i o n on system c h a r a c t e r i s t i c s . The output o f the s i m u l a t i o n was used t o c o n s t r u c t an a n a l y t i c a l a p p r o x i m a t i o n of mean response time which i n t u r n was used t o c o n s t r u c t the o b j e c t i v e f u n c t i o n f o r the o p t i m a l l o c a t i o n problem. T h i s study w i l l be based on t h i s model and w i l l a p p l y t h i s model t o the G.V.R.D. A d e t a i l e d d e s c r i p t i o n o f the s i m u l a t i o n model i s p r o v i d e d i n Chapter 3. 1.2.3. O p t i m i z a t i o n Method t o Locate Ambulances The ambulance l o c a t i o n problem must be c o n s i d e r e d i n the c o n t e x t of a p p r o p r i a t e system performance c r i t e r i a . As noted p r e v i o u s l y , t h i s s tudy w i l l judge the performance of an ambulance system on the b a s i s o f i t s response t i m e . Because the mean i s the most common measure used, the e f f e c t i v e n e s s o f response time w i l l be measured i n terms o f the mean. - 9 -The problem of optimally locating "N" ambulance in a service area can be stated e x p l i c i t l y to be that of selecting a subset "w" of ambulance locations, from the set of a l l feasible locations, so that the mean response time w i l l be minimized. The problem can be stated as: Min R (w) ffl] weW where R(w) i s the system mean response time for the ambulance location set w and W i s the c o l l e c t i o n of a l l fea s i b l e combin-ations of ambulance locations. To solve [1], simulation can be applied. But, i f (W) (ca r d i n a l i t y of W) becomes large, simulation is no longer e f f e c t i v e . It was for this reason that Swoveland et. a l . (10) developed an a n a l y t i c a l approximation of mean response time which was based on the simulation. From this they used a " p r o b a b i l i s t i c enumeration method" to solve [ l ] . In this study i t was found that good solutions could be obtained by i t e r a t i v e l y moving each ambulance to an adjacent location u n t i l the mean response time could not be further improved. This " l o c a l search Algorithm" was applied to solve the ambulance location problem in the G.V.R.D. - 10 -CHAPTER 2 - EXISTING AMBULANCE SERVICES IN G.V.R.D. 2.1. E x i s t i n g Ambulance S e r v i c e s Ambulance s e r v i c e s are c u r r e n t l y b e i n g p r o v i d e d t o the s e v e r a l m u n i c i p a l i t i e s w i t h i n the G.V.R.D. by the ten ambulance companies and f i r e departments l i s t e d i n Table 1. The m a j o r i t y o f ambulance c a l l s are s e r v i c e d by M e t r o p o l i t a n Ambulance Co. (76% o f c a l l s i n the G.V.R.D.), who a l s o p r o v i d e the d i s p a t h i n g s e r v i c e f o r D e l t a , Richmond, S u r r e y and White Rock Ambulance Companies. A s m a l l p a r t o f M e t r o p o l i t a n ' s o p e r a t i o n (about 2% o f a l l c a l l s they s e r v i c e ) i s a back-up s e r v i c e f o r c a l l s o r i g i n a t i n g o u t s i d e o f t h e i r p r i m a r y s e r v i c e r e g i o n . Table 2 g i v e s , f o r each m u n i c i p a l i t y , the pe r c e n t a g e of c a l l s s e r v i c e d by each o f the t e n ambulance s e r v i c e s . With the e x c e p t i o n of M e t r o p o l i t a n Ambulance, a l l o f the ambulance companies o p e r a t e w i t h a c o n s t a n t number o f crews over each hour of the day and over each day o f the week. Metro-p o l i t a n Ambulance o p e r a t e s w i t h n i n e crews d a i l y between 8 a.m. and 6 p.m. and e i g h t crews at a l l o t h e r t i m e s . The c o s t per c a l l t o the user v a r i e s by company and m u n i c i p a l i t y as shown i n Table 1. These charges s h o u l d be compared w i t h the c u r r e n t a c t u a l c o s t per c a l l , which f o r M e t r o p o l i t a n Ambulance i s a p p r o x i m a t e l y $53.00. M e t r o p o l i t a n charges $20.00 p l u s $ l . / m i l e f o r a l l c a l l s which they s e r v i c e i n D e l t a , Richmond, S u r r e y and White Rock and a f l a t $53.00 f o r a l l o t h e r c a l l s o u t s i d e o f Burnaby, Vancouver and New Westmi n s t e r , West Vancouver F i r e Department has h i g h e r r a t e s than those l i s t e d f o r c a l l s o r i g i n a t i n g i n Horseshoe Bay, c a l l s o Van Bby N W Rmd Sry N Vn W Vn. D'el W Rk Coq P Coq P My % • % % % % % % / o A - /a /a A Coquitlam F i re Dept. 67.3 Delta Amb. .1 4.1 1.0 100.0 Metropolitan Amb. 99.6 100.0 96.6 1.5 6.3 4.7 32.7 22.3 6.5 North Vancouver F i re Dept. 9 3 . 0 - 1 . 2 Port Coquitlam Fi re Dept. . ? 7 , 7 Port Moody Fi re Dept. Richmond Amb. .3 1.1 94.1 Surrey Amb. ' 2.3 76.0 4.0 West Vancouver F i re Dept. 2.3 98.8 White Rock Amb. 16.7 96.0 TABLE 2 AMBULANCE SERVICE BY MUNICIPALITY • (Columns Sum To 100%) Company Primary Service Area // Ambulances Leased or Owned v Manned Ainbs. Per Shift Ave. (f Calls /Day Cost/Call to User Coquitlam Fire Dept. Delta Amb. Metropolitan Ambulance North Vancouver Fire Dept. Port Coquitlam Fire Dept. Port Moody Fire Dept. Richmond Amb. Surrey Ambulance West Vancouver F i re Dept. White Rock Ambulance D i s t r i c t of Coquitlam Delta Burnaby, Vancouver, New Westminster North Vancouver, North Vancouver D is t r i c t Port Coquitlam Port Moody Richmond Surrey Wes't Vancouver White Rock, Surrey 1 13 3 2 3 1 3 - 9 2 1 2 1.6 1.4 95.3 6.6 1.2 1.0 4.5 .7.2 3.0 3.0 no cnarge $20 + $ 1/mi. Burnaby: $20 Vancouver: $25 + $ 1/mi. New Westminster: $25 + $ 1/mi North Vancouver Residents $10 For a l l others $20.+ $ 1/mi. no charge no charge $20 + $ 1/mi. $20 -P $' 1/mi. $22.50 $20 +•$l/mi. TOTAL FOR REGION 32 22 125.0 TABLE 1 AMBULANCE COMPANIES SERVING THE G.V.R.D. - 13 -g o i n g to VGH, and c e r t a i n o t h e r u n u s u a l l y long c a l l s . A l t h o ugh the f i r e departments o f the D i s t r i c t o f C o q u i t l a m , P o r t C o q u i t l a m and P o r t Moody do not charge f o r ambulance s e r v i c e , r o u t i n e c a l l s ( e . g . , s c h e d u l e d t r a n s f e r c a l l s between home and h o s p i t a l ) are d i r e c t e d t o M e t r o p o l i t a n Ambulance who charge a f l a t $53.00 f o r each such c a l l . M e t r o p o l i t a n , D e l t a , Richmond, S u r r e y and White Rock Ambulance are each s u b s i d i z e d by the m u n i c i p a l i t i e s w i t h i n t h e i r i p r i m a r y s e r v i c e areas and t o some e x t e n t by c e r t a i n o t h e r m u n i c i p a l i t i e s . For the c a l l s s e r v i c e d by M e t r o p o l i t a n Ambulance i t i s p o s s i b l e to p r o v i d e the f o l l o w i n g rough c l a s s i f i c a t i o n o f c a l l s by type of b i l l i n g : Type of B i l l i n g p e r cent o f a l l b i l l i n g P r i v a t e b i l l i n g s t o i n d i v i d u a l s 47% I n s t i t u t i o n a l ( p r i v a t e companies, w e l f a r e , o l d age a s s i s t a n c e , e t c . ) 33% No b i l l i n g (ambulance not used, e t c . ) 20% A p p r o x i m a t e l y 18% of the b i l l i n g s to i n d i v i d u a l s are w r i t t e n o f f f o r non-payment. 2.2. Demand f o r Ambulance S e r v i c e s 2.2.1. Data C o l l e c t i o n Data was c o l l e c t e d from the t e n d i f f e r e n t ambulance companies d u r i n g October and November of 1973. I n f o r m a t i o n was r e c o r d e d f o r - 14 -a l l c a l l s o c c u r r i n g i n Vancouver, Burnaby, and New Westminster over a 14-day p e r i o d . I n f o r m a t i o n on c a l l s o c c u r r i n g elsewhere was c o l l e c t e d over p e r i o d s o f three, o r f o u r weeks. I t was found t h a t s e a s o n a l e f f e c t s i n the demand f o r ambulance s e r v i c e are not s i g n i f i c a n t (see s e c t i o n 2.2.3. ( a ) ) . Co n s e q u e n t l y , i t was not f e l t n e c e s s a r y to c o l l e c t a d d i t i o n a l data d u r i n g o t h e r p a r t s o f the y e a r . For each o f the 2,227 c a l l s o bserved the f o l l o w i n g i n f o r -mation was r e c o r d e d : time c a l l was r e c e i v e d l o c a t i o n o f c a l l (scene) d e s t i n a t i o n from scene ( e . g . , a h o s p i t a l ) For those c a l l s d i s p a t c h e d by M e t r o p o l i t a n Ambulance the f o l l o w i n g i n f o r m a t i o n was r e c o r d e d f o r each c a l l : t ype o f c a l l ( r e g u l a r , c a n c e l l e d , ambulance not used) time ambulance - l e f t base - a r r i v e d at scene - l e f t scene f o r d e s t i n a t i o n - a r r i v e d at d e s t i n a t i o n - c l e a r e d d e s t i n a t i o n p r i o r i t y o f c a l l ( s i r e n , normal or t r a n s f e r ) For those areas not d i s p a t c h e d by M e t r o p o l i t a n Ambulance ( a c c o u n t i n g f o r o n l y 11% o f the c a l l s w i t h i n the r e g i o n ) the d i s t r i b u t i o n o f types and p r i o r i t i e s o f c a l l s , and t h e i r d i s t r i b u t i o n s o f the v a r i o u s components o f s e r v i c e times were e x t r a p o l a t e d from the e m p i r i c a l d i s t r i b u t i o n s o b t a i n e d from the - 15 -sample of c a l l s d i s p a t c h e d by Metro. A t r a v e l time m a t r i x g i v i n g the t r a v e l times between 6642 p a i r s o f l o c a t i o n s w i t h i n the G.V.R.D., was o b t a i n e d from the U.B.C. H P S t r a n s p o r a t i o n study and then a d j u s t e d somewhat on the b a s i s o f t r a v e l times observed i n our sample o f ambulance c a l l s . 2.2.2. D e f i n i t i o n s The terms d e f i n e d below are used f r e q u e n t l y throughout t h i s r e p o r t : ANU (Ambulance Not Used): the s i t u a t i o n i n which an ambulance reaches the scene but i s not used f o r t r a n s p o r t ( f i r s t - a i d may or may not be a d m i n i s t e r e d ) . Ambulance U t i l i z a t i o n : the per cent o f time an ambulance i s a c t u a l l y s e r v i c i n g c a l l s . C a n c e l i e d C a l l : a c a l l f o r an ambulance which i s c a n c e l l e d b e f o r e the ambulance reaches the scene. High P r i o r i t y C a l l s (Code 3 ) : C a l l s i n which the s i r e n i s used en r o u t e to the scene. Loading Time: the amount o f time spent at the scene (time spent a p p l y i n g f i r s t a i d , moving the p a t i e n t to the ambulance, e t c . ) Non-Primary Response: the s i t u a t i o n i n which a c a l l i s s e r v i c e d by some ambulance o t h e r than the one whose depot i s c l o s e s t to the scene o f the c a l l (because the l a t t e r i s busy and t h e r e f o r e u n a v a i l a b l e t o r e s p o n d ) . Response Time: the e l a p s e d time between a c a l l b e i n g r e c e i v e d by the d i s p a t c h e r and the a r r i v a l o f the ambulance at the scene. _ - 16 -S e r v i c e Time: the t o t a l time spent s e r v i c i n g a c a l l - the e l a p s e d time between an ambulance b e i n g d i s p a t c h e d and the ambulance l e a v i n g the d e s t i n a t i o n to which the i n d i v i d u a l ( s ) was t r a n s p o r t e d . S t a r t - U p Time: the e l a p s e d time between an ambulance b e i n g d i s p a t c h e d and the ambulance d e p a r t i n g from i t s c u r r e n t l o c a t i o n t o proceed to the scene. Time i n Queue: the e l a p s e d time between a c a l l b e i n g r e c e i v e d by the d i s p a t c h e r and the ambulance b e i n g d i s p a t c h e d . T r a n s f e r C a l l : a non-urgent ( o f t e n s cheduled) c a l l i n which a p a t i e n t i s t r a n s p o r t e d between two p o i n t s ( o f t e n between home and h o s p i t a l ) and does not r e c e i v e f i r s t a i d t r e a t m e n t . U n l o a d i n g Time: the e l a p s e d time spent at the p o i n t to which the p a t i e n t ( s ) i s b e i n g t r a n s p o r t e d ( g e n e r a l l y , t h i s i s time spent a d m i t t i n g p a t i e n t s t o emergency wards). 2.2.3. C u r r e n t Demand (a) Temporal V a r i a t i o n s ( i ) S easonal V a r i a t i o n s There are no pronounced s e a s o n a l v a r i a t i o n s i n demand. An e x a m i n a t i o n of monthly c a l l volumes f o r the p e r i o d between August 1965 and November 1973 suggests t h a t a f t e r the t r e n d e f f e c t i s removed i t i s o n l y d u r i n g the months o f December and January t h a t t h e r e i s n o t i c e a b l e i n c r e a s e i n monthly demand. The i n c r e a s e i n demand d u r i n g these two months i s not g r e a t and i s not l a r g e enough t o warrant a s e p a r a t e a n a l y s i s f o r t h a t p a r t o f the y e a r . ( i i ) Day of Week V a r i a t i o n s An e x a m i n a t i o n o f the d i s t r i b u t i o n o f c a l l s by day of the - 17 -week i n d i c a t e d a maximum loa d on the system on F r i d a y and S aturday (about 15% above week-days) and a minimum l o a d on Sunday (about 25% l e s s than on Weekdays). The d i f f e r e n c e i n loads between Tuesday, Wednesday and Thursday i s not s i g n i f i c a n t , ( i i i ) Time of Day V a r i a t i o n s The v a r i a t i o n s i n c a l l r a t e by time of day i s g i v e n f o r the G.V.R.D. as a whole i n F i g u r e 1. From m i d n i g h t u n t i l 6 a.m. t h e r e i s a s t e a d y d e c l i n e i n the demand on the system. At 6 a.m. the demand b e g i n s i n c r e a s i n g and i n c r e a s e s s t e a d i l y u n t i l noon, at which time t h e r e i s a s l i g h t d e c l i n e d u r i n g the l u n c h hour. A f t e r c l i m b i n g t o a h i g h o f n i n e c a l l s per hour at 2 p.m. the demand begi n s to d e c l i n e once a g a i n , except f o r s p u r t s around 8 p.m. and 10 p.m. (b) S p a t i a l D i s t r i b u t i o n o f C a l l O r i g i n The v a r i a t i o n i n c a l l r a t e by p e r i o d of the day, by m u n i c i -p a l i t y , i s g i v e n i n Table 3. G e n e r a l l y , the p e r i o d from 8 a.m. t o 6 p.m. i s the b u s i e s t time o f day w h i l e the p e r i o d from mid-n i g h t u n t i l 8 a.m. i s the s l o w e s t . D e l t a , which i s as busy i n the e a r l y hours o f the morning as i n the r e s t o f the day, i s a n o t a b l e e x c e p t i o n t o the r u l e . Of p a r t i c u l a r i n t e r e s t i s the v a r i a t i o n by m u n i c i p a l i t y i n the average number o f c a l l s per day per thousand p o p u l a t i o n . No s i g n i f i c a n t a s s o c i a t i o n e x i s t s ( i n our data) between the c a l l r a t e per c a p i t a and the c o s t to the i n d i v i d u a l f o r ambulance s e r v i c e . A s i g n i f i c a n t a s s o c i a t i o n does e x i s t between the c a l l r a t e per c a p i t a and the degree of u r b a n i z a t i o n ( d e n s i t y of p o p u l a t i o n , i n d u s t r i a l a c t i v i t y , e n t e r t a i n m e n t f a c i l i t i e s , e t c . ) . For example, FIG. 1 - Average G.V.R.D. Cal l Rate (Calls/Kovr) By Time of Day Average /' calls/hour 12 M-8 AM 8 AM-6 PM 6 PM-12 M Average // calls/day Average // calls/day/ thousand population % of population over 60 years Burnaby .25 . .73 .76 13.9 .10 11.5 Coquitlam .05 , J-2 .13 • 2.4 .04 9.1 Delta .05 .05 .04 1.2 .02 6.5 New Westminster . .14 .35 .35 6.7 .15 17.6 North Vancouver .18 .38 .29 7.0 .07 *9.1 Port Coquitlam .05 .08 .06 1.6 .06 5.7 Richmond .07 .29 .16 4.5 .06 7.7 Surrey .19 .44 .43 8.5 .08 11.0 Vancouver 1.70 4.23 3.02 74.2 . .17 18.7 West Vancouver .08 .16 .12 3.0 .08 15.5 White Rock .04 .11 .05 1.8 .15 37.4 Port Moody .03 .06 . .03 1.10 .08 5.3 TABLE 3 DAILY AMBULANCE DEMAND RATE BY TIME OF DAY AND MUNICIPALITY - 20 -Vancouver has t w i c e the c a l l r a t e per c a p i t a as North and West Vancouver. T h i s r e l a t i o n s h i p i s , however, m o d i f i e d somewhat by the age s t r u c t u r e o f the p o p u l a t i o n . For example, w h i l e White Rock i s no more urban than i t s n e i g h b o u r s , i t s c a l l r a t e per c a p i t a i s n e a r l y as l a r g e as Vancouver's, which i s the h i g h e s t i n the r e g i o n . White Rock's e x c e p t i o n a l l y h i g h c a l l r a t e per c a p i t a i s a p p a r e n t l y e x p l a i n e d i n l a r g e p a r t by i t s r e l a t i v e l y l a r g e p r o p o r t i o n o f r e s i d e n t s who are o ver 60 years o l d (See T able 3 ) . (c) S p a t i a l D i s t r i b u t i o n o f C a l l D e s t i n a t i o n s The d a t a j u s t d i s c u s s e d f o c u s e d on the o r i g i n o f demand. We see i n Table 4 where the ambulances go a f t e r p i c k i n g up t h e i r p a t i e n t s (from l e f t to r i g h t the h o s p i t a l a b b r e v i a t i o n s s t a n d f o r L i o n ' s Gate, S t . P a u l ' s , Royal Columbian, S t . Mary's, Vancouver G e n e r a l , Shaughnessy, Burnaby G e n e r a l , Richmond G e n e r a l , and S u r r e y H o s p i t a l s ) . The emergency rooms o f L i o n ' s Gate, S t . P a u l ' s , Vancouver G e n e r a l and Royal Columbian are the o n l y ones i n the G.V.R.D. which are c l a s s i f i e d as major Emergency F a c i l i t i e s ( i . e . , emergency rooms which have at a l l times both the s t a f f and the f a c i l i t i e s to p r o v i d e v i r t u a l l y any k i n d o f emergency t r e a t -ment n e c e s s a r y ) ( 2 ) . The m a j o r i t y of the 82% o f c a l l s o r i g i n a t i n g i n White Rock t h a t go to " o t h e r " are a d m i t t e d to Peace Arch H o s p i t a l , i n White Rock. Table 5 i s the complement of Table 4, g i v i n g the d i s t r i b u t i o n o f c a l l o r i g i n s by h o s p i t a l . 2.2.4. Demand Growth P a t t e r n s Change i n the l e v e l and s p a t i a l d i s t r i b u t i o n o f demand f o r Average # calls/day LGH SPH RCH SMH VGH SHY . BGH RGH SUK OTHER' Burnaby 13.9 .6% 5.2% 23.9% 1.9% 26.5% 4.5% 29.0% .6% .6% 7.1% Coquitlam 1.2 80.3 3.8 3.8 3.8 7.7 Delta 2.4 7.7 76.9 • 15.4 New Westminster 6.7 1.2 53.0 12.0 4.8 8.4 2.4 1.2 4.8 12.0 North Vancouver 7.0 81.6 4.1 — 2.0 - • 12.2 \ Port Coquitlam 1.6 5.3 84.2 5.3 5.3 Port Moody 1.1 93.8 6.2 . Richmond 4.5 1.5 4.6 1.5 18.5 6.2 1.5 52.3 1.5 12.3 Surrey 8.5 . 21.8 2.3 4.6 1.1 • 1.1 1.1 47.1 20.7 Vancouver 74.2 .7 21.2 2.0 .4 54.3 5.2 • 1.1 .1 15.2 West Vancouver '. 3.0 93.0 2.3 2.3 2.3 White Rock 1.8 9.1 4.5 4.5 81.8 G.V.R.D. Total 125.9 8.7% 13.2% 11.6% 1.5% 35.1% 4.5% 3.9% " 3.9% ' 3.4% 14.2% TABLE 4 DISTRIBUTION OF AMBULANCE DESTINATIONS BY MUNICIPALITY (RCWS SUM TO 100%). LGH SPH RCH SMH VGH SHY BGH RGH SDH Burnaby Coquitlam .8% 4.1% 21.6% 12.3 13.6% 4.5 ; 7.9% .2 10.6% 1.5 77.6% • 2.0 2.0% Delta .5 New Westminster North Vancouver 62.0 .5 2.1 25.7 45.5 .8 10.6 3.4 20.6 • 2.0 4.0 8.0 Port Coquitlam .8 9.4 4.5 Port Moody 8.8 4.5 Richmond Surrey Vancouver . .8 4.7 1.5 90.8 11.1 9.9 4.5 9.1 13.6 2.3 .8 87.7 6.1 1.5 66.7 1.7 1.7 15.5 70.3 2.0 2.1 2.0 82.0 West Vancouver 31.0 .5 .5 • 1.5 White Rock •1.2 1.5 - 2.0 is) TABLE 5 (CoSl?^FS 0 R I G I N B Y HOSPITAL) - 23 -ambulance s e r v i c e over time i s e x t r e m e l y d i f f i c u l t t o f o r e c a s t . We are even u n c e r t a i n as to which f a c t o r s most i n f l u e n c e demand. C e r t a i n l y p o p u l a t i o n , age s t r u c t u r e , income d i s t r i b u t i o n , c o s t o f s e r v i c e t o u s e r , a c c e s s i b i l i t y o f emergency f a c i l i t i e s , the a v a i l a b i l i t y of p r i v a t e and p u b l i c t r a n s p o r t a t i o n f a c i l i t i e s , and the g e n e r a l awareness and p e r c e p t i o n o f the ambulance s e r v i c e are f a c t o r s which a f f e c t demand. However, t h e i r r e l a t i v e importance i n d i v i d u a l l y and i n c o m b i n a t i o n w i t h one another i s unknown. Of one t h i n g we are s u r e : i n the C i t y o f Vancouver the demand f o r ambulance s e r v i c e i s i n c r e a s i n g at a much f a s t e r r a t e than p o p u l a t i o n . Between 1966 and 1971 the annual volume o f c a l l s i n Vancouver i n c r e a s e d 21%; at the same time the p o p u l a t i o n i n c r e a s e d by o n l y 4%. Hence, the c a l l r a t e per y e a r p e r 1000 p o p u l a t i o n i n c r e a s e d by 17% d u r i n g t h a t p e r i o d . The i n c r e a s e s i n demand from y e a r to y e a r were q u i t e e r r a t i c , as shown below: % i n c r e a s e i n number of c a l l s Year from p r e v i o u s y e a r  67 0.3% 68 9.4% 69 6.2% 70 1.0% 71 2.2% 72 6.3% Al t h o u g h l o n g range t r e n d s i n the demand f o r ambulance s e r v i c e are d i f f i c u l t t o f o r e c a s t , the consequent d i f f i c u l t i e s t h a t a r i s e i n the p l a n n i n g and management o f the s e r v i c e are not n e c e s s a r i l y s e v e r e . T h i s i s because ambulance systems are - 24 -h i g h l y l a b o u r i n t e n s i v e ; o n l y a r e l a t i v e l y s m a l l c a p i t a l i n v e s t -ment i s r e q u i r e d t o i n i t i a t e and m a i n t a i n the s e r v i c e . Moreover, many of the c a p i t a l r e s o u r c e s ( e . g . , ambulances) can be i n c r e a s e d i n c r e m e n t a l l y i n response to changes i n demand. The r e m a i n i n g c a p i t a l r e s o u r c e , communications equipment (assuming o f f i c e space i s l e a s e d ) , i s a v a i l a b l e i n d i s c r e t e " s i z e s " , where each " s i z e " i s a p p r o p r i a t e f o r h a n d l i n g a f a i r l y wide range of demand r a t e s . For example, M e t r o p o l i t a n Ambulance now has s u f f i c i e n t communi-c a t i o n c a p a c i t y to d i s p a t c h a l l o f the ambulance c a l l s i n the G.V.R.D. (at the c u r r e n t l e v e l o f demand). 2.3. PRODUCTION AND EVALUATION 2.3.1. P r o d u c t i o n Components Ambulance S e r v i c e i s one sub-system o f the t o t a l h e a l t h care d e l i v e r y system and, as s u c h , i t s e f f e c t i v e n e s s s h o u l d be measured i n terms of the q u a l i t y o f h e a l t h care which i t p r o v i d e s to the p o p u l a t i o n i t s e r v e s . The q u a l i t y o f s e r v i c e p r o v i d e d by an ambulance system i s a f u n c t i o n o f r e s p o n s i v e n e s s , treatment and t r a n s p o r t . The focus o f t h i s r e p o r t i s on the f i r s t o f t h e s e t h r e e components, r e s p o n s i v e n e s s , which ( u n l i k e most o t h e r h e a l t h care sub-systems) i s an i m p o r t a n t c o n s i d e r a t i o n i n both the d e s i g n and o p e r a t i o n of an ambulance s e r v i c e . " R e s p o n s i v e n e s s " i s an i m p r e c i s e term, i n s p i t e o f the f a c t t h a t we a l l know g e n e r a l l y what i t means. For example, i s an ambulance which responds to two c a l l s i n f o u r minutes each more or l e s s " r e s p o n s i v e " than one which responds to one c a l l i n two minutes and to the o t h e r i n s i x minutes? - 25 -In t h i s c h a p t e r we s h a l l p r i m a r i l y be r e p o r t i n g average times f o r the v a r i o u s p r o d u c t i o n components o f the e x i s t i n g ambulance s e r v i c e s which s e r v e the g e o g r a p h i c a l area d i s p a t c h e d by M e t r o p o l i t a n Ambulance: D e l t a , Burnaby, New W e s t m i n s t e r , Richmond, S u r r e y , Vancouver and White Rock.* In Chapter 3 we s h a l l c o n s i d e r the e f f e c t s o f v a r i o u s o t h e r measures o f r e s -p o n s i v e n e s s on the d e s i g n o f an ambulance system. The (average) time spent s e r v i c i n g each c a l l i s the sum o f the i n d i v i d u a l (average) times r e q u i r e d f o r each o f f i v e " p r o -d u c t i o n " components:** s t a r t - u p time t r a v e l time t o scene time at scene t r a n s p o r t time from scene time at h o s p i t a l « s e r v i c e time r-^ ^ response time —^ s t a r t - u p time t r a v e l time to scene time at I t r a v e l time scene \^ . to h o s p i t a l time at h o s p i t a l C a l l ambulance ambulance ambulance ambulance ambulance a r r i v e s , l e a v e s a r r i v e s l e a v e s a r r i v e s l e a v e s ambulance f o r at f o r at h o s p i t a l d i s p a t c h e d scene scene h o s p i t a l h o s p i t a l * Response time d a t a was o n l y a v a i l a b l e from the ambulance s e r v i c e s d i s p a t c h e d by M e t r o p o l i t a n Ambulance (89.3% o f a l l c a l l s i n the G.V.R.D.). ** We s h a l l use the words "scene" to mean the l o c a t i o n to which the ambulance i s d i s p a t c h e d and " h o s p i t a l " t o denote the l o c a t i o n to which the p a t i e n t i s t r a n s p o r t e d (even i f i t i s not a " h o s p i t a l " ) . - 26 -2.3.2. A n a l y s i s o f Components R e c a l l t h a t response time i s d e f i n e d as the e l a p s e d time between the a r r i v a l of a c a l l t o the d i s p a t c h e r and the a r r i v a l of the ambulance at the scene. The c u r r e n t average response time f o r each o f the m u n i c i -p a l i t i e s w i t h i n the b o u n d a r i e s o f M e t r o p o l i t a n Ambulance d i s -p a t c h i n g s e r v i c e are g i v e n i n Table 6. Note t h a t average response times are g i v e n i n each o f t h r e e p e r i o d s o f the day, as w e l l as an o v e r a l l 24 hour average. The column l a b e l e d "90th F r a c t i l e " c o n t a i n s those response times which are exceeded i n o n l y 10% o f the c a l l s . F o r example, 10% o f the c a l l s o r i g i n a t i n g i n Burnaby have a response time g r e a t e r than 18.0 minutes ( c o n v e r s e l y 90% o f the c a l l s o r i g i n a t i n g i n Burnaby have a response time not e x c e e d i n g 18.0 m i n u t e s ) . With the e x c e p t i o n o f Burnaby, the response times d i s p l a y a s t r o n g n e g a t i v e c o r r e l a t i o n w i t h p o p u l a t i o n d e n s i t y . Vancouver, New Westminster and White Rock a l l have f a i r l y dense p o p u l a t i o n s and low response t i m e s . D e l t a , Richmond and S u r r e y are more s p a r s e l y p o p u l a t e d and have l o n g e r response t i m e s . The l o n g response time i n Burnaby i s due t o the f a c t t h a t t h e r e i s c u r r e n t l y ( i . e . , when the d a t a was c o l l e c t e d ) o n l y one ambulance l o c a t e d i n Burnaby. T h i s ambulance, which responds - 27 -to the m a j o r i t y o f the c a l l s o r i g i n a t i n g i n Burnaby, has an average response time of 11.2 m i n u t e s , which i s a t t r i b u t a b l e t o the f a c t t h a t i n terms o f t r a v e l time i t has a v e r y l a r g e geo-g r a p h i c a l a r e a t o co v e r . I t i s i n t e r e s t i n g to compare the average response time w i t h the 90th f r a c t i l e t i m e s . The d i f f e r e n c e s are most d r a m a t i c f o r the m u n i c i p a l i t i e s i n which o n l y a s i n g l e ambulance i s s t a t i o n e d . For these m u n i c i p a l i t i e s the c a l l s w i t h e s p e c i a l l y l o n g response times are p r i m a r i l y those which o c c u r when the c l o s e s t ambulance i s busy w i t h another c a l l . In t h i s case the d i s p a t c h e r e i t h e r h o l d s the c a l l u n t i l the c l o s e s t ambulance becomes f r e e or e l s e d i s p a t c h e s a more d i s t a n t ambulance, which-ever a c t i o n m i n i m i z e s the response t i m e . For example, i t i s not unusual f o r one o f the average 6.7 c a l l s per day i n New Westminster to oc c u r when the o n l y ambulance l o c a t e d i n New Westminster i s busy. In t h i s case an ambulance i s e i t h e r sent from Burnaby, Vancouver, or o c c a s i o n a l l y (3.4% o f a l l c a l l s ) from S u r r e y or Richmond. The s u r p r i s i n g l y low 90th f r a c t i l e time f o r S u r r e y i s due to the f a c t t h a t the New Westminster ambulance p r o v i d e s a back up i n the n o r t h e r n p a r t o f S u r r e y (about one such c a l l every o t h e r day, r e s p r e s e n t i n g about 6% o f a l l S u r r e y c a l l s ) and White Rock Ambulance s e r v e s south S u r r e y and p r o v i d e s a d d i t i o n a l back-up s e r v i c e ( r e s p o n d i n g to 16.7% of a l l S u r r e y c a l l s ) . The average s e r v i c e times vary from a low o f 4.2 minutes i n Vancouver to a h i g h o f about one hour i n D e l t a and i n Richmond. I t i s i m p o r t a n t to keep i n mind t h a t s e r v i c e times - 28 -Average Response Time ( i n minutes) M u n i c i p a l i t y * 12 M-8 AM Time of Day • 8 AM-6 PM 6 PM-12 . O v e r a l l 24 M hours) 9 0 t h F r a c t i l e Average Service Time Burnaby 11.6 12.3 11.3 11.9 • 18.0 49.3 Delta 9.8 9.7 10.5 10.0 18.0 61.6. O New Westminster 6.7 12.8 7.8 ' 10.5 . 22.0 N/A Richmond 9.8 10.2 9.2 10.0 19.0 59.6 Surrey 11.5 10.3 9.3 10.3 16 . 6 49.8 Vancouver 7.2 8.3 7.6 7.9 12.0 42.0 White Rock . 7.0 7.7 6.0 7.3 10.0 44.6 OVERALL 8.3 9.3 8.4 8.9 16.0 46.1 Information not a v a i l a b l e f o r Coquitlam, North Vancouver, Port Coquitlam, Port Moody and West Vancouver. TABLE 6 AVERAGE RESPONSE TIME AND SERVICE TIME BY MUNICIPALITY - 29 -have a s i g n i f i c a n t i n d i r e c t e f f e c t on response t i m e s ; the more time spent s e r v i c i n g c a l l s the g r e a t e r the l i k e l i h o o d t h a t when a c a l l o c c u r s the c l o s e s t ambulance w i l l be busy and a more d i s t a n t ambulance w i l l have to be d i s p a t c h e d . The o v e r a l l f requency d i s t r i b u t i o n o f response time (over 24 hours per day, seven days per week) f o r the a r e a d i s p a t c h e d by M e t r o p o l i t a n Ambulance i s g i v e n i n F i g u r e 2A. The c u m u l a t i v e d i s t r i b u t i o n i s g i v e n i n F i g u r e 2B d i s p l a y s the p e r c e n t a g e o f c a l l s i n which the response time d i d not exceed a s p e c i f i e d v a l u e . For example, we see t h a t f o r 60% o f the c a l l s the response time was not more than about 8% minutes. The frequency and c u m u l a t i v e d i s t r i b u t o r s f o r emergency c a l l s o n l y are g i v e n i n F i g u r e s 3a and 3b. Before examining the v a r i a t i o n s i n average response time by time of day, i t i s i n s t r u c t i v e t o examine Table 7, which c o n t a i n s , by p r o d u c t i o n component, average times by p r i o r i t y of c a l l and by time o f day. As e x p e c t e d , average s t a r t - u p times a r e , by p r i o r i t y , l o west f o r emergency c a l l s , and by time of day are h i g h e s t f o r the c a l l s o c c u r r i n g i n the wee hours o f the morning. Note t h a t on the average, s t a r t - u p time accounts f o r a f u l l 28% o f response t i m e . The average time at the scene i s g r e a t e s t f o r t r a n s f e r c a l l s . T h i s happens d u r i n g the daytime, when the m a j o r i t y o f the t r a n s f e r c a l l s o c c u r . Response times are g r e a t e s t d u r i n g the day when the l o a d on the system i s the h e a v i e s t and when the t r a v e l times are the - 30 -s l o w e s t . As e x p e c t e d , the response time i s lower f o r emergency c a l l s ( i . e . , s i r e n used) than f o r normal and t r a n s f e r c a l l s . The f a c t t h a t the response time f o r t r a n s f e r c a l l s are r e l a t i v e l y low i s p r o b a b l y due p r i n c i p a l l y to the f a c t t h a t t r a n s f e r c a l l s g e n e r a l l y o r i g i n a t e at h o s p i t a l s , which are c e n t r a l l y l o c a t e d . V a r i a t i o n s i n time at the scene by time o f day and by p r i o r i t y are not l a r g e , p a r t i c u l a r l y as they i n d i r e c t l y a f f e c t response t i m e . The time spent w i t h the p a t i e n t at h i s o r her d e s t i n a t i o n (not shown i n F i g u r e d ) v a r i e s c o n s i d e r a b l y , but to a l a r g e e x t e n t i s a f u n c t i o n o f emergency room a d m i t t i n g p r o c e d u r e s , which v a r y from one h o s p i t a l to the n e x t . Average " u n l o a d i n g " times range from 12 to 16 minutes. S e r v i c e times e x h i b i t a s i g n i f i c a n t v a r i a t i o n by p r i o r i t y , but not by time o f day. Because o f f a s t e r response t i m e s , emergency c a l l s have the s h o r t e s t s e r v i c e t i m e s . The r e l a t i v e l y l o n g average s e r v i c e time f o r t r a n s f e r c a l l s i s due to a s i g n i -f i c a n t f r a c t i o n o f t r a n s f e r c a l l s i n which a p a t i e n t i s t r a n s -f e r r e d from one h o s p i t a l to a n o t h e r , o f t e n d i s t a n t h o s p i t a l . R e t u r n i n g now to Table 6, we see t h a t , except f o r New Westminster, t h e r e i s l i t t l e v a r i a t i o n i n average response time by time of day. The l o n g e s t average response time i s between 8 a.m. and 6 p.m. i n a l l m u n c i p a l i t i e s except D e l t a (where i t i s i n the evening) and S u r r e y (where i t i s i n the e a r l y morning h o u r s ) . The v a r i a t i o n i n average response by time o f day i n New Westminster i s e x t r e m e l y l a r g e , undoubtedly due to the f a c t t h a t d u r i n g the day t h e r e i s a g r e a t e r p r o p o r t i o n of c a l l s which o c c u r when the ambulance l o c a t e d i n New Westminster i s e i t h e r F I G . 2 - D i s t r i b u t i o n of Response Time, Current Data, A l l Calls (Burnaby, Vancouver and New Westminster only) FIG. 3)a Frequency D i s t r i b u t i o n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 FIG3jb Cumulative D i s t r i b u t i o n 3, D i s t r i b u t i o n of Response Time, Current Data, Emergency C a l l s Only (Burnaby, Vancouver, and New Westminster only) Time of Day (all priorities) 12AM-8 AM 8 AM-6 PM 6 PM-12 M Transfer Priority Normal Emergency A l l Priorities % of calls - 17.8% 56.1% 26.2% 13.8% 63.8% 22.4% 100.0% Average start-up times 2.6 2.4 2.3 2.2 2.6 2.0 2.4 Average response times 7.9 9.2 8.0 7.8 9.5 7.0 8.7 Average time at scene 8.7 9.5 8.2 9.9 9.2 8.0 9.0 Average service time 44.4 45/5 41.2 50.6 43.9 40.8. 44.0 • TABLE 7 AVERAGE OPERATIONS TIMES (in minutes) BY TYPE AND TIME OF CALL (Burnaby, New Westminster and Vancouver) only - 34 -busy w i t h a n other c a l l w i t h i n New Westminster or i s p r o v i d i n g back-up s e r v i c e t o Burnaby, P o r t C o q u i t l a m or S u r r e y . 2.3.3. Ambulance U t i l i z a t i o n Ambulance u t i l i z a t i o n i s d e f i n e d as the f r a c t i o n o f time t h a t an ambulance i s a c t u a l l y engaged i n p r o v i d i n g s e r v i c e . T h i s does not i n c l u d e the time spent r e t u r n i n g to the ambulance depot a f t e r " c l e a r i n g " the d e s t i n a t i o n o f the p r e v i o u s c a l l . Average u t i l i z a t i o n per ambulance, by time o f day by company i s g i v e n i n T a ble 8. Cbmpr.ny .or Firc.Df.pt. Coquitlam Fire. Dept. Delta Arab. Metropolitan Ambulance North Vancouver Fi r e Dept. Port Coquitlam F i r e Dept. Port: Moody Fi r e Dept. Richmond Amb. Surrey A'.nb. West Vancouver Fire Dept. White Rock Ambulance Primary Servicing Area 0 manned Average. Ambulance U t i l i z a t i o n ambulance Over-12AM-8 AM 8 AM-6 PM 6 PM-12 AM a l l D i s t r i c t of Coquitlam Delta 2, 1 8« 9 Burnaby, Vancouver, and New W c s tmins t e. r North Vancouver, N o r th Va n c ouv e r D i s t r i c t Port Coquitlam Port Moody 1 Richmond i 2 Surrey . • . 1 West Vancouver 2 N/A White Rock, South Surrey 4.0 N/A N/A 6.4 12.2 2.7 5 . 4 N/A 5.2% 9.8% 19.4% 54 .5 8.3 N/A N/A 17.6 35.6 5 . 8 18.7 N/A 2.9% 28.5 6.4 N/A N/A 20.2 4 . 3 5 . 4 " 3.3* 5.8% 38.1 6.7 2.5 ' 4.2. ' 8.9 22.6 4.4 9.8 .* Approximate average u t i l i z a t i o n TABLE 8 . AMBULANCE UTILIZATION BY TIME OF DAY - 36 -CHAPTER 3 - COMPUTER SIMULATION MODEL Thi s c h a p t e r d i s c u s s e s t h r e e main t o p i c s : the use of s i m u l a t i o n i n system a n a l y s i s ; the d e s c r i p t i o n o f computer s i m u l a t i o n models and; the model v a l i d a t i o n f o r the Vancouver ambulance system. 3.1. Use o f S i m u l a t i o n i n System A n a l y s i s Computer s i m u l a t i o n can be d e f i n e d as the use o f a n u m e r i c a l model t o study the b e h a v i o r o f a system as i t operates over t i m e . P a r t s o f the model may be s y m b o l i c , i . e . , p u r e l y m a t h e m a t i c a l . Other p a r t s may be l o g i c a l , such as i n the r e s p r e s e n t a t i o n o f o p e r a t i n g p o l i c i e s . A s i m u l a t i o n model norm-a l l y i s used i n an e x p e r i m e n t a l manner to i n d i c a t e the response o f a system to a s e t o f g i v e n parameter s e t t i n g s . O f t e n , i n the system under s t u d y , the t r a n s i e n t b e h a v i o r i s o f i n t e r e s t i n a d d i t i o n t o the steady s t a t e performance. In the case o f an ambulance system i t may be more i m p o r t a n t t o know how a system responds to a peak l o a d than t o know i t s average performance. F u r t h e r m o r e , two systems h a v i n g a s i m i l a r average performance can have v e r y d i f f e r e n t t r a n s i e n t b e h a v i o r , S i m u l a t i o n i s one o f the few t o o l s a v a i l a b l e f o r e s t i m a t i n g system dynamics. Most a n a l y t i c a l t e c h n i q u e s are o n l y a b l e t o determine s t e a d y s t a t e performance measures. A model used t o a n a l y s e dynamic b e h a v i o r must depend on s t r u c t u r e s t o e x p l a i n how a system moves ahead i n t i m e . T h i s f e a t u r e i s the essence of computer s i m u l a t i o n . - 37 -3.2. General D e s c r i p t i o n o f Computer S i m u l a t i o n Model The ambulance s i m u l a t i o n model c o n s i s t s o f two programs: * an ambulance c a l l s g e n e r a t o r which c r e a t e s an " i n p u t s t r e a m " * the main s i m u l a t o r which s i m u l a t e s the b e h a v i o r o f the system and c r e a t e s an "output s t r e a m " 3.2.1. Ambulance C a l l s G e n e r a t o r The ambulance c a l l s g e n e r a t o r program c r e a t e s an " i n p u t stream" o f a l l ambulance c a l l s w i t h the f o l l o w i n g i n f o r m a t i o n f o r each c a l l : * time o f o c c u r r e n c e * l o c a t i o n o f the i n c i d e n t * p r i o r i t y o f c a l l * time at scene * time at h o s p i t a l * t r a v e l time from l o c a t i o n o f a scene t o the d e s t i n a t i o n ( u s u a l l y a h o s p i t a l ) The time o f a c a l l i s sampled from a P o i s s o n d i s t r i b u t i o n w i t h mean a r r i v a l r a t e s which are a f u n c t i o n o f the time o f day. T h i s i s a c c o m p l i s h e d by s e l e c t i n g i n t e r - a r r i v a l time from an e x p o n e n t i a l d i s t r i b u t i o n . The p r i o r i t y o f c a l l s ( i . e . , emergency, n o r m a l , t r a n s f e r ) are based on an e m p i r i c a l d i s t r i b u t i o n . S i m i l a r l y , time at the scene and time at the h o s p i t a l are a l s o g e n e rated from an e m p i r i c a l d i s t r i b u t i o n . 3.2.2. Main S i m u l a t o r The main s i m u l a t o r c r e a t e s the "output stream" which t o -g e t h e r w i t h the " i n p u t stream" p r o v i d e s d e t a i l e d i n f o r m a t i o n about each c a l l p r o c e s s e d by the s i m u l a t i o n . - 38 -The output stream p r o v i d e s f o r each c a l l : * the d i s p a t c h d e l a y * the ambulance d i s p a t c h e d to the c a l l * the l o c a t i o n o f the ambulance when d i s p a t c h e d * the response t i m e . The s t r u c t u r e o f the s i m u l a t i o n program f o l l o w s the t y p i c a l sequence o f e v e n t s . The ambulance s e r v i c e b e g i n s when n o t i f i c a t i o n o f an i n c i d e n t o c c u r s and ends when the ambulance r e t u r n s to i t s s t a t i o n . N o t i f i c a t i o n The i n p u t stream t o the model i s a s i m u l a t e d s e r i e s o f i n c i d e n t o c c u r r e n c e s as they are r e p o r t e d to the agency (ambulance d i s p a t c h e r ) . With the n o t i f i c a t i o n , i n f o r m a t i o n i s p r o v i d e d about the i n c i d e n t ( i . e . , l o c a t i o n , type o f i n c i d e n t , e t c . ) . Ambulance A v a i l a b i l i t y A f t e r an i n c i d e n t has been r e p o r t e d , and b e f o r e an ambulance i s d i s p a t c h e d , the d i s p a t c h e r must check the c u r r e n t a v a i l a b i l i t y o f i t s v e h i c l e s . Each v e h i c l e at any p o i n t i n time i s i n one o f the f o l l o w i n g s t a t e s : * f r e e at i t s home base * on the way t o a c a l l * busy w i t h a p a t i e n t * f r e e and r e t u r n i n g t o home base With t h i s i n f o r m a t i o n , a s e t o f i m m e d i a t e l y a v a i l a b l e or soon to be a v a i l a b l e v e h i c l e s can be i d e n t i f i e d f o r s e l e c t i o n o f a v e h i c l e t o respond t o the i n c i d e n t . - 39 -V e h i c l e S e l e c t i o n and D i s p a t c h A d e c i s i o n i s next made to s e l e c t from the s e t o f a v a i l -a b l e ambulances. The ambulance t o be d i s p a t c h e d to the scene i s d i s p a t c h e d by one o f the f o l l o w i n g r u l e s : (a) the c l o s e s t a v a i l a b l e ambulance ( c l o s e s t i n time) responds. T h i s r u l e i s c a l l e d c l o s e s t ambulance response o r CAR or (b) the ambulance a s s i g n e d t o a d i s t r i c t s e r v e s i t s own d i s t r i c t f i r s t even i f i t i s s t i l l r e t u r n i n g from a c a l l and i s c u r r e n t l y o u t s i d e the d i s t r i c t . Hence, t h i s r u l e i s c a l l e d r e g i o n a l i z e d response or RR. A r r i v a l at the Scene The a r r i v a l time o f an ambulance at the scene o f an i n c i d e n t i s s c h e d u l e d i n s i m u l a t e d time by adding the t r a v e l time from the c u r r e n t l o c a t i o n o f the ambulance t o the i n c i d e n t l o c a t i o n t o -geth e r w i t h the time the ambulance takes l e a v i n g from home base. For emergency c a l l s which use a s i r e n the t r a v e l time i s m u l t i p l i e d by a g i v e n f r a c t i o n (<CL) On the Scene Care Once the ambulance a r r i v e s at the scene, a f u r t h e r d e l a y i s i n c u r r e d to account f o r f i r s t a i d a s s i s t a n c e and p o s s i b l e e x t r a -c t i o n i f n e c e s s a r y . To account f o r the cases o f c a n c e l l a t i o n o f c a l l s or cases where ambulances are not r e q u i r e d at a l l , a c e r t a i n p r o p o r t i o n o f c a l l s i s randomly c a n c e l l e d . The i n d i c a t i o n o f c a n c e l l a t i o n i s done on the b a s i s o f the number o f c a n c e l l e d c a l l s from e m p i r i c a l d a t a . I f the c a n c e l l a t i o n time i s l a r g e r than response t i m e , the c a l l i s c l a s s i f i e d as ANU (ambulance not used) - 40 -c a l l ; otherwise the c a l l i s c l a s s i f i e d as cancelled. The ambulance turns back to the station after the appropriate simulated time and at that moment is freed to respond to other c a l l s . D e s t i n a t i o n from the Scene For those cases that require transporation of the injured person to a medical f a c i l i t y , the question of which hospital to select arises. The selection is done on the basis of empirical p r o b a b i l i t i e s of destination for each source area. The time of the ambulance's a r r i v a l at hospital is computed in the simulation by adding to the time of c a l l to the sum of response time, time at scene and travel time from the scene to the hos p i t a l . Ambulance Return to the Station The ambulance travels along the shortest path to i t s home base after completing various a c t i v i t i e s in the hospital such as admitting the patient, processing records, etc. The ambulance can be dispatched for another assignment while at a hospital or on the way home. 3 . 3 . Model Validation for the Vancouver Ambulance System To validate any kind of model, i t is necessary to demonstrate that the model represents a true abstraction of r e a l i t y . It i s not enough that a model gives results observable in the real world. The model should actually behave l i k e the real wor1d,particularly in i t s dynamic operation in time; otherwise one cannot be certain that the results obtained under s l i g h t l y different conditions w i l l be useful. An approach suggested in an a r t i c l e by Naylor and Finger (7) - 41 -(1966) c a l l e d " M u l t i - S t a g e V e r i f i c a t i o n " i s used to v a l i d a t e the s i m u l a t i o n model. T h i s method o f v e r i f i c a t i o n c o n s i s t s o f the f o l l o w i n g s t a g e s : * the f i r s t s t a g e c a l l s f o r the f o r m u l a t i o n of a s e t of p o s t u l a t e s o r hypotheses d e s c r i b i n g the b e h a v i o u r of the system under s t u d y . * the second stage c a l l s f o r an attempt to v e r i f y as many of these p o s t u l a t e s as one can u s i n g a v a i l a b l e s t a t i s t -i c a l t e s t s . * the f i n a l s t age c o n s i s t s o f t e s t i n g the model's a b i l i t y t o p r e d i c t the b e h a v i o r o f the system under s t u d y . 3.3.1. General V a l i d i t y of the Model The most i m p o r t a n t assumption made from the model i s t h a t the d i s t r i b u t i o n o f " c a l l s a r r i v i n g " i s a t y p i c a l P o i s s o n d i s t r i b u t i o n . Other assumptions, such as those made a b o u t . l o a d -i n g and u n l o a d i n g time and c a n c e l l a t i o n s , are based on e m p i r i c a l d i s t r i b u t i o n s . 3.3.2. S t a t i s t i c a l Test f o r P o i s s o n A r r i v a l s To t e s t t h a t the a r r i v i n g d i s t r i b u t i o n f o l l o w s a P o i s s o n d i s t r i b u t i o n , t he f o l l o w i n g theorem (8) w i l l be a p p l i e d . Suppose one has observed the p r o c e s s over a p e r i o d o f l e n g t h T, d u r i n g which n events have o c c u r r e d . For j = l , . . ., n, l e t V. denote the time (measured from the s t a r t o f the p e r i o d o f o b s e r v a t i o n ) at which the j event o c c u r r e d . I f the event has o c c u r r e d i n a c c o r d w i t h a P o i s s o n P r o c e s s , the random v a r i a b l e s U^, . . . , u"n are independent and u n i f o r m l y d i s t r i b u t e d o v er i n t e r v a l 0 t o T. Consequently one method of t e s t i n g whether the events are of P o i s s o n type i s to t e s t whether the o b s e r v a t i o n s U , . . ., U are independ-- 42 -ent and u n i f o r m l y d i s t r i b u t e d o v er 0 to T. The t e s t was conducted f o r t h r e e p e r i o d s o f the day c o r r e s -ponding to t h r e e ambulance s h i f t s : 12 m i d n i g h t - 8 a.m.; 8 a.m. -6 p.m.; 6 p.m. - 12 m i d n i g h t . To t e s t t h a t the o b s e r v a t i o n U j , . . . , U n are independent and u n i f o r m l y d i s t r i b u t e d , the Cramer Von Mises t e s t was employed at a 10% s i g n i f i c a n c e l e v e l . The t e s t was based on a sample o f f o u r t e e n days from M e t r o p o l i t a n Ambulance Company, For each day the t h r e e p e r i o d s time were t e s t e d . The r e s u l t between 8 a.m. - 6 p.m. arid 6 p.m. - 12 m i d n i g h t a l l f o u r t e e n t e s t s were accep t e d and 12 a.m. - 8 a.m. t h i r t e e n t e s t s were a c c e p t e d . At 10% s i g n i f i c a n c e l e v e l f o r a t r u e d i s t r i b u t i o n ( u n i f o r m ) , acceptance o f 12.6 t e s t s are e x p e c t e d . Hence, the assumption t h a t U j i s a U n i f o r m d i s t r i b u t i o n (and hence t h a t a r r i v a l s f o l l o w a P o i s s o n d i s t r i b u t i o n ) i s not r e j e c t e d . 3.3.3. D e t e r m i n i n g the Length o f the S i m u l a t o r The problem o f d e t e r m i n i n g the l e n g t h o f the s i m u l a t i o n i s one o f s e l e c t i n g a sample s i z e o f s u f f i c i e n t magnitude t o p r o v i d e the d e s i r e d degree of a c c u r a c y i n the mean response t i m e . F i v e d i f f e r e n t t o t a l s of c a l l s (500, 750, 1000, 1250 and 1500) were g e n e r a t e d . Each number was generated f i v e times w i t h d i f f e r e n t random number seeds. The s t a n d a r d d e v i a t i o n between average response times i n each number o f c a l l s was c a l c u l a t e d . F i g u r e 5 shows the v a l u e o f s t a n d a r d d e v i a t i o n . Note t h a t as the number o f c a l l s i n c r e a s e s , the s t a n d a r d d e v i a t i o n d e c r e a s e s . 3.3.4. Correspondence o f Real World and S i m u l a t e d Data One o f the most i m p o r t a n t t e s t s of v a l i d i t y i s the a b i l i t y - 43 -of the model t o p r e d i c t response time a c c u r a t e l y ( e m p i r i c a l response times from the d a t a were not used i n the c r e a t i o n o f the model). A h i s t o r i c a l v a l i d a t i o n was performed by comparing response time d a t a from the r e a l w o r l d w i t h the model r e s u l t s f o r the same time p e r i o d . C h i - s q u a r e t e s t s were used to t e s t the degree of s i m i l a r i t y on the d i s t r i b u t i o n o f response t i m e . The C h i - s q u a r e t e s t i s a c l a s s i c a l s t a t i s t i c a l method which can be used f o r t e s t i n g the degree o f s i m i l a r i t y between a d i s t r i b u t i o n g e n e r ated by a model and fr e q u e n c y d i s t r i b u t i o n o f observed d a t a . The d a t a are summarized i n t o c a t e g o r i e s and the d i f f e r e n c e i n r e l a t i v e f r e q u e n c i e s found i n each c a t e g o r y are used t o d e r i v e a s t a t i s t i c . S i n c e a c a t e g o r y must have at l e a s t f i v e o b s e r v a t i o n s , the t e s t i s not u s e f u l , t o t e s t o b s e r v a t i o n s at the extreme t a i l s o f a d i s t r i b u t i o n . The d i s t r i b u t i o n of response time f o r M e t r o p o l i t a n Ambulance Company o f Vancouver i s compared to the s i m u l a t i o n r e s u l t s i n F i g u r e 4. From the d a t a i n Table 9, the Chi-Square s t a t i s t i c at a .05 s i g n i f i c a n c e l e v e l w i t h 16 degrees o f freedom, i s equal t o 26.22. Thus, the h y p o t h e s i s t h a t the two samples ( a c t u a l and s i m u l a t e d ) came from the same p o p u l a t i o n i s a c c e p t e d . - 46 -Actual^ F r e q .(f) Simulated F r e q . ( f ) A. f - f f 1 - 2 4.32 4 .32 0.02 2- 3 12.48 7 5.48 2.40 3 - 4 26.88 28 1.12 0.04 4 - 5 34.56 39 4.44 0.57 5 - 6 47.04 55 7.-96 •:ii;-34 6- 7 50.40 47 3.40 0.22 7 - 8 32.64 48 15.36 7.22 8 - 9 34.56 45 10.44 3.15 9-10 30.72 40 9.28 2.80 10-11 33.60 26 7.60 1.71 11-12 38.88 30 8.88 2.02 12-13, 27.84 19 8.84 2.80 13-14 15.36 13 2.36 0.36 14-15 19.20 16 3.20 0.53 15-16 .11.52 12 0.48 0.01 16-17 11.04 9 2.04 0.37 17> 48.96 . 42 16.96 0.98 25.20 Table 9 " : ~ CHI Square Goodness of F i t Test of Response Time P r o b a b i l i t y D i s t r i b u t i o n - 48 -CHAPTER 4 - THE AMBULANCE LOCATION PROBLEM In this chapter the problem of minimizing the mean response time of the ambulance system as a function of the location of the ambulance depots is considered. We s h a l l assume that the closest ambulance dispatch rule i s used. 4.1. Essential Features in the Location of Ambulances The following features of ambulance systems were noted during the computer simulation studies. ambulances are not always awaiting a c a l l at th e i r station when a c a l l for their service is received, the p r o b a b i l i t y that an ambulance is at i t s depot depends upon the incident rate. the response time for particular c a l l depends upon the number of ambulances which are busy when the c a l l is received. since during peak periods the hospital becomes a departure point for ambulances responding to c a l l s , the locations of hospitals affect the mean response time. These features suggest that the location of ambulances could be considered as a stochastic location problem in that the assign-ment of c a l l s to servers is a dynamic process and a function of the number of busy ambulances at the time c a l l s are received. 4.2. An Analytical Approximation to Mean Response Time Let N be a c o l l e c t i o n of nodes ( d i s t r i c t s ) and K be the number of ambulances ( i . e . , ambulance depots) to assign to node locations. An assignment p is a mapping from (1 . . . K) to N. Hence, for any JL e ( l . the location of s t a t e d as f i n d i n g an assignment peT such t h a t V (f>) = V (p) f o r - 49 -ambulance A, Let T denote the c o l l e c t i o n o f a l l a s s i g n m e n t s , o f which K t h e r e are |T|=(N) where jT| i s the c a r d i n a l i t y of the s e t T . D e f i n e V (^) as the mean response time i n the a b s t r a c t ambulance l o c a t i o n model. The problem of o p t i m i z i n g the ambulance l o c a t i o n s can be p"eT such t h a t V (^) = V (^ a l l ^ e T . T h e r e f o r e , an a n a l y t i c a l a p p r o x i m a t i o n i s the mean response time f u n c t i o n V would be d e s i r a b l e . Swoveland e t . a l . (10) have made the f o l l o w i n g a p p r o x i m a t i o n which i s based on s i m u l a t i o n r e s u l t s : K o v (p ) * Z w c 1^) d (>(i»q).i) • • - t i l \ i e N q=l ' where N - the number o f nodes K - the number o f ambulances "^=> ( i , q ) - the l o c a t i o n of qth c l o s e s t ambulance t o node i under assignment ^ • w ( i , q ) = the f r a c t i o n o f c a l l s which o c c u r at node i and s e r v i c e d by the qth c l o s e s t ambulance. A m a t r i x (w ( i , q ) ) was o b t a i n e d from a number o f s i m u l a t i o n runs w i t h a wide mix o f ass i g n m e n t s . Because i t was found t h a t the v a l u e w ( i , q ) produced by one assignment d i d not d i f f e r s i g n i f i c a n t l y from those produced by any o t h e r assignment, i t was determined t h a t the v a l u e s o f w ( i , q ) are independent o f ^  . A comparison of e x p r e s s i o n (1) w i t h the mean response times o b t a i n e d from s i m u l a t i o n was made w i t h a v a r i e t y of ambulance assignments. The r e s u l t s o b t a i n e d by Swoveland et a l (10) show - 50 -t h a t the v a l u e from e x p r e s s i o n (1 ) was a r e a s o n a b l e a p p r o x i m a t i o n of mean response t i m e . 4 . 3 . A L o c a l Search A l g o r i t h m M i n i m i z i n g v over T w i t h v d e f i n e d as i n e x p r e s s i o n [1 ] c o n s i d e r e d . For example, w i t h N = 82 and K = 20, t h e r e would 2 0 be |T|= 82 assignments t o enumerate. At each s t e p of the a l g o r i t h m , one ambulance i s moved from i t s c u r r e n t l o c a t i o n to an a d j a c e n t l o c a t i o n u n t i l such a movement e f f e c t s a r e d u c t i o n i n the mean response t i m e , as computed by e x p r e s s i o n [ 1 ] . The a l g o r i t h m t e r m i n a t e s when a l o c a l minimum i s reached; i . e . , when no f u r t h e r r e d u c t i o n i n mean response time can be a c h i e v e d by moving a s i n g l e ambulance from i t s c u r r e n t l o c a t i o n t o any of i t s a d j a c e n t l o c a t i o n s . Let Aj C N denote t h a t c o l l e c t i o n o f nodes which are d e f i n e d to be " a d j a c e n t " node j . We say the assignment p i s a d j a c e n t to w i t h complete enumeration i s too e x p e n s i v e an approach to be assignment p i f f o r some J(> e f o r a l l i f t . The a l g o r i t h m i s as f o l l o w s : Step 0 C a l c u l a t e the mean response time f o r an i n i t i a l assignment po. Set k = 0 and go t o Step 1 . Step 1 Search f o r an assignment which i s a d j a c e n t to and f o r which v (pV) < v (PL . ) . I f no such assignment e x i s t s , t e r m i n a t e o t h e r w i s e go to Step 2. Step 2 Set one and go to Step 1 . One may proceed from Step 1 to Step 2 as soon as the f i r s t found f o r which v A l t e r -- 51 -natively, before proceeding to step 2 one may enumerate a l l adjacent assignments (or some minimal number of assignments) to find the one for which the marginal improvement in the objective is maximum. 4.4. Experimental Evaluation of the Algorithm Recall that expression [ l ] determines the system mean response time by summing the mean response time for each node multiplied by the probability of an incident at that node. Consequently, one would expects that the algorithm would tend to select the locations close to the nodes with a high pr o b a b i l i t y of an incident. Thus the nodes with low pr o b a b i l i t y of an incident would tend to suffer a high mean response time, and the desirable ambulance locations depend on the sp a t i a l d i s t r i b u t i o n of demand. If the incident rate were to increase one would expect ambulances to be in the i d l e state ( i . e . , at th e i r station awaiting a c a l l ) less frequently. As shown in figure 6 1the pr o b a b i l i t y of c a l l s being served by the closest ambulance decreases as the rate of incidents increase. However, the p r o b a b i l i t i e s that c a l l s are served by 2nd and 3rd closest ambulances both increased. These observations lead one to expect that the desirable ambulance locations are a function of the level of demand within the system. In the algorithm we have discovered that when optimizing ambulance location, low demand areas generally suffer a high mean response time. In an e f f o r t to reduce the variance in average response times between high demand and low demand, the travel time matrix was replaced by a travel time matrix in which each element was raised to the kth power (k = 2,3,4). The - 52 -ambulances were then l o c a t e d by m i n i m i z i n g the average a d j u s t e d response t i m e . The a d j u s t e d t r a v e l times were found t o have p l a c e d a g r e a t e r emphasis on c a l l s h a v i n g l o n g e r response t i m e s . I t appeared t h a t the a l g o r i t h m would tend to d i s p e r s e ambulances more and more as k i s i n c r e a s e d . 4.5. S e n s i t i v i t y o f A l g o r i t h m t o Choice o f S t a r t i n g L o c a t i o n s As i n p u t f o r t h i s a l g o r i t h m i s a s e t o f s t a r t i n g l o c a t i o n s , one may ask how the c h o i c e o f s t a r t i n g l o c a t i o n s a f f e c t s the mean response time a s s o c i a t e d w i t h the f i n a l l o c a t i o n s . S i n c e the a l g o r i t h m i s a l o c a l s e a r c h a l g o r i t h m , two s e t s o f d i f f e r e n t s t a r t i n g l o c a t i o n s would not n e c e s s a r i l y converge t o the same se t of f i n a l l o c a t i o n s . We e x perimented w i t h these a l g o r i t h m f o r numerous d i f f e r e n t s t a r t i n g s e t s o f l o c a t i o n s and found t h a t the d i f f e r e n c e s i n c o n f i g u r a t i o n s f o r f i n a l s s e t s o f l o c a t i o n s was not as much as f o r s t a r t i n g l o c a t i o n s ( i . e . , the f i n a l s e t s o f l o c a t i o n s are q u i t e s i m i l a r , i f not the same). In t h e s e r u n s , the d i f f e r e n c e i n mean response time between d i f f e r e n t s t a r t i n g s e t s of l o c a t i o n s i s between 1 and 2 m i n u t e s . But the d i f f e r e n c e i n the f i n a l s e t s o f f i n a l s e t s o f l o c a t i o n s are s t i l l f a i r l y dependent on the i n i t i a l l o c a t i o n s . Thus, i t i s n e c e s s a r y to run experiments s t a r t i n g from a number o f s t a r t i n g p o i n t s t o i n c r e a s e one's l e v e l o f assurance t h a t one has found the b e s t s e t o f l o c a t i o n s . JEff ect_ofwGrobabUity.of_.an, inciden£.cl is patch by. k-.th. closes t. .Ambulance, (k-l^).—. .4 .3 Prob:"'4th'closest Rfltp r a i l r>A-r hnnr - 54 -CHAPTER 5 - DESIGN OF THE EXPERIMENT I n t r o d u c t i o n • The experiments conducted f o l l o w e d the f o l l o w i n g p r o c e s s : * run s i m u l a t i o n w i t h a number of d i f f e r e n t assignment t o c r e a t e (W(i,q)) m a t r i x . * use the (W(i,q)) m a t r i x and l o c a l s e a r c h a l g o r i t h m to determine the o p t i m a l l o c a t i o n s . * run the s i m u l a t i o n w i t h the o p t i m a l l o c a t i o n s t o d e r i v e the s t a t i s t i c a l output i . e . , average response t i m e . I f the experiments w i t h the s i m u l a t i o n model are w e l l d e s i g n e d , they s h o u l d be u s e f u l i n e x p l a i n i n g the b e h a v i o r o f an ambulance system. Two r e q u i r e m e n t s must be c o n s i d e r e d . 1 . The experiments must be performed under c o n t r o l l e d c o n d i t i o n s so t h a t the e f f e c t o f one v a r i a b l e on system performance can be s t u d i e d w i t h o u t c o n f o u n d i n g i t s e f f e c t w i t h s i m u l t a n e o u s changes i n the o t h e r system v a r i a b l e s . 2. The ambulance system s e l e c t e d f o r study s h o u l d be an a b s t r a c t i o n of r e a l i t y so t h a t o n l y the i m p o r t a n t f e a t u r e s o f the system are modeled. T h i s w i l l h i g h -l i g h t the changes i n b e h a v i o r caused by expe r i m e n t -a t i o n w i t h the system v a r i a b l e s . I t i s i m p o r t a n t t o l i m i t the number o f s t o c h a s t i c f e a t u r e s i n the model i n o r d e r to sharpen the system response to changes i n parameter o r p o l i c i e s . For example, the s t o c h a s t i c f e a t u r e of s t a r t up time at home base can be e l i m i n a t e d by u s i n g mean s t a r t up time w i t h o u t s e r i o u s l y a f f e c t i n g - 55 -the r e a l i s m o f the model. F i n a l l y , e x p e r i m e n t a t i o n w i l l s t u d y the f e a t u r e s t h a t are b e l i e v e d t o a f f e c t response time most d i r e c t l y , d e s p i t e the f a c t t h a t many o t h e r i n t e r e s t i n g system v a r i a b l e s have not been c o n s i d e r e d ( i . e . , l e v e l o f ambulance a t t e n d a n t t r a i n i n g , equipment,„carried on b o a r d , s c r e e n i n g c a l l s and p r i o r i t y d i s p a t c h i n g ) . 5.2. Experiment Design The p l a n was based on the s y s t e m a t i c v a r i a t i o n o f the f o l l o w i n g : * Number o f ambulance * L o c a t i o n o f ambulance * D i s p a t c h i n g P o l i c y * A r r i v a l r a t e of ambulance c a l l s The o p t i m a l l o c a t i o n o f the ambulance was found by u s i n g the l o c a l s e a r c h a l g o r i t h m . The s i m u l a t i o n was used to d e s c r i b e i n the p r e v i o u s c h a p t e r t o check t h a t r e s u l t s were ind e e d l o c a l l y o p t i m a l . Because both the l e v e l and s p a t i a l d i s t r i b u t i o n o f demand va r y by time o f day, i t was n e c e s s a r y t o conduct s e p a r a t e , experiments d u r i n g d i f f e r e n t p a r t s o f the day. The day was a r b i t r a r i l y d i v i d e d i n t o t h r e e p e r i o d s : 8 a.m. to 6 p.m.; 6 p.m. to 12 m i d n i g h t ; and 12 m i d n i g h t t o 8 a.m. Because t h i s d i v i s i o n was an a r b i t r a r y one, i t does not f o l l o w t h a t s h i f t changes, o r changes i n the numbers o f ambulances, s h o u l d o n l y o c c u r at the hour o f 8 a.m., 6 p.m., and m i d n i g h t . The l o c a t i o n o f ambulances p l a y an i m p o r t a n t p a r t i n d e t e r -m i n i n g response t i m e . F o r t h i s r e a s o n , i t was n e c e s s a r y to l o c a t e the ambulances o p t i m a l l y i n each e x p e r i m e n t . The o p t i m a l ambulance - 56 -l o c a t i o n was o b t a i n e d u s i n g t h a t l o c a l s e a r c h a l g o r i t h m which was d e s c r i b e d i n the p r e v i o u s c h a p t e r . Although we have recommended s p e c i f i c g e o g r a p h i c a l l o c a t i o n s o f ambulance depots ( e . g . , 41st and G r a n v i l l e ) f o r v a r i o u s numbers of ambulance depots at v a r i o u s times o f day, we would not expect our recommended c o n d i g u r a t i o n s to be adopted and implemented w i t h -out c o n s i d e r a b l e a l t e r a t i o n and a d d i t i o n a l e x p e r i m e n t a t i o n . There are f i v e main reasons f o r t h i s : S m a l l adjustments i n i n d i v i d u a l l o c a t i o n s are o f t e n n e c e s s a r y t o s e c u r e adequate accommodation f o r the ambulance depot, to account f o r l o c a l t r a f f i c c o n d i t i o n s , e t c . C o n s i d e r a t i o n s o t h e r than response time may i n f l u e n c e the l o c a t i o n s o f ambulance de p o t s . For example, f o r reasons o f t r a i n i n g and l i a i s o n , i t may be d e s i r a b l e to l o c a t e a c e r t a i n number o f ambulance depots a d j a c e n t to emergency rooms. Some of a l l o f the depot l o c a t i o n s s e l e c t e d f o r the e v e n i n g and e a r l y morning p e r i o d s may be r e s t r i c t e d to the depot l o c a t i o n s s e l e c t e d f o r the daytime p e r i o d . Our o b j e c t i v e of m i n i m i z i n g average response time may be d i s p u t e d . For example, i t may be f e l t t h a t a s m a l l improvement i n average response time i n one o f the out-l y i n g , s p a r s e l y p o p u l a t e d r e g i o n s i s worth a r e l a t i v e l y l a r g e r i n c r e a s e i n average response time f o r the r e g i o n as a whole. T h i s i s c l e a r l y a p o l i t i c a l d e c i s i o n . - 57 -S i g n i f i c a n t i n c r e a s e i n c e r t a i n t r a v e l times d u r i n g rush hours may n e c e s s i t a t e a re-deployment o f some ambulances. For example, i n response to the f a c t t h a t back-up s e r v i c e to the North Shore and t o Richmond d e t e r i o r a t e s i g n i f i -c a n t l y d u r i n g the a f t e r n o o n r u s h h o u r s , i t may be wise to t e m p o r a r i l y move a d d i t i o n a l u n i t s i n t o these areas at such t i m e s . 5.3. S i m u l a t i o n S t a t i s t i c s 1. I n c i d e n t S t a t i s t i c s F o r each i n c i d e n t , the response t i m e , w a i t i n g t i m e , and time to h o s p i t a l are c o l l e c t e d . Then, the f o l l o w i n g system s t a t i s t i c s are c a l c u l a t e d : * Mean * S t a n d a r d D e v i a t i o n * Means by M u n i c i p a l i t i e s * 90th f r a c t i l e v a l u e by M u n i c i p a l i t i e s (90% o f c a l l s have v a l u e s l e s s or equal than 90th f r a c t i l e v a l u e ) . 2. Ambulance S t a t i s t i c s The f o l l o w i n g s t a t i s t i c s were c o l l e c t e d f o r each ambulance: * Per c e n t a g e o f c a l l s t o the system answered by the ambulance * Mean time i n queues, response t i m e , time t o h o s p i t a l , s e r v i c e time * Mean U t i l i z a t i o n - 58 -CHAPTER 6 - EXPERIMENT RESULTS AND THEIR IMPLICATIONS In t h i s c h a p t e r the r e s u l t s o f the experiments d i s c u s s e d i n Chapter 5 are r e p o r t e d . V a r y i n g the number of ambulances has the e f f e c t o f v a r y i n g the average response time o v e r a l l and i n each M u n i c i p a l i t y . T h e r e f o r e , the s e c t i o n d e a l i n g w i t h the number o f ambulances and l o c a t i o n s i s d i v i d e d as f o l l o w s : * o p t i m a l l o c a t i o n s w i t h d i f f e r e n t number of ambulances i n system * M u n i c i p a l v a r i a t i o n due t o v a r i a t i o n o f number of ambulances 6.1. Optimal Ambulance L o c a t i o n With r e s p e c t t o the o b j e c t i v e o f m i n i m i z i n g average response t i m e , we have determined (near) o p t i m a l ambulance depot l o c a t i o n s u s i n g the a l g o r i t h m d e s c r i b e d i n Chapter 4 f o r t w e l v e d i f f e r e n t c o m b i n a t i o n s o f time o f day and number of ambulances. For the p e r i o d 8 a.m. - 6 p.m. we have o p t i m a l l y l o c a t e d 9,12,15,18,21,24 and 27 ambulance de p o t s ; f o r the p e r i o d 6 a.m. - 12 m i d n i g h t we have l o c a t e d 15,18 and 21 d e p o t s ; and f o r the p e r i o d 12 m i d n i g h t -•,.8 a.m. we have l o c a t e d 12,15 and 18 d e p o t s . The a c t u a l l o c a t i o n s a r e l i s t e d i n Table 14 which i s i n the Appendix. A comparison o f the c u r r e n t and o p t i m a l deployment of 21 ambulances (8a.m. - 6 p.m.) i s shown on Maps 1 and 2. D i s t r i b u t i o n o f response time f o r a l l c a l l s and f o r emer-gency c a l l s o n l y , f o r each o f the t h r e e p e r i o d s of the day, are g i v e n i n F i g u r e s 7, 8 and 9. One f i n d s , f o r example, t h a t when 15 ambulances are d e p l o y e d d u r i n g the p e r i o d 8 a.m. - 6 p.m. ( F i g . 7 ) 40% o f a l l emergency c a l l s have a response time which does not 59 " exceed 2.8 minutes. S i m i l a r l y , a 4.0 minute response time i s a c h i e v e d by 54%, 62%, 73% and 81% o f a l l emergency c a l l s , f o r 12, 15, 18 and 21 ambulances, r e s p e c t i v e l y . As mentioned p r e v i o u s l y , a b a s i c s t a r t - u p time o f 2.4 minutes i s charged f o r a l l c a l l s i n which the ambulance i s d i s p a t c h e d from i t s depot. The s t a r t - u p time i s zero when the ambulance i s d i s p a t c h e d w h i l e "on the r o a d " . T h i s averages out to an average s t a r t - u p time o f 1.9 minutes. Both the s t a r t - u p time and the t r a v e l time to the scene are decreased by 40% f o r those c a l l s i n which the s i r e n i s used. The t r a v e l times from the scene t o the h o s p i t a l are not reduced f o r emergency c a l l s s i n c e fewer than 1% o f a l l c a l l s r e q u i r e the s i r e n to be used on the way to the h o s p i t a l . C o n s i d e r i n g the e f f e c t o f these assumptions on response time i t i s c l e a r t h a t t h e r e are at l e a s t two p o l i c i e s which can be e s t a b l i s h e d which would have a d i r e c t , p r e d i c t a b l e e f f e c t on response t i m e : * reduce s t a r t - u p times and * use the s i r e n more o f t e n I f the s i r e n were used f o r a l 1 o f those n o n - t r a n s f e r c a l l s f o r which i t i s not now used (64% o f a l l c a l l s ) , average response time would decrease by about 2 minutes f o r c a l l s i n t h a t group and about 1 minute o v e r a l l . There a r e , o f course arguments a g a i n s t the i n -d i s c r i m i n a t e use o f the s i r e n which must be c o n s i d e r e d i n e s t -a b l i s h i n g p o l i c y . Average response times f o r each o f the t h r e e p e r i o d s o f the day and f o r v a r i o u s numbers of ambulances are d i s p l a y e d i n F i g u r e s 20 _! J J I J L _ J 1 1 • 10% 20% 30% 40% 50% 60% 70% 80% 90% F i g . 7 b Emergency C a l l s Only F i g . 7 D i s t r i b u t i o n of Response Time for the e n t i r e G.V.R.D. with Optimal Placement of Ambulances (8 a.m.* - 6 p.m.) - 61 -12 Ambulances 10% 20% 30% 40% 50% 60% 70% 80% 90% F i g . 8a A l l C a l l s 10 4> e •H H CO OJ co u CO 3 C C O *H O. g CO ^ <u OS 12 Ambulances 10% 20% 30% 40% 50% 60% 70% 80% 90% F i g . 8b Emergency C a l l s Only F i g . 8 D i s t r i b u t i o n of Response Time f o r the e n t i r e G.V.R.D. w i t h Optimal Placement of Ambulances (6 p.m. - 12 mid.) - 62 -- J L _ J L _ J _ J . ^ ^ 10% 20% 30% 40% 50% .60% . 70% 80% 90% F i g . 9fli A l l C a l l s I J . ! 1__ ! I I I l l 10% 20% 30% 40% 50% 60% 70% 80% 90% . F i g - 9 t Emergency C a l l s Only Fig. 9 D i s t r i b u t i o n of Response Time for the e n t i r e G.V.R.D. with -Optimal Placement of Ambulances (12 mid. - 8 a.m.) 1. Current Placement of Ambulances - 65 -10 and 11. F i g u r e 10 c o n t a i n s average response times f o r a l l c a l l s : F i g u r e 11 c o n t a i n s average f o r emergency c a l l s ( s i r e n ) o n l y . Although i t i s not our f u n c t i o n t o recommend to the G.V.R.D. what p o i n t to choose on these c u r v e s , i t i s apparent t h a t one s h o u l d p r o b a b l y have between 15 and 24 ambulances d u r i n g the day (8 a.m. - 6 p.m.) between 12 and 21 ambulances i n the e v e n i n g (6 p.m. - m i d n i g h t ) , and between 9 and 18 i n the e a r l y morning (midnight - 8 a.m.). Any fewer than 15 ambulances d u r i n g the day lea d s to a s e r i o u s d e t e r i o r a t i o n i n average response t i m e . Moreover, from F i g u r e 4 one sees t h a t the f r a c t i o n o f c a l l s w i t h e x t r e m e l y l o n g response times i s p a r t i c u l a r l y s e n s i t i v e t o numbers o f ambulances i n the 12-15 range. On the o t h e r hand, o n l y a v e r y s l i g h t improvement i n average response time d u r i n g the day i s o b t a i n e d as a r e s u l t of g o i n g from 24 to 27 ambulances. A s i m i l a r , though l e s s d r a m a t i c , d e t e r i o r a t i o n i n average response time o c c u r s when fewer than 12 ambulances are deployed d u r i n g the e v e n i n g or fewer than 9 i n the e a r l y morning. F i g u r e s 12, 13 and 14 d i s p l a y the r e l a t i o n s h i p between average response t i m e s , 90th f r a c t i l e response t i m e s , s e r v i c e times and ambulance u t i l i z a t i o n , f o r each number o f ambulances i n each p e r i o d of the day. The 90th f r a c t i l e curves g i v e those response times which are met or b e t t e r e d by 90% o f a l l c a l l s . The (average) ambulance u t i l i z a t i o n (which i s the f r a c t i o n o f time each ambul-ance i s busy) i s read from the v e r t i c a l s c a l e on the r i g h t of the graph. - 66 -at w CO <V c O 3 D. C in . H 00 TO M (!) > < 1 8 16 1 4 12 t 10 4 t © i 8 a.m. - 6 p.m. 6 p.m. - 12 mid. 12 mid.- 8 a.m. Number of Ambulances Fig. 1 0 Average Response Time by Number oj_Ambulances b yJ^me_of Day, for A l l C a l l s / - 67 -8 a.m. 6 p.m. 12 mid. 6 p. ID . 12 mid. 8 a.m. e •t-i H <1> « W dl. C u O 3 O. C erf 60 CO u > 12 15 18 21 24 27 Number of Ambulances Fig. 11 Average Response Time by N-• mber of Ambulances by Time of Day, Emergency C a l l s Only - 68 -5 0 AO 30 20 10 s e r v i c e time u t i l i z a t i o n ' 90th f r a c t i l e response time \ 40% o CO •H 4-1 P> CU u c 3 J 10% 12 1 5 18 21 24 27 Number of Ambulances F i g . 12 E f f e c t of Number of Ambulances -on Response Time, Service Time and Ambulance U t i l i z a t i o n (8 a.m' - 6 p.m.) 01 3 C •H X c •i-t OJ e H C <u 40 30 20 10 - 69 -30% 20% H service time u t i l i z a t i o n 90th f r a c t i l e response time 10% 15 18 21 F i g . 13 E f f e c t of Number of Ambulances on Response Time, Service Time and Ambulance U t i l i z a t i o n (6 p.m. - 12 micl.) " c o •r-l •U 03 N •H *-! o c 0} D 6 < - 70 -40 30 20 10 service time u t i l i z a t i o n 90th f r a c t i l e response time 12 1 5 1 8 20% 10%. F i g . 14 E f f e c t of Number of Ambulances on Response Time, Service Time and Ambulance U t i l i z a t i o n (12 mid. - 8 a.m.) c o CO N • r - l r-H • H (J c co-rn •g - 71 -6.2. V a r i a t i o n s i n Response Time by M u n i c i p a l i t y Average response time and 90th f r a c t i l e times by M u n i c i p a l i t y , are g i v e n i n Tab l e s 10, 11 and 12 ( f o r a l l c a l l s ) , f o r v a r i o u s numbers o f ambulances. We see i n Table 9, f o r example, t h a t w i t h 21 ambulances, West Vancouver has an average response time d u r i n g the day o f 10.0 minutes compared w i t h a r e g i o n a l average 6.6. N i n e t y p e r c e n t o f c a l l s o r i g i n a t i n g i n West Vancouver r e c e i v e response w i t h i n 15.8 minutes or l e s s . From th e s e t a b l e s we d i s c o v e r e d t h a t the m u n i c i p a l i t y w i t h low demands s u f f e r e d a h i g h average response time a l t h o u g h t h e r e were areas q u i t e o f t e n are d i s p a t c h e d to h i g h demand a r e a s . For example, West Vancouver ambulances answer o n l y 70% o f West Van-couver c a l l s . The r e s t o f the c a l l s i t answers are from North Vancouver. 30% o f West Vancouver c a l l s are s e r v i c e d by more d i s t a n t ambulances ( i . e . , North Vancouver or Van c o u v e r ) . One way t o s o l v e t h i s problem i s to employ a " r e g i o n a l d i s p a t c h " r u l e which w i l l be d i s c u s s e d i n S e c t i o n 6.3. F i g u r e 15 g i v e s a comparison between the b e s t and worst M u n i c i p a l average response time f o r each number o f ambulance. The i n f o r m a t i o n used to c o n s t r u c t t h i s graph i s i n Table 10. Except f o r the case of 18 ambulances, Vancouver has the b e s t M u n i c i p a l average response time f o r each number o f ambulances (about 18% b e t t e r than the G.V.R.D. a v e r a g e ) . Except f o r the case of 12 ambulances, P o r t C o q u i t l a m has the worst M u n i c i p a l average response time (about 90% above the G.V.R.D. a v e r a g e ) . You w i l l r e c a l l from Chapter 4 ( S e c t i o n 4.4.) t h a t the v a r i a n c e between a h i g h demand a r e a and low demand area c o u l d be r i u c i D e r 0 1 i i i i i u u x a u u e t s Ave ir 9 12 • 15 18 21 24 27 Municipality c a l l s /hr. FT ART 90 FT ART 90 FT ART 90 FT ART 90 FT ART; 90 ART 9 0 ^ FT ART 90 Burnaby .73 18.8 36.8 12.3 19.8 10.2 16.0 8.9 13.2 8.2 12.2 7.6 10.5 6.6 9.9 Coquitlam .12 22.7 44.7 13.4 26.7 9.1 19.0 9.1 13.3 8.8 10.9 8.1 9.6 8.4 8.6 Delta .05 26.9 30.6 18.4 18.7 12.1 15.0 10.7 12.6 11.1 18.3 10.5 12.7 10.5 12.7 New Westminster .35 20.1 41.7 10.3 17.3 9.5 15.4 7.7 12.1 7.1 9.7 6.8 8.9 6.8 8.9 North Vancouver .38 21.2 45.3 11.5 25.9 9.0 18.5 8.3 16.4 6.7 13.3 6.6 12.1 6.2 10.7 Port Coquitlam .08 35.2 53.6 17.0 32.5 13.9 25.4 12.8 21.8 12.3 21.4 11.6 17.0 11.6 15.5 Port Moody .06 22.5 39.1 18.7 26.5 6.6 13.3 5.7 5.3 5.5 5.3 5.5 5.3 5.5 5.3 Richmond .29 21.7 52.9 12.5 21.2 9.7 18.2 7.1 15.0 7.1 11.0 7.1 9.7 6.9 7.0 Surrey .44 25.8 46,8 14.5 30.0 12.1 18.7 11.4 17.8 11.2 18.0 9.1 10.8 9.1 12.0 Vancouver 4.23 13.9 32.2 8.3 12.8 6.6 13.2 5.9 10.1 5.2 7.9 5.0 7.6 5.0 8.0 West Vancouver J.6 20.6 43.2 18.5 35.8 11.1 18.0 10.3 17.8 10.0 15.8 9.9 15.2 9.2 13.5 White Rock .11 33.8 53.3 18.7 39.3 7.4 17.6 7.2 17.4 7.4 18.5 5.5 5.0 5.5 5.0. Overall 7.0 17.4 '40.6 10.4 19.5 8.1 15.8 7.2 12.0 6.6 10.7 6.1 10.4 6.0 9.7 Average Service #. Time 52.3 •45.3 42.8 41.9 41.2 40.8 40.6 Average U t i l i z a t i o n 67% 41% 33% ' 25% 21% 18% 17% Table 11 Average and 90 F r a c t i l e Response Times by Municipality and by "umber of Ambulances (8am - 6pm, a l l c a l l s ) - 73 -Number of Ambulances -• Ave // calls /nr. 15 18 21 Municipality ART FT 90 ART FT 90 ART 90 Burnaby .76 9.2 13.9 8.5 12.6 7.9 11.5 Coquitlam .13 9.9 15.6 9.2 15.4 8.6 10.7 Delta .04 13.8 15.0 9.5 10.4 11.3 11.9 • New Westminster .35 6.5 8.9 ' 6.7 9.0 6.4 8.8 North Vancouver .29 6.6 10.0 7.2 12.4 6.4 9.8 Port Coquitlam .06 15.9 17.0? 11.3 15.9 5.9 7.1 Port Moody .03 16.9 19.0 5.7 5.3 11.5 12.7 Richmond . .16 8.8 '"' 15.5 S.7 15.1 6.7 10.1 Surrey .43 6.8 6.9 6.8 6.8 6.8 6.8 Vancouver 3.02 6.0 10.2 5.7 9.8 5.2 8.7 West Vancouver .12 10.6 17.4 10.0 15.2 10.6 16.1 White Rock .05 4.7 4.8 3.8 4.0 3.8 4.0 Overall 5.4 7.1 12.0 6.7 11.3 6.0 10.3 Ave. Service Time 40. 7 40.2 39. 7 Ave. Utilization 24% 20% 17% Table 10 Average and 90th Fractile Response Times by Municipality and by Number of Ambulances (6pm - midnight, a l l calls) \ Number, of Ambulances Municipality Ave // calls /hr. 12 15 18 ART 90 FT ART 90 FT ART FT 90 Burnaby .25 9. 9 14 .5 .8.2 10 .7 7. 9 10.2 Coquitlam .05 8. 6 9 .4 11.3 15 .4 8. 5 9.6 Delta .05 15. 5 15 .0 9.6 10 .4 11. 8 12.0 New Westminster .14 6. 7 7 .5 7.9 11 .9 7. 7 9.7 North Vancouver .18 7. 2 10 .5 6.8 9 .4 .' 6. 3 8.7 Port Coquitlam .05 16. 6 17 .5 11.6 11 .1 10. 8 11.1 Port Moody .03 20. 2 20 .5 7.8 15 .6 9-9 16.0-Richmond .07 9. 0 18 .0 8.5 15 .0 5. 6 7.5 Surrey .19 11. 6 12 .2 11.3 12 .1 11. 6 12.1 Vancouver 1.70 6. 3 10 .3 5.8 9 .5 5. 1 8.0 West Vancouver .08 9. 6 17 .4 8.7 15 .0 8. 7 16.0 White Rock .04 3. 8 4 .5 4.7 4 .8 3. 8 4.0 Overall 2.8 7.3 12 .0 6.8 12 .0 6. 0 10.0 Ave. Service Time - 41.8 40.9 40.5 Ave. Utiliza-tion 16% - 12% 10% Table 12 Average and 90th Fractile Response Times by Municipality and by Number of Ambulances (midnight - 8 am, a l l calls) - 75 -maximum municipal average response time GVRD average response time minimum municipal average response <u E •rH H a) co CO CIJ C J-> O 2 a- c CO *H P i c . rt u > '12 15 18 21 24 27 Number of Ambulances Fig.1 5 Municipal V a r i a t i o n i n Average Response Time by Number of  Ambulances for A.11 C a l l s (8 a.m. - 6 p.m.) - 76 -reduced by a d j u s t i n g t r a v e l t i m e . Two experiments were performed i n an e f f o r t to reduce the v a r i a n c e i n average response times between M u n i c i p a l i e s . Both were f o r 21 ambulances, d u r i n g the daytime p e r i o d . In the f i r s t case the t r a v e l time m a t r i x was r e p l a c e d by a m a t r i x of squared t r a v e l t i m e s ; i n the second case a m a t r i x o f cubed t r a v e l times was used. Ambulances were then l o c a t e d i n such a way as t o m i n i m i z e average a d j u s t e d response t i m e s . The e f f e c t of the squared and cubed t r a v e l times was to p l a c e a g r e a t e r emphasis on c a l l s h a v i n g l o n g e r response t i m e s . The r e s u l t a n t ambulance c o n f i g u r a t i o n s are shown i n Maps 3 and 4. These s h o u l d be compared w i t h the o r i g i n a l o p t i m a l c o n f i g u r a t i o n d i s p l a y e d i n Map 2. A comparison o f average response time by m u n i c i p a l i t y f o r the t h r e e 21 ambulance c o n f i g u r a t i o n s produced u s i n g the o r i g i n a l t r a v e l t i m e s , squared t r a v e l time and cubed t r a v e l t i m e s , i s g i v e n i n Table 13. 6.3. D i s p a t c h Rules The d i s p a t c h r u l e c u r r e n t l y employed i n Burnaby, New West-m i n s t e r , and Vancouver, i n a l l but the b u s i e s t p e r i o d s o f the day, i s the " r e g i o n a l d i s p a t c h " r u l e . Each o f e i g h t ambulances s e r v i n g these t h r e e M u n i c i p a l i t i e s i s a s s i g n e d a t e r r i t o r y over which i t has p r i n c i p a l r e s p o n s i b 1 i t y . Whenever a c a l l o c c u r s , the ambulance a s s i g n e d to the t e r r i t o r y o f the c a l l i s d i s p a t c h e d ( i f a v a i l a b l e ) , even i f t h a t ambulance i s not the " c l o s e s t " ( i n terms o f response time) one to the c a l l . There are two reasons f o r employing t h i s d i s p a t c h r u l e : a. i t s i m p l i f i e s d i s p a t c h i n g d e c i s i o n s f o r the d i s p a t c h e r s and, b. i t tends to e q u a l i z e the wo r k l o a d among the ambulance crews. - 77 -Municipality True Travel Times Squared Travel Times Cubed Travel Times Burnaby Coquitlam Delta New Westminster North Vancouver Port Coquitlam Port Moody Richmond Surrey Vancouver West Vancouver White Rock Overall 8.2 8.8 11.1 7.1 6.7 12.3 5.5 7.1 11.2 5.2 10.0 7.4 6.6 8.6 8.7 13.9 8.1 6.6 7.6 11.3 6.8 9.4 6.6 ,9.7 6.4 7.3 8.4 8.7 12.5 7.9 6.6 7.6 11.3 6.4 9.8 6.8 9.7 10.6 7.4 Table 13 Average Response Times by Municipality, with Optimal Placement of 21 Ambulances With True-, Squared, and Cubed Travel Times (8 am - 6 pm) / o -MAP 3. Optimal Placement of 21 Ambulances with Squared Travel Times (8 a.m. - 6 p.m.) s MAP 4. Optimal Placement of 21 Ambulances with Cubed Travel Times (8 a.m. - 6 p.m.) - 80 -We have experimented with one other dispatch rule, the "closest" dispatch rule, in which the "closest" available ambulance is always dispatched. For the regions as a whole the "closest" dispatch rule yields average response times which are roughly 12% lower than average response times produced under the "regional" dispatch rule (a decrease from 8.3 to 7.3 minutes for period 8 a.m. - 6 p.m. Although the regional average response time i s better when the "closest" dispatch rule i s used, the variance of average response time between Municipalities is greater. Therefore, the choice of dispatch rules depends on the objective of the administration. 6.4. S e n s i t i v i t y of Response Times to Changes in Demand We have performed a small number of experiments to gain some appreciation for the a b i l i t y of the system to deal with increased demand levels. Increased demand levels may occur in the short run or in the long run. For example, we would expect some increase in regional demand i f the service is provided without charge, or for a nominal charge. For a daytime deployment of 21 ambulance the e l a s t i c i t y of response time, as a function of c a l l rate per hour, i s .36. This means that i f the c a l l rate goes up from 7.0 c a l l s / h r by 100% to 14 c a l l s / h r , the average response time goes up 36% (from 6.5 to 8.8 minutes). S i m i l a r l y , i f the c a l l rate goes up by R percent (for R 200%) the response time would increase by about .36 x R percent. Hence, at the 21 ambulance l e v e l , the system can handle a considerable increase in demand with only a moderate deterior-ation in response time. - 81 -From-another p o i n t o f v i e w, i t was found t h a t i f the c a l l r a t e were t o i n c r e a s e by R p e r c e n t , then (.36) x R a d d i t i o n a l ambulances would be r e q u i r e d to a c h i e v e the same average response time as o b t a i n e d w i t h 21 ambulances at the e x i s t i n g c a l l r a t e . For example, i f the c a l l r a t e were t o double ( i ; ; e . , R=100%) , then .36 (100%) = 36% more ambulances - t h a t i s , r o u g h l y 8 ambulances (.36 x 21 = 7.6 ambulances) - would have to be added to m a i n t a i n the same l e v e l o f response t i m e . 6.5. T r a d e o f f s Between Annual O p e r a t i n g Cost andKResponse Time E s t i m a t e s o f t o t a l annual o p e r a t i n g c o s t ( i n 1974 d o l l a r s ) f o r v a r i o u s l e v e l s o f average response time are d i s p l a y e d i n F i g u r e 13, f o r each average response time l e v e l ( f o r a l l c a l l s or f o r emergency c a l l s o n l y ) , the annual o p e r a t i n g c o s t was computed from the f o r m u l a * Cost i n thousands of d o l l a r s = 621 + 125 (.416N, + .25N + a e 33N ) , when, m . $621,000 i s the e s t i m a t e d f i x e d c o s t per y e a r ( s a l a r i e s o f management, d i s p a t c h e r s and s t a f f o t h e r than ambulance a t t e n d a n t s and o t h e r management r e l a t e d c o s t s which do not v a r y w i t h the number o f ambulances). . $125,000 i s the e s t i m a t e d c o s t o f manning one ambulance w i t h two a t t e n d a n t s , 24 hours a day, 365 days a y e a r . Of t h i s t o t a l $105,000 goes to wages and f r i n g e b e n e f i t s f o r ambulance d r i v e r s and a t t e n d a n t s . The r e m a i n i n g $20,000 goes f o r d e p r e c i a t i o n and o p e r a t i n g expenses a s s o c i a t e d w i t h the ambulance, the c o s t o f f i r s t a i d s u p p l i e s and equipment, and r e n t and u t i l i t i e s f o r the ambulance depot. * A l l c o s t e s t i m a t e s were o b t a i n e d from M e t r o p o l i t a n Ambulance Company's a c t u a l and f o r e c a s t e d c o s t s . - 82 -3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 Average Regional Response Time ( i n minutes) Fig. T6 Tradeoff Between Average Response Times and Annual Operating Cost - 83 -the numbers .416, .25, and .333 a r e , r e s p e c t i v e l y , the f r a c t i o n s o f the 24 hour day which f i l l i n the t h r e e p e r i o d s 8 a.m. - 6 p.m. (10/24 = .416), 6 p.m. - m i d n i g h t (6/24 = .25) and m i d n i g h t - 8 a.m. (8/24 = .333). N J , N , and N a r e , r e s p e c t i v e l y , the numbers o f ° e m ambulances r e q u i r e d d u r i n g the day (8 a.m. - 6 p.m.), evenin g (6 p.m. - 12 m i d n i g h t ) and e a r l y morning (mid-n i g h t - 8 a.m.) to m a i n t a i n the g i v e n l e v e l o f average response time 24 hours a day (from F i g u r e s 7 and 8 ) . C o n s i d e r , f o r example, the c a l c u l a t i o n of the annual o p e r a t -i n g c o s t r e q u i r e d t o m a i n t a i n an average response time f o r emergency c a l l s o n l y , o f 4.3 m i n u t e s , 24 hours a day. We see from F i g u r e 8 t h a t to m a i n t a i n t h i s average response time about 19.2 ambulances would be needed d u r i n g the day, 16.5 would be needed i n the e v e n i n g , and about 14.2 would be r e q u i r e d i n the e a r l y morning ( a l t h o u g h f r a c t i o n a l numbers o f ambulances cannot a c t u a l l y be d e p l o y e d , f r a c t i o n a l v a l u e s s u f f i c e f o r t h i s c a l c u l a t i o n ) . Hence, N D = 19.2 N A = 16.5 e N = 14.2 m C o n s e q u e n t l y , the annual c o s t i n thousands o f d o l l a r s e q u a l s 621 + 125 (.416 (19.2) + .25(16.5) + .333(14.2) = 2727, or r o u g h l y 2.73 m i l l i o n , which i s the v a l u e on the v e r t i c a l s c a l e o f F i g u r e 13 which c o r r e s p o n d s to an average r e g i o n a l response time o f 4.3 minutes on t h e curve f o r emergency c a l l s o n l y . - 84 -We do not wish to convey the i m p l i c i t assumption that i t is necessarily desirable to have the same level of average response time at a l l times of the day (which was the assumption made for determining the two curves in Figure 13). In p a r t i c u l a r , we should point out that i f one were to add one more ambulance for one hour per day, i t would have a much greater impact on response times i f added during the daytime than i f added in either the evening or early morning hours. To i l l u s t r a t e this point, consider the effect of adding one ambulance for one hour in each of the three periods of the day, to those numbers of ambulances needed to provide an 8 minute average response time, 24 hours a day. We find from Figure 7 that the addition of one ambulance decreases average response time by roughly 2/3 of 1 minute in each of the three periods of the day. However, this reduction is applied to more c a l l s per hour during the daytime than during the evening or early morning. S p e c i f i c a l l y , the t o t a l reduction in response time per hour is 4.7 minutes during 8 a.m. - 6 p.m., (2/3 x 7.0 calls/hr) 3.6 minutes during 6 p.m. midnight (2/3 x 5.4 calls/hr) and only 1.9 minutes during 12 mid-night - 8 a.m. (2/3 x 2.8 c a l l s / h r ) . 6.6. Summary and Conclusions The results of the simulation experiments on the model of the Vancouver ambulance systems have provided some interesting insights into ambulance system behavior. The most noteworthy conclusions and results can be b r i e f l y summarized $s follows: * The long-range planning of ambulance service must regard ambulance service as an integral part of the t o t a l - 85 -emergency h e a l t h care system. Improvements s h o u l d be made i n the c o - o r d i n a t i o n of community emergency r e s o u r c e s . Ambulance s e r v i c e s h o u l d be p r o v i d e d on a r e g i o n a l b as i s . Ambulances s h o u l d be l o c a t e d s t r a t e g i c a l l y t o m i n i m i z e response t i m e . For a g i v e n d e s i r e d l e v e l o f response t i m e , the number and l o c a t i o n o f ambulances s h o u l d be d i f f e r e n t i n d i f f e r e n t p e r i o d s o f the day. P e r i o d i c r e view o f the l e v e l and s p a t i a l d i s t r i b u t i o n o f demand s h o u l d be made and adjustments s h o u l d be made i n the number and l o c a t i o n s o f ambulances. The d e t e r m i n a t i o n o f the p r o p e r response time l e v e l s h o u l d be based i n p a r t on an a n a l y s i s o f the t r a d e o f f between cost and response t i m e . 86 Time of Day Number of Ambulances 8am-6pm Marine D r / l l t h St Capilano Rd/Marine Dr 3rd St/Lonsdale 13th St/Lonsdale Burrard/W.Georgia Main/Hastings 41st/Nanaimo Knight/41st Fraser/41st 'GranvMle'/'Alst MacDonald/Broadway Burrard/Broadway Granville/Broadway Cambie/Broadway Main/Kingsway Fraser/Kingsway Clark/Broadway \ 6-Nanaimo/Broadway 0ak/70th G r a n v i l l e / G e o r g i a Hastings/Boundary Kingsway/Imperial Grandview Hgwy/fioundary Lougheed/Boundary Boundary/S.E. Marine Dr. Barnett/Willingdon 6th Av./12 St Brunet/Columbia Grant McConachleway/499 Hgwy No 5 Rd/499 Hgwy Barnett/Gatensbury Barnett/Woodland Dr. Roper/Johnston Rd Scott Rd/Ladner Trunk Rd Old.Yale Rd/King George Hgwy Tsawassen/Ladner Trunk Rd L,adner Trunk/K.G. Hgwy Table 14 (WVan) (NVan) (NVan) (NVan) (Van) (Van) (Van) (Van) (Van) (Van) (Van) (Van) (Van) (Van) (Van) (Van) (Van) (Van) (Van) (Van) (Van) (Van/Bby) (Van) (Van/Bby) (Bby) (Bby) (NWest) (NWest) (Rich) (Rich) (PMdy) (PCoq) (WRock) (Sry) (Sry) (Lad/Sry) (Sry) \ \ / V v V V V V V V V V v \ V V V-V V V V V V V \ V V V V V V V V s. V V V \ V V \ V V V V V V V V V V V V V V \ V V V V V \ V \ V \ V \ V V V V V V V V V V V V V V \ -V V V V V V V \ \ V V V \ V V \ V V V \ 1 V 1 — 6pm-midnight I5_ 18" 2] mi^lnjlgji^^^m . V V V V V V V V V V V V V V V V V V V V V V V V V V V V V \ V N \ V \ V V \ / Ny V V V V V \ V V \ 1 V V V V V V V V V V V V, V V V V V V \ V V V V V V V V V V \, V V V V V V V V V \ V V V V V Optimal Placement of Ambulance Depots by Time of Day and Number of Ambulances - 87 -REFERENCES 1 . B e l l , C o l i n E., "Optimal P l a n n i n g of an Emergency M e d i c a l S e r v i c e " , U n i v e r s i t y o f C a l i f o r n i a at I r v i n e ( u n p u b l i s h e d r e p o r t ) . 2. Emergency M e d i c a l S e r v i c e s , Report t o G.V.R.D. A d v i s o r y Committee by the Emergency M e d i c a l Subcommittee, Vancouver Canada, A p r i l , 1970. 3. F i t z s i m m o n s , J . A., "Emergency M e d i c a l Systems: A s i m u l a t i o n study and Computerized Method f o r Deployment o f Ambulances", U n i v e r s i t y o f C a l i f o r n i a , Ph.D. t h e s i s , Los A n g e l e s , 1970. 4. F o u l k e s , R. G. , " H e a l t h S e c u r i t y f o r B r i t i s h Columbian'.,', Queen's P r i n t e r , V i c t o r i a , Canada, December, 1973. 5. G i b s o n , Bugbee, Anderson and Odin, "Emergency M e d i c a l S e r v i c e s i n the Chicago A r e a " , U n i v e r s i t y o f C h i c a g o , C h i c a g o , 1971. 6. Morse, P h i l l i p M., Queues, I n v e n t o r i e s and Maintenance, W i l e y , New York, 1958. 7. N a y l o r , T. H. and F i n g e r , J . M., " V e r i f i c a t i o n o f Computer Models", 'Management S c i e n c e , V o l . 14, No. 2 (October 1967), Page B-92 t o B-101. 8. P a r z e n , Emanuel, S t o c h a s t i c P r o c e s s e s , Holden Day, San F r a n c i s c o , 1962. 9. Savas, E. S., " S i m u l a t i o n and C o s t - E f f e e t i v e n e s s A n a l y s i s o f New York's Emergency Ambulance S e r v i c e " , Management  S c i e n c e , V o l . 15, No. 12, (August 1969), page B-608 to B-627. 10. Swoveland, C , Uyeno D., V e r t i n s k y I . and V i c k s o n R., "Ambulance L o c a t i o n P r o b a b i l i s t i c Enumeration Approach", Management S c i e n c e , A p p l i c a t i o n P a r t 2, V o l . 20, (December 1973), page 686 to 698. 

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