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Steady state single channel queues Low, Siew Nghee 1968

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STEADY STATE SINGLE CHANNEL QUEUES by SLEW NGHEE LOW B.E. (Hon.), C i v i l , U niversity of Malaya, 1966 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF BUSINESS ADMINISTRATION i n the Department of BUSINESS ADMINISTRATION We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1968 In p r e s e n t i n g 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 r e q u i r e m e n t s f o r an advanced deg ree at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I ag ree t h a t 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 r e f e r e n c e and s t u d y . I f u r t h e r ag ree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y pu rposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n -t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depar tment o f Business A d m i n i s t r a t i o n The U n i v e r s i t y o f B r i t i s h Co l umb ia Vancouve r 8, Canada Date ABSTRACT This thesis extends-' the application of waiting line theory to situations where both arr i v a l rate and service rate distributions are arbitrary or i. on-random. It does so only for single channel, single phase, steady state, i n f i n i t e queues with no feed-back. Previous work by A.K. Erlang had shorn that queueing characteristics could be predicted for one case of an arbitrary service rate distribution, the constant service time. Also, F. Pollaczek had shown that,'where arri v a l rates are random, queue lengths and waiting times were independent of the form of the service rate distribution, being functions of the coe-f f i c i e n t of variance squared. But a l l of the works- assumed random arrivals around a stable mean arri v a l rate and, except for the constant service time case, most applications were limited to cases where both arr i v a l and service rates were random. This restriction has limited applications severely and has required that most analysis of queueing characteristics be done by simulation. „ \ This study develops and proves by inference the hypothesis that system length is dependent on these factors only: the square of the coefficient of variance of the inter-a r r i v a l time distribution, C 2 , the square of the coefficient 2 of variance of the service time distribution, C , and the ratio of mean arr i v a l rate to mean service rate, p . Through a combination of calculation and simulation a set of curves has been developed c o v e r i n g v a l u e s , C from 0 t o 6 , C_ EL S from 0 t o 6 and o f p from 0.1 to 0 .9 . These curves p e r m i t the p r e d i c t i o n o f system l e n g t h , and then o f average queue l e n g t h and w a i t i n g time, f o r any case where o n l y the mean —and v a r i a n c e o f the a r r i v a l and s e r v i c e time d i s t r i b u t o n s are known, even though n o t h i n g i s known about the form o f the d i s t r l b u t i o n s . ' > -In the usage o f the set o f graphs ( f i g u r e s 10 -29) , the . f o l l o w i n g steps are a l l t h a t i s r e q u i r e d t o o b t a i n the n e c e s s a r y c h a r a c t e r i s t i c s : a) C a l c u l a t e the average i n t e r a r r i v a l time, — = ( T o t a l time of o b s e r v a t i o n / T o t a l number o f A. a r r i v a l s ) b) C a l c u l a t e the v a r i a n c e f o r i n t e r a r r i v a l times, n ± 2 V a r ( t ) = v ( t -—) / T o t a l number o f a r r i v a l s a i = l i x c) C a l c u l a t e the f r a c t i o n a l c o e f f i c i e n t o f v a r i a n c e squared f o r i n t e r a r r i v a l time d i s t r i b u t i o n , C 2 = V a r ( t a ) / ( l / x ) 2 d) C a l c u l a t e the average s e r v i c e time, i _ = ( T o t a l time s e r v i c e f a c i l i t y i s i n o p e r a t i o n / T o t a l number s e r v i c e d ) e) C a l c u l a t e the v a r i a n c e f o r s e r v i c e times, m , 2 V a r ( t ) = 2 — ) / T o t a l number s e r v i c e d S j=l so * i i i Calculate the f r a c t i o n a l c o e f f i c i e n t of variance squared f o r service time d i s t r i b u t i o n , C 2 = V a r ( t s ) / ( 1 / U ) 2 . Calculate the u t i l i z a t i o n factor, p = (Average service time/Average i n t e r a r r i v a l time) 2 2 With the values p C , and C~" , read from the set of graphs (figures 10-29) the v e r t i c l e a x i s , L . Compute Lq, W and Wq . i v TABLE OF CONTENTS Page CHAPTER 1. INTRODUCTION.... , .... 1 Introduction H i s t o r i c a l B r i e f Objective ' • CHAPTER 2. DERIVATIONS OF DISTRIBUTIONS AND THEIR ' • PROPERTIES . 9 Derivations of D i s t r i b u t i o n s ; Poisson, Exponential, Erlangian and Hyperexponential D i s t r i b u t i o n s Properties of D i s t r i b u t i o n s ; Exponential, Erlangian and Hyperexponential CHAPTER 3. EXPONENTIAL INTERARRIVAL TIME DISTRIBUTION: EXPONENTIAL SERVICE TIME DISTRIBUTION 24 Erlang's Model CHAPTER 4. EXPONENTIAL INTERARRIVAL TIME DISTRIBUTION: . ARBITRARY SERVICE TIME DISTRIBUTION... 27 The Pollaczek-Khintchine Formula CHAPTER 5- ARBITRARY•INTERARRIVAL TIME DISTRIBUTION: EXPONENTIAL SERVICE TIME DISTRIBUTION ?4 p Relationship between L , and p and C A n a l y t i c a l Approaches f o r Exponential Service Time D i s t r i b u t i o n and Erlangian or Hyperexponential I n t e r a r r i v a l Time Di s t r i b u t i o n s Construction of Tables and Graphs f o r Single Channel with A r b i t r a r y I n t e r a r r i v a l Time Dist-r i b u t i o n and Exponential Service Time D i s t r i -. bution Testing of Graphs CHAPTER 6. ARBITRARY INTERARRIVAL TIME DISTRIBUTION: ARBITRARY SERVICE TIME DISTRIBUTION 57 2 2 Relationship between L and P , C and C cl S Construction of Graphs Testing of Graphs CHAPTER 7. CONCLUSION... 85 Usage of Graphs ^ • V APPENDIX I. DISTRIBUTIONS FOR TESTING OF GRAPHS. 89 APPENDIX II. FLOW CHART:'G.P.S.S. I l l . 9 4 APPENDIX III. FLOW CHARTS 95 APPENDIX rv. VALUES'-OF L •: - GiVEN C 2 , C 2 , and p. t104 BIBLIOGRAPHY 110 v i LIST OF TABLES Page TABLE I . PROPERTIES OF DISTRIBUTIONS.. 21 TABLE I I . VALUES OF L FOR EXPONENTIAL INTERARRIVAL TIME' DISTRIBUTION AND ARBITRARY SERVICE TIME DIST-RIBUTION.... 31 TABLE I I I A . VALUES OF L FOR ARBITRARY INTERARRIVAL TIME DISTRIBUTION AND EXPONENTIAL SERVICE TIME DISTRIBUTION . 4 9 TABLE I I I B . VALUES OF L FOR ARBITRARY INTERARRIVAL TIME DISTRIBUTION AND EXPONENTIAL SERVICE TIME . DISTRIBUTION 50 TABLE IV. TEST RESULTS FOR ARBITRARY INTERARRIVAL TIME DISTRIBUTIONS AND EXPONENTIAL SERVICE TIME DISTRIBUTIONS 54 TABLE V. TEST RESULTS FOR ARBITRARY INTERARRIVAL TIME DISTRIBUTIONS AND ARBITRARY SERVICE TIME DISTRIBUTIONS . . o . . 82 v i i LIST OF FIGURES Page FIGURE 1. ERLANGIAN DISTRIBUTION FACILITY OF k PHASES 12 FIGURE 2 . HYPEREXPONENTIAL DISTRIBUTION FACILITY 14 FIGURE 3- CUMULATIVE DISTRIBUTIONS FOR ERLANGIAN AND HYPEREXPONENTIAL DISTRIBUTIONS , 22 FIGURE 4. . ERLANGIAN AND HYPEREXPONENTIAL DISTRIBUTIONS. 23 FIGURES 5 ^ 6 . VALUES OF L ; INTERARRIVAL TIME DISTRIBUT-IONS: EXPONENTIAL. SERVICE TIME DISTRIBUTIONS: ARBITRARY ..32-33 FIGURES 7/8. VALUES OF L ; INTERARRIVAL TIME DISTRIBUT-IONS: ARBITRARY ( SERVICE TIME DISTRIBUTIONS: EXPONENTIAL...... 5 1 - 5 2 FIGURE 9. ILLUSTRATION OF DERIVATIONS FOR ARBITRARY INTERARRIVAL AND SERVICE TIME DISTRIBUTIONS.. 6 0 FIGURES 1 0 - 2 9 . VALUES OF L j INTERARRIVAL TIME DISTRIBUT-IONS: ARBITRARY SERVICE TIME DISTRIBUTIONS: ARBITRARY 6l-80 v i i i ACKNOWLEDGEMENT I would l i k e t o express my a p p r e c i a t i o n t o Dr. J . S w i r l e s f o r h i s a s s i s t a n c e i n guidance and r e a d i n g over t h e p r o o f s , Mr. Sam Tan ana Dr. Khien V o n g s u r i y a f o r t h e i r a i d w i t h the computer programs, and Dr. J . Couthald and the U n i v e r s i t y o f B r i t i s h Columbia's computer s t a f f f o r a l l the a s s i s t a n c e t h a t they had g i v e n me. To P r o f e s s o r Hugh W i l k i n s o n , I am e s p e c i a l l y indebted f o r h i s s u g g e s t i o n s , guidance, c o n s t r u c t i v e c r i t i c i s m , and • the thousand and one l i t t l e t h i n g s t h a t go t o making the paper what i t i s . 1. CHAPTER I INTRODUCTION Queues f o r s e r v i c e o f one k i n d or another are p r e s e n t i n many d i f f e r e n t f i e l d s o f a c t i v i t y . The mechanism o f . t h e queueing processes i s v e r y simple and can be broken down i n t o elements, each o f which has the f o l l o w i n g b a s i c behaviour. A sequence of u n i t s or customers a r r i v e s at some f a c i l i t y or counter, which s e r v i c e s each u n i t and e v e n t u a l l y d i s c h a r g e s i t . D e c i s i o n s r e g a r d i n g the amount o f s e r v i c e c a p a c i t y must be made f r e q u e n t l y i n i n d u s t r y . U n f o r t u n a t e l y , the amount o f c a p a c i t y t o p r o v i d e u s u a l l y cannot be determined by estimat-i n g the average flow. I f the mean c a p a c i t y i s l e s s than the average flow, a " t r a f f i c jam" w i l l b u i l d up u n t i l e i t h e r flow i s reduced or c a p a c i t y i s i n c r e a s e d . Even i f mean capacity-i s g r e a t e r than average flow, t r a n s i e n t and, i n c e r t a i n cases, permanent " t r a f f i c jams" can occur because the a c t u a l flow or the a c t u a l c a p a c i t y to handle the flow f l u c t u a t e s , b e i n g some-times l a r g e r and sometimes s m a l l e r than i t s mean v a l u e . Hence, c o n g e s t i o n w i l l occur, from time t o time i f t h e r e are s u f f i c i e n t i r r e g u l a r i t i e s i n the system even though average c a p a c i t y i s more than adequate. G e n e r a l l y the t h e o r y o f queues d e a l s w i t h the i n v e s t i g a t i o n of the s t o c h a s t i c law of d i f f e r e n t p rocesses a r i s i n g i n c o n n e c t i o n w i t h mass s e r v i c i n g i n cases when random f l u c t u a t i o n s o c c u r . However, the p r a c t i c a l aim i n i n v e s t i g a t i n g a system w i t h c o n g e s t i o n i s u s u a l l y t o improve the system. In the study o f queues, the important c h a r a c t e r i s t i c s a r e : . a) Wq , the mean queueing time o f a customer. The queue-i n g time i s d e f i n e d as the time a customer has t o wait b e f o r e . b e i n g attended t o . b) W , the mean w a i t i n g time of a customer. The w a i t i n g time i s d e f i n e d as the queueing time and the s e r v i c e time o f a customer. c) Lq , the mean queue l e n g t h o f the system. The mean queue l e n g t h i s d e f i n e d as the average number o f customers i n the queue e x c l u d i n g the one.in s e r v i c e . d) L , the mean w a i t i n g l e n g t h o f the system. The mean 1 w a i t i n g l e n g t h i s d e f i n e d as the average number o f c u s t -omers i n the system i n c l u d i n g the one i n s e r v i c e ; e) p , the u t i l i z a t i o n f a c t o r o f the s e r v i c e mechanism.. In order t o p r e d i c t one or more o f these c h a r a c t e r -i s t i c s , i t i s n e c e s s a r y to s p e c i f y the system s u f f i c i e n t l y f u l l y . In o r d e r t o s p e c i f y a system completely, i t i s necess-a r y t o d e s c r i b e the elements o f the queue s u f f i c i e n t l y . A system has f o u r elements: a) The i n p u t element i s d e s c r i b e d by the f o l l o w i n g f a c t o r s ; ( l ) The i n i t i a l i n p u t p o p u l a t i o n . I t i s r e q u i r e d t o know whether the customers come from a f i n i t e or i n f i n i t e p o p u l a t i o n . I t i s obvious t h a t no popu-l a t i o n can be i n f i n i t e . However, when i t i s s u f f i -c i e n t l y l a r g e , the e r r o r i n assuming i n f i n i t e popu-l a t i o n i s n e g l i g i b l e . Messrs. L.G. Peck and R.N. Hazelwood in their presentation, Finite Queue- ing Tables, for the Operations Research Society of America in 1958 presented a maximum of 250 customers in his tables for f i n i t e population. (2) The input variation. Customers can arrive singly or in batches according to a certain distribution. (3) The time factor. The input rate can be transient or stationary. "The theory of stationary queues is very important because most of the queueing processes-are ergodic; i.e., starting from any i n i t i a l state, the process tends towards equilibrium irrespective of the i n i t i a l state. In state of equilibrium the process shows only s t a t i s t i c a l fluctuation with no tendency to a certain state. Many queueing processes rapidly approach equilibrium and this explains why one can apply with success the stationary approxi-mation. However, the investigation of the transient behaviour of queueing processes i s also important, not only from the point of view of the theory but also in the applications." 1 (4) The interarrival time distribution. It is necessary to know the type of input distribution or interarrival time distribution. One is dependent on the other. The interarrival time distribution may either be arbitrary or i t may follow an algebraic expression. The most important algebraic expressions in queue-ing theory are the exponential interarrival time distribution and i t s hybrids, the Erlangian and hyper-exponential distributions. The exponential inter-a r r i v a l time distribution is obtained from the assump-tion that the arri v a l is "pure random". -The Erlangian Lajos Takacs, Introduction to the Theory of Queues (Oxford University Press, Inc., New York, 1962) p. 6. / 4. d i s t r i b u t i o n s p r o v i d e a f a m i l y o f d i s t r i b u t i o n s which ranges from the "pure random" to the completely r e g u l a r constant i n t e r a r r i v a l time, w h i l e the hyper-e x p o n e n t i a l i n t e r a r r i v a l time d i s t r i b u t i o n s p r o v i d e another f a m i l y which i s "more random" than the "pure random". F i g u r e s 3 and 4 give the cumulative graph-' i c a l r e p r e s e n t a t i o n o f the d i s t r i b u t i o n s and the frequency g r a p h i c a l r e p r e s e n t a t i o n o f the d i s t r i -b u t i o n s r e s p e c t i v e l y . "The s i m p l e s t a r r i v a l p a t t e r n , mathematic-a l l y , and the most commonly u s e f u l one i n a p p l i c a t i o n s , i s when the a r r i v a l s are completely random." 1 (5) The customers behaviour b e f o r e j o i n i n g the queue. The f o l l o w i n g are v a r i o u s p o s s i b i l i t e s : (£) Lost c a l l s . Customers may not be able t o wait and' hence the presence of any queue w i l l r e -s u l t i n l o s t c a l l s . ( i i ) B a l k i n g . A r r i v i n g customers may not j o i n the queue because o f the l e n g t h o f the e x i s t i n g queue. T h i s i s considered as b a l k i n g . ( i i i ) Delayed c a l l s . I t may be p o s s i b l e t o d e l a y the customers so t h a t when they a r r i v e t h e r e w i l l be no queue. ( i v ) Imperfect i n f o r m a t i o n about the l i n e t o be j o i n e d . In a m u l t i p l e queue o p e r a t i o n t h e r e may be i n s u f f i c i e n t or i mperfect i n f o r m a t i o n about the v a r i o u s queues and/or -channel. •'•Cox, D.R. and W.L. Smith, Queues (Methuen and Co. L t d . , London, 1965) p. 5.- . 5. (v) J o i n the nea r e s t queue. The customer a r r i v -i n g at the s e r v i c e mechanism w i l l j o i n the near e s t queue r e g a r d l e s s o f i t s r e l a t i v e l e n g t h . ( v i ) C o l l u s i o n o f customers. S e v e r a l customers may be i n c o l l u s i o n whereby one person may wait i n l i n e whi?e the r e s t are then f r e e t o atten d t o other t h i n g s . (6) The c o n t r o l o f customers before a r r i v a l s . There can be c o n t r o l or no c o n t r o l . • ,;b) The f o l l o w i n g f a c t o r s d e s c r i b e the w a i t i n g l i n e element: (1) The type o f queue d i s c i p l i n e . The queue d i s c i p l i n e can be on ( i ) F i r s t c o m e - f i r s t served. ( i i ) A l l o c a t i o n t o d e f i n i t e channels. ( i i i ) Ordered queues. ( i v ) L a s t c o m e - f i r s t served. (v) Random s e l e c t i o n f o r s e r v i c e . ( v i ) P r i o r i t i e s , preemptive or nonpreemptive. (2) The p r o p e r t i e s o f the l i n e and customers a f t e r j o i n -i n g the queue. The queue or queues can be o f ( i ) A s p e c i a l i z e d type where a p a r t i c u l a r need o f customers are met or g e n e r a l type where a l l the needs o f customers are met. ( i i ) F i n i t e or I n f i n i t e maximum l e n g t h . and the customers can be d e s c r i b e d as r e n e g i n g or non-reneging. ' c) The components t o d e s c r i b e the s e r v i c e element a r e : • 6. (1 ) The s e r v i c e time 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 can be a r b i t r a r y or f o l l o w an a l g e b r a i c e x p r e s s i o n . "For many a p p l i c a t i o n s the assumption of random a r r i v a l s i s r easonable; however, the assumption of e x p o n e n t i a l d i s t r i b u t i o n s of s e r v i c e time i s o f t e n unsatisfactory.""'" P "Nelson's study i n d i c a t e d t h a t other math-e m a t i c a l d i s t r i b u t i o n s such as the h y p e r e x p o n e n t i a l and the E r l a n g i a n d i s t r i b u t i o n s are b e t t e r d e s c r i p t -i o n s o f the a c t u a l d i s t r i b u t i o n . " (2) The s e r v i c i n g v a r i a t i o n . S e r v i c i n g can be done s i n g l y or i n batches, a c c o r d i n g t o a p r o b a b i l i t y d i s t r i b u t i o n . (3) The number o f channels. T h i s can be f i x e d or v a r y i n g . ( 4 ) The number o f phases i n the channels. ( 5 ) The type o f channels. The channels can be o f a s p e c i a l o r g e n e r a l type. d) The output element. The output can e i t h e r go o f f or c y c l e . H i s t o r i c a l B r i e f The o r i g i n o f queueing t h e o r y i s to be found i n t e l e -4 phone-network c o n g e s t i o n problems. E r l a n g i n 1917 p u b l i s h e d h i s paper where he assumes t h a t i n p u t i s P o i s s o n and s e r v i c e time d i s t r i b u t i o n e i t h e r e x p o n e n t i a l or c o n s t a n t . H i s models 1 I b i d . , p. 5 0 . p Nelson, R.T., An E m p i r i c a l Study o f A r r i v a l , S e r v i c e  Time, and W a i t i n g Time D i s t r i b u t i o n s o f a Job Shop P r o d u c t i o n  Process (Research Report No. 6, Management Sciences Research P r o j e c t , U.C.L.A., 1 9 5 9 ) . -5 •^Buffa, E.S., Models f o r P r o d u c t i o n and Operations Management (John W i l e y and Sons, Inc., New York, 1966) p. 252. 4 Brockmeyer, E., H.L. Halstrom, and A. Jensen, The  L i f e and Works of A.K. E r l a n g (Copenhagen Telephone Company, Copenhagen, 1 9 4 3 ) . -are f o r a delay system with an a r b i t r a r y number of channels f o r exponential service timo d i s t r i b u t i o n and one, two, or three channels for constant service time d i s t r i b u t i o n . Erlang's work simulated other works i n the f i e l d by T.C. F r y 1 , E.C. Molina 2, and G.F. O'Dell 5. This, as described by S y s k i 4 , i s the " E r l a n g - 0 ' D e l l " period, and the works were b a s i c a l l y concerned v/ith proving or disproving Erlang's r e s u l t s . In the early t h i r t i e s , F. Pollaczek developed his v/ell known formula f o r a single channel with Poisson input and a r b i -t r a r y holding time. This formula i s known as the Pollaczek-Khlntchine formula. From then on numerous works had been done on queue-ing theory. T.L. Saaty^ provided a bibliography of about 9 0 0 a r t i c l e s i n 1 9 6 1 . This quantity has grown tremendously from then. Queues of a l l types are being studied, but i t i s re-gretted that most of the r e s u l t s are extremely complex and hence do not provide a very good base f o r business applications: Fry, T.C, The Theory of P r o b a b i l i t y as Applied to  Problems of Congestion, i n "Probability and Its Engineering  Uses" (D. Van Nostrand, Inc., Princeton, N.J., 192ti). p Molina, E.C, Application of the Theory of Probabi- l i t i e s Applied to Telephone Trunking Problems ( B e l l System Tech. Journal, v o l . b, 1 9 2 7 J p. 4 5 1 - 4 9 4 . : •^O'Dell, G.F., T h e o r i t i c a l P r i n c i p l e s of the T r a f f i c  Capacity of Automatic Switches (P.O. Elec. Congrs. J., v o l . 1 3 , 1 9 2 0 J p. 2 0 9 - 2 2 3 . 4 Syski, R., The Theory of Congestion i n L o s t - c a l l  Systems (A.T.E. J., v o l . 9 , 1 9 5 3 ) p. 1 8 2 - 2 1 5 . c; -^Saaty, T.L., Elements of Queueing Theory with Appli- cations (McGra\'/-HilI Book Co., Inc., New York, 1961). "We did not cover any of the multiple-phase situations involving a r r i v a l and/or service d i s t r i b u t i o n s other than Ppisson, and queue d i s c i p l i n e s other than f i r s t come-first served. It i s f e l t that the more complex structures are best handled through the technique of computer simulation. 1 , 1 It i s the objective of t h i s thesis to provide a set of graphs to enable the businessman to predict the necessary c h a r a c t e r i s t i c s of a single channel queue with a r b i t r a r y i n t e r -a r r i v a l time and service time d i s t r i b u t i o n s , and with custom-ers a r r i v i n g from an i n f i n i t e population. It i s assumed that the queue and service f a c i l i t y are i n s t a t i s t i c a l equilibrium. . Buffa, E.S., Models for Production and Operations  Management (John Wiley and Sons, Inc., New York, 1966) p.266. CHAPTER 2 DERIVATIONS OP DISTRIBUTIONS AND THEIR PROPERTIES D e r i v a t i o n s of D i s t r i b u t i o n s This s e c t i o n s h a l l d e a l w i t h the d e r i v a t i o n of the Poisson and expo n e n t i a l d i s t r i b u t i o n s from random a r r i v a l s and thence t o d e r i v e the E r l a n g i a n and hyperexponential d i s t r i b u t -ions v i a s i m u l a t i o n i n which only the ex p o n e n t i a l element i s used. I t i s necessary to s p e c i f y the mechanism by which the E r l a n g i a n and hyperexponential d i s t r i b u t i o n s are obtained so th a t we may be able t o o b t a i n the c h a r a c t e r i s t i c s of queues w i t h these d i s t r i b u t i o n s f o r i n t e r a r r i v a l time and s e r v i c e time. Figure 1 shows the u t i l i z a t i o n of the expo n e n t i a l d i s t r i b u t i o n t o simulate the E r l a n g i a n d i s t r i b u t i o n while Figure 2 i s used t o simulate the hyperexponential d i s t r i b u t i o n . Appendix I I shows how t h i s p r o p e r t y i s u t i l i z e d t o simulate the queues w i t h Erlangian/hyperexponential i n t e r a r r i v a l time d i s t r i b u t i o n and Erlangian/hyperexponential s e r v i c e time d i s t r i b u t i o n . P oisson and Expon e n t i a l D i s t r i b u t i o n In the study of queues, the Poisson and i t s counter-p a r t , the e x p o n e n t i a l , d i s t r i b u t i o n s are of prime importance. This i s due t o the p r o p e r t i e s of these d i s t r i b u t i o n s . I t i s only the expo n e n t i a l f a c i l i t i e s and Poisson a r r i v a l s that give r i s e to simple l i n e a r equations f o r d e t a i l e d balance of t r a n s i -t i o n s between s t a t e s , independent of time. Assuming that the a r r i v a l s are random v/ith an average 10. r a t e o f X from an i n f i n i t e p o p u l a t i o n . In a s h o r t I n t e r v a l o f time &t , P r o b a b i l i t y o f 1 a r r i v a l , ?^ = P r o b a b i l i t y o f 0 a r r i v a l , P Q = 1-X-At-O.At 'where O.^t i s such t h a t i i m . ^ J = 0 &t-0 A P r o b a b i l i t y o f n a r r i v a l s , P n = O.At _ 0 as At _ 0 In the p e r i o d t , where t = m.At , By the use o f the b i n o m i a l p r o b a b i l i t y law, P . mi ' ( x . A t ) r ( l - X . A t ) m ~ r  r r ( m - r ) i C o n s i d e r ^ t _ 0 and m = t / ^ t , m— co . T h e r e f o r e ? r • 1 1 " ^ ( r " ' ^ ' = [ ( ^ r ] l i m - ^ l i m ( l - ^ ) m l i m ( l - ^ ) " r ' m_oom (m-r)i nu*> m m - » m A l s o l i m 5LL_ = l i m m.(m-l).(m-2)....(m-r+l) m-oo m r(m-r)I m - c o m * m * m m l l m B.1±m I S i l i . i i m lH£2l l i M (m-r+1)-w, „ m ™ _ m „, „ m m = 1 l i m . ( l - - ^ ) r = l " r = 1 m -Xt l i m . ( l - ^ ) = lim.(l-Fy) y where y = ^ m_oo , y_o m 11. ^ - x t = [ l i m . ( l + y ) y ] y-o 1 But l i m . ( l + y ) y i s the d e f i n i t i o n o f e y-0 T h e r e f o r e l i r a . ( l + ^ ) = e " X t x m m_ oo T h e r e f o r e P = ^^1* e~ x t r rl T h i s i s the POISSON d i s t r i b u t i o n . P - -Xt _ -Xt T h i s i s the p r o b a b i l i t y t h a t i n t e r a r r i v a l time i s g r e a t e r than T h i s i s so because the p r o b a b i l i t y o f no a r r i v a l s i n time, t i s the p r o b a b i l i t y t h a t i n t e r a r r i v a l time i s g r e a t e r than t . 'Hence P Q = j a ( t ) . d t where a ( t ) i s the i n t e r a r r i v a l time t d i s t r i b u t i o n . a ( t ) . d t - f a ( t ) d t 0 J 0 But i O 0 J a ( t ) . d t = 1 as t h i s i s the t o t a l p r o b a b i l i t y . 0 T h e r e f o r e Hence a ( t ) = d ^ 1 ^ " X t ) Th e r e f o r e a ( t ) = X e " X t The p r o b a b i l i t y t h a t i n t e r t ' - r r i v a l time i s g r e a t e r than t , A ( t ) = f°°a(t)dt =.1\ = e " X t t a ( t ) i s the EXPONENTIAL d i s t r i b u t i o n and A ( t ) i s the cumulative EXPONENTIAL d i s t r i b u t i o n . E r l a n g i a n D i s t r i b u t i o n With a chosen s e t of e x p o n e n t i a l phases and a p p r o p r i -ate r u l e s f o r t r a n s i t i o n as re p r e s e n t e d by F i g u r e 1, the E r l a n g i a n d i s t r i b u t i o n can be si m u l a t e d . u n i t s e n t e r exp. u n i t s depart phase k phase k-1 phase k-2 phase 1 F i g u r e 1: E r l a n g i a n d i s t r i b u t i o n f a c i l i t y o f k phases. R u l e s : a) There are k phases, w i t h each phase h a v i n g an e x p o n e n t i a l d i s t r i b u t i o n o f r a t e ky . b) No u n i t s are t o be i n t r o d u c e d i n t o the f a c i l i t y u n t i l the p r e v i o u s u n i t has completed a l l k phases, P r o b a b i l i t y t h a t u n i t passes through a l l k phases between time t and t+dt = P r o b a b i l i t y o f u n i t i n phase one s t a r t i n g i n time x^ and f i n i s h i n g i n time t . x P r o b a b i l i t y o f u n i t i n phase two s t a r t i n g i n time x^ and f i n i s h i n g i n time x 2 13. x P r o b a b i l i t y o f u n i t i n phase k s t a r t i n g i n time 0 and f i n i s h i n g i n time x, K. t - g t - x ) x 2 -k\/x -x ) / k - 1 ( X f c . - V * * . ) = [J Qkae ^ d x 2 j k^e. d • ? &Xy\ He d: By s u c c e s s i v e i n t e g r a t i o n , s ( t ) = ( k i a t ) k - 1 [ e - k W t / ( k - l ) i ] k w T h i s i s the ERLANGIAN d i s t r i b u t i o n . The cumulative ERLANGIAN d i s t r i b u t i o n S ( t ) = P* s ( t ) . d t J t = .e'W Y ( k u t ) n / n l • n=0 • The f o l l o w i n g s i t u a t i o n generates an E r l a n g i a n a r r i -v a l r a t e . Consider a depot w i t h a constant queue o f customers e n t e r i n g one at a time i n t o a c o n s t a n t l y busy mechanism o f ^ phases, each w i t h an e x p o n e n t i a l time d i s t r i b u t i o n and mean s e r v i c e r a t e i\ from the mechanism a f t e r completing a l l i phases and j o i n another queue. No customer e n t e r s the f i r s t phase u n t i l the p r e v i o u s customer has completed a l l i phases. Customers l e a v i n g t h i s mechanism j o i n a second queue. The i n t e r a r r i v a l time d i s t r i b u t i o n f o r t h i s second s e r v i c e f a c i l i t y i s an i phase E r l a n g i a n . With the same form of a n a l y s i s used p r e v i o u s l y f o r the s e r v i c e time d i s t r i b u t i o n , the i n t e r a r r i v a l time d i s t r i -14. b u t i o n f u n c t i o n , a ( t ) , can be determined: a ( t ) = U x t ) ^ 1 [ e - - t X t / ( t - l ) . ' 3 t X hence the cumulative i n t e r a r r i v a l time d i s t r i b u t i o n , A ( t ) = e"tXt s U x t ) n / n J n=0 Hyp e r e x p o n e n t i a l D i s t r i b u t i o n The h y p e r e x p o n e n t i a l d i s t r i b u t i o n can be si m u l a t e d w i t h e x p o n e n t i a l d i s t r i b u t i o n by assuming t h a t the s e r v i c e channel i s made up o f two independent branches, one o f r a t e 2au 3 the o t h e r o f r a t e 2(1-CT)J. (where 0 _< rj < -|) • When a u n i t e n t e r s f o r s e r v i c e , i t i s as s i g n e d t o one o f the two branches at random, the choice going to the 2QJJI branch, on the average, a o f the time, and going t o the 2(1-oOp, branch, (1-a-) o f the times, on the average. When i t e n t e r s one o f the branches no u n i t s are t o be i n t r o d u c e d i n t o the f a c i l i t y u n t i l the p r e v i o u s u n i t has completed i t s s e r v i c e . F i g u r e 2 g i v e s a d i a g r a m a t i c r e p r e s e n t a t i o n o f the f a c i l i t y . -u n i t s >. depart F i g u r e 2: Hyperexponential d i s t r i b u t i o n f a c i l i t y . The p r o b a b i l i t y t h a t a u n i t w i l l complete s e r v i c e i n time between t and t+At Queue ' • - 15. = o ( P r o b a b i l i t y t h a t a unit' w i l l complete s e r v i c e i n time between t and t+^t i n the 2<j\x branch) + ( 1 - o ) ( P r o b a b i l i t y t h a t a u n i t w i l l complete s e r v i c e i n time between t and t+At i n the 2(l-cr)|j. b r a n c h ) . T h e r e f o r e s ( t ) . d t = (a)(2 a u)e"^ 2M- a^ t.dt , + ( l - a ) [ 2 ( l - a ) | a ] e - [ 2 ( 1 - a ) ^ ] t . d t = [2a2ue~2^t + 2 ( l - a ) 2 | J L e - 2 ( 1 - o ) M t ] d t Here j p l a y s the same r o l e as i n t e g e r s k and g p l a y w i t h the E r l a n g i a n d i s t r i b u t i o n . The q u a n t i t y measures the depart-ure from pure random. ^Therefore 0 = \ - " 2(1+3) ^  s i n c e -° .<'o < "| S ( t ) = P* s ( t ) d t J t = a e - 2 0 ^ + ( l - a ) e " 2 ( 1 _ C T ^ t A depot w i t h a constant queue o f customers e n t e r i n g one at a time i n t o a mechanism, w i t h two branches, one w i t h an e x p o n e n t i a l time d i s t r i b u t i o n o f r a t e 2<j\ and the other w i t h an e x p o n e n t i a l d i s t r i b u t i o n o f r a t e 2 ( l - a ) \ . When a u n i t e n t e r s a mechanism i t i s a s s i g n e d to one o f the two branches a t random, the choice going to the 2oX branch, on the average a o f the time, and to the 2 ( l - a ) x branch ( l - a ) o f the time. 16. Only one u n i t may be i n the system at a time. The u n i t l e a v e s the mechanism t o j o i n a second queue. The i n t e r a r r i v a l time d i s t r i b u t i o n f o r t h i s second queue i s h y p e r - e x p o n e n t i a l . With the same form o f a n a l y s i s as f o r the s e r v i c e time d i s t r i b u t i o n above, the i n t e r a r r i v a l time d i s t r i b u t i o n , a ( t ) , may be determined: a ( t ) = [ 2 a 2 x e - " 2 ^ t + ^ l - a ) ^ - ^ 1 - ^ 2 ^ ] and the cumulative i n t e r a r r i v a l time d i s t r i b u t i o n , A ( t ) = a e " 2 " 1 * + ( l - a ) e - 2 ( 1 - a h t where a = \ - jl^ - 2(1+3) ^  s i n c e 0 < 0 <. \ P r o p e r t i e s o f D i s t r i b u t i o n s S i n c e the f r a c t i o n a l c o e f f i c i e n t s o f v a r i a n c e squared — o f — i n t e r a r r i v a l - t i m e - d i s t r i b u t i o n and s e r v i c e time d i s t r i b u t i o n w i l l be- o f prime importance i n the c o n s t r u c t i o n of the graphs, t h i s p r o p e r t y w i l l be d e r i v e d f o r the e x p o n e n t i a l , E r l a n g i a n , ..and h y p e r e x p o n e n t i a l d i s t r i b u t i o n s . . . L e t f ( t ) be the continuous p r o b a b i l i t y d i s t r i b u t i o n of time t a be the average r a t e . A v ( t ) be the average time. V a r ( t ) be the v a r i a n c e o f d i s t r i b u t i o n f ( t ) . 2 C be the f r a c t i o n a l c o e f f i c i e n t o f v a r i a n c e squared. A v ( t ) = f t . f ( t ) . d t V a r ( t ) = r°° [ t - A v ( t ) ] 2 f ( t ) . dt = j"* t 2 f ( t ) d t - 2 A v(t) f e°tf(t).dt + [ A v ( t ) ] 2 r°°f(t).dt °o  Jo Jo = f 0 ° t 2 f ( t ) d t - 2 [ A v ( t ) ] 2 + [ A v ( t ) ] !  0 = f * t 2 f ( t ) d t - [ A v ( t ) ] 2 Jo C 2 = V a r ( t ) / [ A v ( t ) ] 2 E x p o n e n t i a l D i s t r i b u t i o n f ( t ) = d e _ a t = r V a t d t — i [ e " a t ] Jo l a dt P - a t - a t ' CO - CO 04- ~ T <-. 2 J % e - * « . ( i j 2 . | ( i ) . ( I ) ; : 18, , 2 C 2 - - 1 2 - = ! ( ~ ) 2 E r l a n g i a n D i s t r i b u t i o n s f ( t ) = . ( e ^ t ) ^ 1 [ e - ^ t / ( / - l ) i ] ^ a A v ( t ) = r t ( 4 a t ) * - 1 [ e - * a t / ( z , - l ) i ] £ a . d t 0 = f ( t a t ) * [ e * a t / U - l ) J ] d t By s u c c e s s i v e i n t e g r a t i o n by p a r t s , r t v ^ a t . d t = - ^ - + 1 T h e r e f o r e Av(t) = - = ^_ 4+1 £ a 1 a V a r ( t ) = f * t 2 ( ^ a t ) ^ - 1 [ e - ^ t / ( ^ - l ) i ] ^ a . d t - (.£)' 2 = r t ( t a t ) « [ e - * a t / ( z - l ) l ] a t - (§) " ( , a ) 2 ( a ) - ^ ( i ) 2 " ( i ) 2 - ^ - l ) H y p e r e x p o n e n t i a l D i s t r i b u t i o n f ( t ) = 2 C T 2 a e - 2 a ^ + 2(1-a)2™2^1-^ A v ( t ) = f t [ 2 a 2 a e - 2 a ^ + 2 ( l - a ) 2 a e - 2 a ( 1 ^ ) t ] d t 0 = f t [ 2 C r 2 a e - 2 ^ t ] d t + f t t ^ l - a ) ^ " ^ 1 " I n t e g r a t i n g by p a r t s A v ( t ) = . 0 ( 1 ) + ( I - a ) ( | ) . - < £ > V a r ( t ) = p t 2 [ 2 a 2 a e - 2 a ^ + ^ l - o ^ a e " 2 ^ 1 " ^ ] d t J 0 0 J 0 ( 2 a g ) 2 [ 2 ( l - f f ) a ] 2 ' a i _ . [ 2q + 2(1-pr) 1 , i a 2 4 a 2 4 ( l - a ) 2 1 r 2 - 2 g + 2 a - 4 a + 4 r T 2 1 1_ r 2 " 4 ^ ? 2 ! a 2 L 4 a ( i - 0 y J 1 r 1- 2rr+2fT2 -j a 2 L 2 a ( l - C T j 1 hi a 3 - [^Sff] - a - | - - s i n c e 0 21. E x p o n e n t i a l E r l a n g i a n A v ( t ) V a r ( t ) C 2 Comments 1 1_ a • 2 a 1 1- 1 a 2 » H y p e r e x p o n e n t i a l — J L . I ' _ 1 _ / f l __, „ Table I . P r o p e r t i e s of D i s t r i b u t i o n s ji;:|+itl-i±i±i: • CHAPTER 3 • EXPONENTIAL INTERARRIVAL TIME DISTRIBUTION: EXPONENTIAL SERVICE TIME DISTRIBUTON Erlang's Model The model i s for steady state i n which the customers come from an i n f i n i t e population with an exponential i n t e r -a r r i v a l time d i s t r i b u t i o n of rate \ ' to a service channel where the service time d i s t r i b u t i o n i s exponential with mean rate ^ . The u t i l i z a t i o n factor p = < 1 . Pn(t+A.t) : P r o b a b i l i t y of n customers In the waiting l i n e i n the system' at the time t+At . P (t) : P r o b a b i l i t y of n customers i n the waiting l i n e in the systemat the time t fThe .state p r o b a b i l i t i e s equations are: I n i t i a l equation: P 0 ( t + A t ) = P 0 ( t ) [ l - u ] + P-^tJuAt 1 Queue equation: P n(t+At) Pn(t)[l-(x+^)At] + P^ ^ t J u t + P n + 1 ( t ) l i A t 2 Therefore ^ 0 dt = - \P 0 + VL\ and dP. = - U+n)Pn + \P. + uP. n n - l n+l dt Since the system i s i n steady state: 0 = 0 a n d -J± = 0 dt ~ " dt Hence - \ P Q + = 0 T h e r e f o r e P , = A P A 1 ix 0 .'" Ppo A l s o -(x+n)P n + XP n_ x + U P n + I = < D i v i d i n g by p. , • - ( p + l ) P n + P P n _ x + P n + 1 = 0 Let n = 1 , P 2 = ( P + l ^ - PP Q But from (3) , P-j^  = PP Q T h e r e f o r e P 2 = P 2 P Q L e t n = 2 , . P 5 = ( p + l ) P 2 - PP X But from ( 3 ) , P 1 = PP Q and'(4), T h e r e f o r e P ^ = P ^ P Q L e t n = 3 , P4 = ( p + l ) P 5 - pP 2 But from (4), ? 2 = p 2 P Q and ( 5 ) 4 T h e r e f o r e P^ = P P Q By d e d u c t i o n P = p N P ~ n 0 2 6 . But P n - + P., + P - + . . . + P + . . . + P-o = 1 0 1 2 n w i . e . % P =. 1 or s P n P n = 1 n=0 n n=0. u T h e r e f o r e P = ? c.n = TTZ-T . . n=0 ^ P J T h e r e f o r e P n = ( l - p ) .....8 L = ? n P n = J „ ( i - p ) p n « i l i P l £ n=0 n n=0 ((1-Pf) P T h e r e f o r e L = -r-^ r \ i - P ; L q = S ( n - l ) P n •- I n P n - S P(„) - ^ y - P -H n=l n=l n=l v ' P 2 ( l - P l ....10 The main l i m i t a t i o n o f E r l a n g ' s model f o r steady s t a t e s i n g l e channel a n a l y s i s i s i t s r e s t r i c t i o n t o exponent-i a l i n t e r a r r i v a l time d i s t r i b u t i o n and e x p o n e n t i a l s e r v i c e time d i s t r i b u t i o n . 27. CHAPTER 4 . EXPONENTIAL INTERARRIVAL TIME DISTRIBUTION: • ARBITRARY SERVICE TIME DISTRIBUTION The Po.llacze'K <- Khintchlne Formula The Pollaczek - Khintchlne formula also deals with steady'state single channel with exponential i n t e r a r r i v a l time d i s t r i b u t i o n . However, i t goes one step further by consider-ing service time d i s t r i b u t i o n as a r b i t r a r y . A r r i v a l s occur at random by a Poisson process with the rate \ per unit time,' to a waiting l i n e , i n s t a t i s t i c a l equilibrium before a single service f a c i l i t y . The a r r i v a l s are to be served by an a r b i t r a r y service time d i s t r i b u t i o n f a c i l i t y at the rate jy, per unit time f i r s t come-first served basis. The u t i l i z a t i o n factor, Suppose that the departing customer leaves q i n l i n e , including the one i n service, whose service time i s t . Let r customers arrive during t h i s time t . I f the next departing, customer leaves q' customers behind, then q' = q - l + r + j where [0 i f q > 0 6 {l i f q = 0 Therefore E(q') = E(q) - E ( l ) + E(r) + E(s) But since i t i s i n steady state or s t a t i s t i c a l equilibrium, E(q') = E(q) Therefore E(s) = E ( l ) - E(r) = 1 - E(r) But during the service time of length t 28. we have E ( r ) = (\t) r" E ( r2 ) = s r 2 - ^ ] - e"** = [ E ( r ) ] 2 + E ( r ) = ( X t ) 2 + ( X t ) r=0 Le t s ( t ) be the s e r v i c e time d i s t r i b u t i o n w i t h mean — E ( r ) = g x t s ( t ) = X S t s ( t ) t=0 t=0 But j t s ( t ) i s the mean and i s equal t o — t=0 V-T h e r e f o r e E ( r ) = = p A l s o E ( r 2 ) = | [xt) 2 + (xt)]s ( t ) t=0 X 2 S t 2 s ( t ) + X S t s ( t ) t=0 t=0 2 X 2 ? ( t - i ) s ( t ) + - £ t s ( t ) - ? s ( t ) + p t=o . u t=0 ^ t=0 X 2 V a r ( t ) + 2 p 2 - p 2 + p = \ 2 V a r ( t ) + P 2 + p ' Squaring both s i d e s o f the equ a t i o n q' = q - l + r + 5 q ' 2 = ( q - l + r+ s ) 2 But s i n c e & = 1 when q = 0 and 6 = 0 when q _> 1 p T h e r e f o r e 6 = 6 and q ( l - f i ) = q Hence q ' 2 = q 2 - 2 q ( l - r ) + (r - 1 . ) ' 2 +. fi(2r-l) But since i t i s s t a t i s t i c a l equilibrium, E(q' 2) = E(q 2) . Hence E(q' 2) - E(q 2) = 2E(.q)E(r-l) + E[ ( r - l ) 2 ] +' E( S)E( 2 r - l ) = 0 i . e . E ( q ) = M i - i g ^ M 2 - i l - E(r^)-2E(r)+E(l)+E( g) [ 2 E ( r ) - E ( l ) 3 2 L E ( l ) - E ( r J J X 2Var(t)+P 2 + P-2P+l+(l-p)(2P-l) • 211=7] - . n 2 , X 2Var(t) . - P + p + 2(l-p) But the f r a c t i o n a l c o e f f i c i e n t of variance squared of service time, c 2 = Var(t) s -• 2 p 2(l-fC 2) Therefore L = E(q) = p + — 2 ( l - P ) P 2 ( I + C 2 ) Lq = L - p = 2 ( l _ p ) 2 Therefore L and Lq are functions of p and C_ only. Since L i s a function of the u t i l i z a t i o n f actor, p ,. and the square of the c o e f f i c i e n t of variance of service rate 2 d i s t r i b u t i o n , C , only i t i s necessary only to vary p 2 and C g to compute the d i f f e r e n t values of L i n the equation P 2(l+C2) L = P + -2(Trpf-Table II was obtained'by substituting values of p from 0.1 to 0 . 9 and C from 0 to 6 in the above equation, s 2 Using C as the ordinate and L as the abscissa £ and varying P , Figure 5 was constructed. Knowing that the interarrival time distribution is exponential, i.e. arrivals are random, and also the coefficient • p of variance squared of service time, C g , and the u t i l i z a t i o n factor, p , the average length of the system, L , can be read from the verticle axis. The values of the average queue length, Lq-, the average waiting time in the system, W , and the average queue time, Wq , can be easily computed once L is known,:as they are a l l functions of L , p and the average arrival .rate,. X , only. The relationships are listed below. 1 Lq = (L-p) Values of L with given P for exponential interarrival time distribution and arbitrary service time distribution with a 2 fractional coefficient of variance squared, C_ 0 4 . 0.05 o.io 0.15 0.20 0.25 0.30 0.35 0 .40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 O.85 O.90 0.95 O.051316 0.105556 0.163235 0.225000 0.291667 0.364286 0.438846 0.533333 0.634091 0.750000 0.886111 1.050000 1.253571 1.516667 1.875000 2.400000 3.258333 4.950000' 9.975000 0.052632 0.111111 0.176471 0.250000 0.333333 0.428572 0.527692 O.666667 O.818182 1.000000 1.222222 1.500000 1.857142 2.333333 3.000000 4.000000 5.666667 9.000000 19.00000 0.053948 0.116667 O.189706 0.275000 0.375000 O.492858 0.616538 0.800000 1.002273 1.250000 1.558333 1.950000 2.560713 3.150000 4.125000 5.600000 8.075000 13.05000 28.02500 0.055264 0.122222 0.202941 0.300000 0.416667 0.557144 0.705384 0.933333 1.186364 1.500000 1.894444 2.400000 3.064284 3.966667 5.250000 7.200000 10.48333 17.10000 37.05000 O.056580 0.127777 0.216176 0.325000 0.458333 0.621430 O.794230 1.066667 1.370455 1.750000 2.230555 2.850000 3.667855 4.783333 6.375000 8.800000 12.89166 21.15000 46.07500 O .057896 0.13J353 0.229411 0.350000 0.500000 0.685716 O .883076 1.200000 1.554546 2.000000 2.566666 3.300000 4.271426 5.600000 7.500000 10.40000 15.30000 25.20000 55.10000 O.O592I2 0.133888 0.242646 0.375000 0.541667 0.750000 0.971922 1.333333 1.738637 2.250000 2.902777 3.750000 4.874997 6.416667 8.625000 12.00000 17.70833 29.25000 64.12500 Table I I Values o f L f o r E x p o n e n t i a l I n t e r a r r i v a l Time D i s t r i b u t i o n and A r b i t r a r y S e r v i c e Time D i s t r i b u t i o n H 32. K E U F F E L & E S S E R C O . CHAPTER 5 •ARBITRARY INTERARRIVAL TIME DISTRIBUTION: EXPONENTIAL SERVICE TIME DISTRIBUTION R e l a t i o n s h i p -between L and p and C The P o l l a c z e k - K h i n t c h i n e formula p r o v i d e s a r e l a t i o n -p s h i p between L , p and C f o r the e x p o n e n t i a l i n t e r a r r i v a l time d i s t r i b u t i o n and a r b i t r a r y s e r v i c e time d i s t r i b u t i o n . The f o l l o w i n g a n a l y s i s w i l l attempt to shoi* t h a t , f o r the a r b i t r a r y i n t e r a r r i v a l time d i s t r i b u t i o n and e x p o n e n t i a l time d i s t r i b u t i o n , 2 L i s a; f u n c t i o n o f P and C~" o n l y . c l The i n t e r a r r i v a l time d i s t r i b u t i o n , a ( t ) , i s a r b i t r a r y w i t h a mean r a t e of \ per u n i t , t o a w a i t i n g l i n e , i n s t a t -i s t i c a l e q u i l i b r i u m before a s i n g l e f a c i l i t y . The a r r i v a l s are then served by an e x p o n e n t i a l s e r v i c e time d i s t r i b u t i o n channel at the r a t e o f p. per u n i t time f i r s t c o m e - f i r s t served d i s -c i p l i n e . The u t i l i z a t i o n f a c t o r , p = < 1 . Suppose t h a t an a r r i v i n g customer f i n d s q i n l i n e , i n c l u d i n g the one i n s e r v i c e and h i m s e l f . The time b e f o r e the a r r i v a l o f the next customer i s t . L e t r customers depart d u r i n g t h i s time t . I f the next a r r i v i n g customer f i n d s q' customers, i n c l u d i n g the one i n s e r v i c e and h i m s e l f , t h e n q' = q + l . - r + 6 where 0 i f q > r 6 = r - q i f q < r E(q') = E(q) + E ( l ) - E ( r ) + E( 6 ) But s i n c e i t i s i n s t a t i s t i c a l e q u i l i b r i u m or steady s t a t e , 35. E(q') = E(q) T h e r e f o r e E(h) = 1 - E(r) But d u r i n g an a r r i v a l time .of l e n g t h t 3 we have E ( r ) = £? ^ ^ r ^ - e ~ u t = ut r=0 r * where A M > / r , i s the Pois.son d i s t r i b u t i o n , T h e r e f o r e E ( r ) = ^ t E(r 2) = ? r 2 1 ^ e"^ = ( w t ) 2 + (^t) r=0 r * L e t a ( t ) be the i n t e r a r r i v a l time d i s t r i b u t i o n v/ith mean i . e . f t a ( t ) d t = i E(r) = f° n.ta(t)dt = n f* t a ( t ) d t X P A l s o -E(r*) = p [ ( p : t ) 2 + ( ^ t ) ] a ( t ) d t Jo = u 2 f * t 2 a ( t ) d t + n t a ( t ) d t J 0 J 0 u 2 ( t - i ) 2 a ( t ) d t + n 2 J * | t a ( t . ) d t -J o X *• H 2 V a r ( t ) + - 2 V - 4 + ¥ X X^ x 36. p 2 V a r ( t ) + K + | S q u a r i n g q' = q - l + r + $ on "both s i d e s . q ' 2 = ( q - l + r + 6 ) 2 q ' 2 = q 2 + 1 + r 2 + 6 2 - 2q + 2rq + 2&q - 2r - 2s + 2re Hence, E ( q ' 2 ) = E ( q 2 ) + E ( l ) + E ( r 2 ) + E ( 6 ) 2 - 2E(q) + 2 E ( r ) E ( q ) + • • + 2E (6)E(q) - 2E(r) - 2 E ( 6 ) + 2 E ( r ) E ( s ) But s i n c e i t i s i n s t a t i s t i c a l e q u i l i b r i u m , E ( q ' 2 ) = E ( q 2 ) T h e r e f o r e 0 = 1 + E ( r 2 ) + E ( f i 2 ) - 2E(q) + 2 E ( r ) E ( q ) + 2 E ( 6 ) E ( q ) - 2E(8) + 2 E ( r ) E ( 6 ) . T h e r e f o r e E(q) = 1 + E ( r 2 ) + E ( ^ ) - 2 E ( ^ r ^ -2E^)+2E( r )E( &) T h e r e f o r e L = E(q) i s a f u n c t i o n o f E ( r 2 ) , E ( 6 2 ) 3 E ( r ) and E ( 5 ) o n l y . A l s o E ( r ) = p , ; E(*) = 1 - E ( r ) = 1 - P j .. E ( r 2 ) = w 2 V a r ( t ) + 1 + i P 2 P o p T h e r e f o r e L i s a f u n c t i o n o f p and E( & ) o n l y . S i n c e & = 0 i f q > r and 6 = r - q i f q < r T h e r e f o r e 6 2 = 0 i f q > r 57. . 6 2 = ( r - q ) 2 i f q < r 2 2 2 Hence s i s a f u n c t i o n o f r , q , r and q o n l y , or . E ( 6 2 ) i s a f u n c t i o n o f E ( r 2 ) , E ( q 2 ) , E ( r ) and E(q) o n l y . But (q'-q) '= (1-r-fs) Squaring "both s i d e s , q ' 2 - 2qq' + q 2 = 1 + r 2 + s 2 - 2r + 2& - 2rs P T h e r e f o r e E ( q ' ). i s a f u n c t i o n o f E ( r ) , E(6) } E(q) and E ( r 2 ) o n l y . S i n c e E ( q ' 2 ) -= E ( q 2 ) and: E(q') = E(q) T h e r e f o r e E ( a 2 ) i s a f u n c t i o n o f E ( r ) , E ( s ) , E(q) and E ( r 2 ) o n l y . P p T h e r e f o r e E(6 ) i s a f u n c t i o n of E ( r ) , E(q) and E ( r ) o n l y as. E ( s ) = 1 - E ( r ) , and s i n c e E ( r ) , E(q) and E ( r 2 ) o „ are f u n c t i o n s o f p and C o n l y , a p T h e r e f o r e L i s a f u n c t i o n o f P and C Q o n l y . I t i s u n f o r t u n a t e t h a t no" simple e q u a t i o n can be o b t a i n e d d i r e c t l y , but a s e t of t a b l e s or graphs can be'set up through two b a s i c approaches: ( i ) Morse's method: S i m u l a t i o n of n o n - e x p o n e n t i a l d i s t r i b u t i o n s , E r l a n g i a n and h y p e r e x p o n e n t i a l d i s t r i b u t i o n s , v i a the use o f e x p o n e n t i a l phases and branches. ( i i ) Monte C a r l o or General Purpose S i m u l a t i o n System I I I t o o b t a i n v a r i o u s L w i t h d i f f e r e n t p's and C 2 « s 5 8 . A n a l y t i c a l Approaches f o r E x p o n e n t i a l S e r v i c e Time D i s t r i b u t i o n  and E r l a n g i a n or Hyperexponential I n t e r a r r i v a l Time D i s t r i b u t i o n s As p r e v i o u s l y mentioned,.there are two main t e c h -niques v i a which a set o f t a b l e s or graphs can be set up to enable one t o o b t a i n the c h a r a c t e r i s t i c s o f a queue, g i v e n t h a t the s e r v i c e time d i s t r i b u t i o n i s e x p o n e n t i a l , an u t i l i z a t i o n f a c t o r o f P , and an i n t e r a r r i v a l time d i s t r i b u t i o n which i s a r b i t r a r y and having a f r a c t i o n a l c o e f f i c i e n t o f v a r i a n c e 2 squared, c a • As P.M. Morse p r o v i d e d an a n a l y t i c a l approach, I s h a l l use the former technique f o r the f o l l o w i n g reasons: ( I ) The v a l u e s obtained are d e f i n i t e and not an approximate. ( i i ) Much l e s s computer time i s needed f o r computation. E r l a n g i a n I n t e r a r r i v a l Time D i s t r i b u t i o n U n i t s , from a depot w i t h an i n f i n i t e p o p u l a t i o n , passes through I phases t o j o i n a queue f o r s e r v i c e . The s e r v i c e time d i s t r i b u t i o n i s e x p o n e n t i a l . No u n i t s are t o enter the phases u n t i l the p r e v i o u s one has completed a l l •& phases. The u n i t s s t a y i n each phase f o r an i n t e r v a l o f time. T h i s i n t e r v a l o f time i s e x p o n e n t i a l l y d i s t r i b u t e d . The average r a t e f o r each phase i s l\. . The average r a t e f o r s e r v i c e i s w . As had p r e v i o u s l y been shown, the r e s u l t o f u s i n g a channel w i t h t phases, each w i t h time i n t e r v a l e x p o n e n t i a l l y d i s t r i b u t e d and average r a t e i\ , i s an E r l a n g i a n i n t e r a r r i v a l time d i s t r i b u t i o n . L e t P ( t ) be the p r o b a b i l i t y o f the s t a t e where a u n i t i s i n phase s and t h e r e are n u n i t s i n ' t h e queue and s e r v i c e f a c i l i t y at the time t . Since a t l e a s t one u n i t must be i n the channel, 1 _< s < I . In the time i n t e r v a l t t o (t+At) where &t - 0 ; I n i t i a l e q u a t i o n s ; P ^ 0 ( t + A t ) s [ l - U x ) 4 t ) ] P ^ 0 ( t ) + n A t P - ^ t ) I PS 0(.t+Mt) = " [ l - ( 4 \ ) A t ] P s ^ 0 ( t ) + . | j A t P s ^ 1 ( t ) + + ^ P s . - l . 0 ( t ) 2 Queue eq u a t i o n s ; P l j n ( t + A t ) = [ l - ( t \ + W ) A t ] P l j n ( t ) + 4 X A t P ^ n _ 1 ( t ) + + M A t P ^ n + 1 ( t ) 3 P S j n ( t + A t ) = [ l - ( * X + n ) A t ] P S i I 1 ( t ) + < U A t P s _ l j n ( t ) + d P l 0 T h e r e f o r e ^ = ixT-^ Q + VP1 ^ 5 dP 0 = U P S j 0 + w P S j l - ^ X P s _ l j 0 , . . . . 6 dP ~ W = ^ P s , n - 1 + MPi-n+1 " ( * ^ + u ) p l , n ••••' 7 dP ~tlr = ^ P s - l , n + UPs^+l " U\+»K,n . 8 S i n c e the queue i s i n s t a t i s t i c a l e q u i l i b r i u m or steady s t a t e , the l e f t hand s i d e o f equations 5 to 8 are zero . Hence l\Plj0 + w p i i = 0 ' ^ P s , 0 " + H P B , 1 - ' ^ s - 1 , 0 = 0 ^ n - l + »\n+l ' ( W u > p l ^ P s - l , n + M p s ; n + 1 - U*+u)ps L e t P s , n = S u b s t i t u t i n g i n ( 9 ) ( W ' J U + U ) B 1 = 0 ( 1 0 ) (*p-(u)B s = tPBg^-L (.11) uj(^p+l-(.u)B1 = ApB^' (12 ) (tp+l-m)Bs = i p B s _ 1 From ( 1 2 ) B g ' B s - 1 -CP (tP+l-ou). B s - 1 -Cp (Ap+l-u)) B s - 2 B 2 = Up+l-uj} B l s - 1 s " L U P+l-i») J b l Hence BE = [_*P_] B. * t - 1 T h e r e f o r e B^ = i-(lp+l^ \ D i v i d i n g ( 15 ) by ( 1 7 ) , _ r IP / «> = i- » n i l J • UJ t o_ 1 T h e r e f o r e UJ = u and B = u B n s 1 S u b s t i t u t i n g B„ = u s _ 1 B • i n P = B uj n s . . i s,n s s,n 1 • , . . . S u b s t i t u t i n g i n (12) * P P s - l , n + Ps,n+1 " ( * P + I ) p s , n = 0 becomes i^u*1^-2 + B 1 u * n + * + s - 1 - ( t p + l ) B ] L u t n + s - 1 = 0 T h e r e f o r e t p + u A + 1 - ( t p + l ) u = 0 = (u-l)(-u*+u* + 1+.. ,+u2+u-4p) • . . . l 8 "One r o o t i s u = 1 and i s d i s c a r d e d . S i n c e by Des c a r t e ' s r u l e o f s i g n , t h e r e are one or two root's l e f t i n which one roo t i s l e s s than one but g r e a t e r than zero and p o s s i b l y another r o o t which i s l e s s than z e r o . D i s c a r d the n e g a t i v e r o o t . T h e r e f o r e 0 .< u < 1 m M u l t i p l y i n g (18) by J£ m /1 t | \ 1 . m-A-1 A l s o u * = IP - u. - i i 2 - ... - u1'1 U s i n g the above r e l a t i o n s and the i n i t i a l e q u a t i ons, 42. -IP' P2,0 = B l u B JLP B. i ( ^ - 1 + u ^ 2 + u ^ S + . . . . + u 2 ) P1.0 = B l u ° " ^(u^1+u^".2+.ufc'"3+....+.u2+u) B Adding p = B £ u m - - i [ U-l)^" 1+( A-2)u t" 2+...+2u 2+u) m=0 *P p - t " 1 m B l I " 1 m x m=0 " m=l " B l t l T u - ^ ~ T p - C ( 1 . U ) 2 ~ ^ B l p ( l - u ) P n = S J X P s , n - B l (1-u) CO £ P. n=0 n 1 = P + ? P = B (J-TP).^ + S^-° n=l n 1 p ( l - u ) + 1-u B x u ^ p ( l - u ) Hence P 0 = 1 ~ P '> p n = P ( l - u * ) n ( n - 1 ) z L = £ nP = — £ n=0 n ( l - u * ) Lq = Z ( n - l ) P n = - J ^ _ 4 J . H y p e r e x p o n e n t i a l I n t e r a r r i v a l Time D i s t r i b u t i o n U n i t s from a depot, with' an i n f i n i t e p o p u l a t i o n , pass through a channel, w i t h two branches, t o j o i n a queue. The s e r v i c e time d i s t r i b u t i o n i s e x p o n e n t i a l . The u n i t s e n t o r one branch a of the time. I t s p e r i o d i n t h i s branch i s ex-p o n e n t i a l l y d i s t r i b u t e d w i t h a mean o f -g-^ . They e n t e r the oth e r branch (l-o) of the time. I t s p e r i o d i n t h i s branch i s e x p o n e n t i a l l y d i s t r i b u t e d w i t h a mean o f -0/ ., 1 — . No u n i t s are t o e n t e r the channel i f one o f the branches i s o c c u p i e d . As has p r e v i o u s l y been shown, the r e s u l t o f u s i n g t h i s mechanism i s a h y p e r e x p o n e n t i a l i n t e r a r r i v a l time d i s t r i -b u t i o n . L e t P^-n b e t h e P r o b a b i l i t y o f the s t a t e where a u n i t i s i n branch 1 and there are n u n i t s i n the queue and s e r v i c e f a c i l i t y , and F0 be the p r o b a b i l i t y o f the s t a t e where a u n i t i s a branch 2 and t h e r e are n u n i t s i n queue and s e r v i c e f a c i l i t y . In the time i n t e r v a l t t o t+At where &t -» 0 I n i t i a l e q u ations; P l j 0 ( t + A t ) = [ l - ( 2 a X ) A t ] P 1 ^ 0 ( t ) + •uAtP 1 x ( t ) .....1 P ^ 0 ( t + A t ) = [ l - 2 ( l - c ) X A t ] P 2 Q ( t ) + »LtP21(t) .....2 Queue equations; P l j n ( t + A t ) = a ( 2 a x ) A t P 1 ^ n _ 1 ( t ) + 2 a ( l - < r ) X A t P 2 ^ n _ 1 ( t ) + ' ' ^ t P l , n + l ( * ) + U - ( y + 2 a \ ) A t ] P l j n ' . ' . . . . . 3 44, P 2 ^ n ( t + A t ) = ( l - a ) ( 2 o x ) A + P ^ n - l ^ ' + U - o ^ U t P g ^ C t ) '+ M A t P 2 ^ n + 1 ( t ) + ; i - t 2 ( l - o J x + u } A t ] P 2 i I 1 ( t ) 4 d P l 0 T h e r e f o r e d ^ = WP ] L x - 2 a X P 1 0 5 i F = » P 2 , 1 " 2( l-ahP 2 > 0 .....6 -aH" 2^l,n-l + 2 ^ l - a h P 2 > n . 1 + + * P l , n + l " ^ + 2 ^ ) p i , n • — ' 7 dP, I t 1 1 =^2^l-a)xPljn_1 +. 2 ( l - a ) 2 X P 2 j n _ 1 + • + ^ P2,n+1 " U + 2 ( 1 - a ) x ] P 2 j n 8 Si n c e the queue i s i n - s t a t i s t i c a l e q u i l i b r i u m or steady s t a t e , the l e f t hand s i d e s o f equations ( 5 ) t o ( 8 ) are z e r o . ^ 1 , 1 > 2 ^ 1 , 0 = ° ••••• 9 ^ P 2 , l = 2(l-a ) x P 2 ^ 0 = 0 ....10 2 < ^ P l , n - l + 2°(1-°^2,n-l + ^ l , n + l " ^ + 2 ^ P l , n = 0 - - " 1 1 2 a ( l - a ) x P l j n _ 1 + 2(l-a)2XP2.3n_1 + ^ n + 1 -- [n+'2(l-a)'X]P2 n = 0 ....12 L e t P s , n = V * " 1 S u b s t i t u t i n g i n (11) 2C T 2X'i)n* + 2a(l-a)X(Juri"2B2 + - ( j j ^ o x ) ^ " 3 ^ = 0 ( 1 2 ) 2 C T ( 1 - a )X u j n " 2 B x + 2 ( 1- 0 ) 2 Xu> n ~% + - [ ^ + 2 ( 1-cr)X I t / 1 " " ^ = C 14 and t h s s e c u l a r e q u a t i o n from ( 1 3 ) and (14) i s « j ( u r l ) [ ( j j 2 - ( l + 2 p ) u r f 2 p - 4 C T ( l - a ) p ( l - p ) ] = 0 The. r o o t s <y = 0 and « j = 1 are d i s c a r d e d . The other r o o t s are w = i + p ± - ( 2 a - D 2 p ( l - p ) ] There i s o n l y one r o o t which i s g r e a t e r than 0 and l e s s than 1 . T h i s i s u, = \ + p - - ( 2 C T - l ) 2 p ( l - p ) ] D i s c a r d the other' r o o t which i s g r e a t e r than one. S u b s t i t u t i n g i n ( 1 1 ) and ( 1 2 ) , we o b t a i n B 0 = *{l-m)+2aP(w-o)B = 1-CT+2CTP-m B # / > # 1 3 S u b s t i t u t i n g ( 1 5 ) and P g n = B ^ 1 1 " 1 i n (11) P l j a - B , / - 1 S u b s t i t u t i n g P = B a / 1 - 1 i n s j n s P = P + P 0 r l , 0 r 2 , 0 46 V B 2 2 a p + 2 ( 1 - a ) p B 1 ( l - g ) + B 2 a _ 2 _ 2 c r f 2 f 7 p _ t 2 a ( l - a ) p L 2 a ( l - a ) p J 1 P = P + P r n . r l , n + r 2 , n n - l = ( B - L + B ^ C I ^ 1 = C l + 2 g p - « , 3 B n - l P Q + P 2 + P 2 +...+ P n +...+ P^ = 1 i . e . P Q + j P n = 1 n=l r2-2a+2aP-(u 3 B + [ l + 2 a P - a J ] B n = ± . . 2aCl-aJp 1 a 1 n * 0 S i n c e £ tu n = y i - and ( l - o O (2p-aO = 4 C T ( l - a ) p ( l - p ) n=0 • L" U ) T h e r e f o r e B, = f | < | ^ P ^ ) . Hence P n = 1 - p and P n = ^ t^ g-gHi-^^p-^^-l n 2a(2-2a+2CTp-(jjJ = (1-ai) pa/1 1 00 oj p L = E nP = E nP = 3-5-n=0 n n=0 i - u) Lq = ? ( n - l ) P n = £ S n=l n •L~ 47. I t s hould be noted from the d e r i v a t i o n o f the h y p e r e x p o n e n t i a l t h a t j = [ l / 2 a ( l - o ) ] - 1 or CT. = \ - y [ | " 2(1+3) 1 C o n s t r u c t i o n o f Tables and Graphs f o r S i n g l e Channel w i t h A r b i t r a r y I n t e r a r r i v a l Time D i s t r i b u t i o n and E x p o n e n t i a l S e r v i c e Time D i s t r i b u t i o n „• S i n c e the average w a i t i n g l e n g t h o f a queue w i t h a r b i t r a r y i n t e r a r r i v a l time d i s t r i b u t i o n and e x p o n e n t i a l s e r v i c e time d i s t r i b u t i o n i s o n l y dependent on the u t i l i z a t i o n f a c t o r , 2 p , and the f r a c t i o n a l c o e f f i c i e n t o f v a r i a n c e squared, C , the p l o t w i l l have an o r d i n a t e i n term of the f r a c t i o n a l c o e f f i -c i e n t o f v a r i a n c e squared and the a b s c i s s c a w i l l be the average w a i t i n g l e n g t h , f o r v a r y i n g v a l u e s o f u t i l i z a t i o n f a c t o r . With the a i d of the U n i v e r s i t y o f B r i t i s h Columbia I.B.M. 7040, L was computed f o r v a r y i n g P and i f o r the E r l a n g i a n i n t e r a r r i v a l time d i s t r i b u t i o n s and p and j f o r the h y p e r e x p o n e n t i a l i n t e r a r r i v a l time d i s t r i b u t i o n s . F o r the E r l a n g i a n d i s t r i b u t i o n s , the formulae where 0 _< u < 1 i n u* + u ^ " 1 + ... + u + u = pi and f o r the h y p e r e x p o n e n t i a l d i s t r i b u t i o n s , the formulae 48. re 0 < c u < 1 i n u, = \.+. P - - ( 2 ( j-.l) 2P(l-p) ] whe and a = \ - A\ - 2(1+57] were used t o compute L . Table I I and F i g u r e 7 g i v e the 2 computed v a l u e s o f L a g a i n s t C f o r v a r i o u s v a l u e s o f p . cl I t i s not n e c e s s a r y t o p r o v i d e graphs f o r the o t h e r c h a r a c t e r i s t i c s as the r e l a t i o n s h i p between L and L q , W Wq and the busy p e r i o d can be e a s i l y computed. Lq = (L-p) Wq - iV»l A. • Busy p e r i o d = u t i l i z a t i o n f a c t o r = p 0 1/20 1/10 1/5 1/4 • 1/3 1/2 0.05 0.050000 0.050001 0.050016 0.050039 0.050112 0.050423 0.10 0.100030 0.100098 o.ioo4i9 0.100684 0.101282 0.103006 0.15 0.150485 0.150936 0.152299 0.153193 0.154952 0.159300 0.20 0.201409 0.202450• 0.203758 0.207050 0.208982 0.212547 O.220696 0.25 0.257390 0.260045 0.266148 0.260524 0.275535 O.288675 0.30 0.312793 0.316926 0.321409 ' 0.331210 0.336448 0.345576 0.364984 0.35 0.382884 0.389692 0.404137 , 0.411694 0.424684 0.451800 0.40 0.448129 0.457463 0.467149 0.487312 0.497714 . 0.515430 0.551956 0.45 0.543486 0.556699 O.583866 0.597749 0.621244 0.669264 0.50 0.627410 0.644771 0.662321 O.698098 . 0.716262 0.746863 0.809017 0.55 0.766733 0:789668 0.836146 0.859635 0.899078. O.978831 o.6o 0.887837 0.917412 0.947151 1.007165 1.037394 1.088038 1.190100 0.65 1.109375 1.147924 1.225490 1.264465 1.329653 1.460707 0.70 1.313074 1.363507 1.413863 1.514969 I.565685 1.650409 1.820436 0.75 1.717353 1.784292 1.918494 1.985729 2.097951 2.322876 0.80 2.153432 2.245948 2.337815 2.521805 2.613907 2.767538 3.075184 0.85 3.124321 3.257793 3.524927 3.658572 3.88l4ll 4.327373 0.90 4.651163 4.877497 5.094249 5.527877 5.744745 6.106251 6.829455 0.95 10.13049 10.59720 11.53067 11.99744 12.77540 14.33144 TABLE I I I A Values o f L f o r A r b i t r a r y I n t e r a r r i v a l Time D i s t r i b u t i o n and E x p o n e n t i a l S e r v i c e Time D i s t r i b u t i o n 1 2 3 4 5 6 7 0 . 0 5 0 . 0 5 2 6 3 2 0 .053538 0 . 0 5 4 0 1 6 . 0 . 0 5 4 3 1 0 0 . 0 5 4 5 1 0 0 . 0 5 4 6 5 5 0 .054764 0 .10 0.111111 0.115069 0.117265 0.118664 0.119633 . 0.120345 0.120890 0 .15 0.176471 0.186215 0 .191922 0.195686 O.198360 0.200358 0.201910 0 .20 0.250000 0.269008 O.280776 0.288839 0.294727 0.299225 0.302776 0 .25 0.333333 0.366025 0.387426 0.402700 0 .414421 0 .423232 O.430501 0.30 0.428571 0.480566 0.516539 0 .543344 0.564268 O.581139 0.595075 0 .35 0.538462 0.616922 0.674217 0.718837 . 0.754991 O.785098 O.810681 0 .40 0.666667 0.780776 0.868517 0.939902 1.000000. I.051785 1.097168 0 .45 0.818182 0 .979321 . 1 .110243 1.220867 1.317532 1.403700 1.481617 0.50 1.000000 : 1 .224745 1 .414214 1.581139 1.732051 I.870829 2.000000 0.55 1.222222 1.530892 1.801408 2.047726 2.276984 2.493411> 2.699754 0.60 1.500000 1.921165 2.302776 2.659852 3.000000 3.327677 3.645751 0.65 1.857143 2.431426 2.966403 3.477840 3.973555 4.458039 4 .934122 0.70 2.333333 3.121320 3.871924 4.601136 5.316624 6.022657 ' 6.721841 0 .75 3.000000 4.098076 5.162278 6.208099 7.242641 8.269696 9.291503 o.8o 4.000000 5.576034 7.123105 8.655354 10.17891 11.69690 1 3 . 2 1 1 1 0 0.85 3.666667 8.055216 10.42089 1 2 . 7 7 5 5 5 15.12404 17.46870 19.81082 0.90 9.000000 13.03562 17.05538 21.06797 25.07669 29.08309 33.08799. 0 .95 18.99999 28.01722 37.02628 46.03187 55.03566 64.03839 7 3 . 0 4 0 4 4 TABLE I I I B Values o f L f o r A r b i t r a r y I n t e r a r r i v a l Time D i s t r i b u t i o n and E x p o n e n t i a l S e r v i c e Time D i s t r i b u t i o n U l o Testing of Graph . 1 The v a l i d i t y of the graph w i l l now be tested with various a r b i t r a r y i n t e r a r r i v a l time d i s t r i b u t i o n s , and expo-n e n t i a l service time d i s t r i b u t i o n with the aid of the Uni v e r s i t y of B r i t i s h Columbia I.B.M. 7040 computer. A p. of 0.8 were used for the d i f f e r e n t combinations of a r b i t r a r y i n t e r a r r i v a l time d i s t r i b u t i o n and service time d i s t r i b u t i o n . The program G.P.S.S. I l l was used to simulate the queue. A run of ten thousand transactions were simulated f o r each of the combination. The flow chart for the program i s as shown i n Appendix I I I . The d i s t r i b u t i o n s used and the r e s u l t s from G.P.S.S. I l l and from the graph are shown i n Table IV. Appendix I/x : The d i s t r i b u t i o n as given i n Appendix I, section x 3 e.g. Appendix I/a i s the d i s t r i b u t i o n as depicted i n Appendix I, section a . p : The u t i l i z a t i o n factor used. pc : The u t i l i z a t i o n of the f a c i l i t y as obtained from G.P.S.S. II Lqc : The average length of queue as obtained from G.P.S.S. I I I . Lc : The average waiting length as obtained from G.P.S.S. I l l = Lqc + pc . Lp : The average waiting length i n the system as obtained from graph with u t i l i z a t i o n factor = 0.8 = p . Lpc : The average waiting length i n the system as obtained from graph v/ith u t i l i z a t i o n factor = pc . I n t e r a r r i v a l time d i s t r i b u t i o n ° l S e r v i c e time d i s t r i b u t i o n C 2 s P pc Lqc Lc L P Lp |Lpc-L< (1) Appendix I/a 0.4046 E x p o n e n t i a l 1 0.8 0.8330 3.39 4.22 2.90 3.68 •54 (2) Appendix I/b 1 E x p o n e n t i a l 1 0.8 0.7412 2.47 3.21 : 4.00 3.52 . .31 (5) Appendix I/c 5.8 E x p o n e n t i a l 1 0.8 0.7054 7.33 8.04 11.40 7.47 •57 ( 4 ) Appendix l / d 1.6 E x p o n e n t i a l 1 0.8 0.7420 3.39 4.13 4.85 3.55 .58 (5) Appendix I/e 0.3333 E x p o n e n t i a l 1 0.8 0.8087 2.61 3.42 2.77 2.99 .43 (6) E x p o n e n t i a l 1 E x p o n e n t i a l 1 0.8. 0.8032 3.68 4.48 4.00 4.17 .31 TABLE IV T e s t R e s u l t s f o r A r b i t r a r y I n t e r a r r i v a l Time D i s t r i b u t i o n s and' E x p o n e n t i a l S e r v i c e Time D i s t r i b u t i o n s 55. The following factors w i l l account for the difference between Lc , Lp and Lpc p a) The of the actual run of 10,000 transactions 2 is not equal to tne C of an in f i n i t e number of runs as used in"the construction of the graph. b) pc and p . are not equal. c) The C of the actual run of 10,000 transactions ' s 3 2 i s not equal to the C g of an in f i n i t e number of runs as used in the construction of the graph. d) The accuracy of the random number generator. It can be noted that in the case where the inter-ar r i v a l time and service time distributions are exponential, the difference of the simulated L and that of the computed L. is 12.08 percent.-Average . p c = (4.6335/6) = 0.7723. Since average pc < p i t is probable that £ LpC < 2 Lc and £ Lc < £ Lp i f the assumptions in the usage of the graph are correct. This is borne out by the facts: S Lc = 27.50 , E L p = 29.92 and £ Lpc = 25.38 , then 2> Lpc < £ Lc < £ Lp . Furthermore we shall examine the differences between Lc and Lpc . We use Lpc instead of Lp for comparison because pc is the actual u t i l i z a t i o n factor of the simulation. The differences between Lc and Lpc for the six sets of simu-lation are shown in the f i n a l column of Table IV. The average difference is 0.46. When we use exponential interarrival and . 56. service time distributions and where we were sure of Lpc because i t was calculated from the classical Erlang's formula there was a difference between Lc and Lpc of 0.31. Hence, the value of 0.46 average difference is not disturbing. It is necessary for further works to be done on the testing of the graphs. This, unfortunately, is outside the scope of the limited time available in the presentation of this thesis. 57. . CHAPTER 6 ARBITRARY INTERARRIVAL TIME DISTRIBUTION: ARBITRARY SERVICE TIME DISTRIBUTION When i n t e r a r r i v a l time d i s t r i b u t i o n i s e x p o n e n t i a l and s e r v i c e time d i s t r i b u t i o n i s a r b i t r a r y , the average l e n g t h p i n the system i s a f u n c t i o n o f p and C o n l y as proved by s the P o l l a c z e k - K h i n t c h l n e formula; and when i n t e r a r r i v a l time d i s t r i b u t i o n i s a r b i t r a r y and s e r v i c e time d i s t r i b u t i o n i s e x p o n e n t i a l , the average l e n g t h i n the system i s a f u n c t i o n o f 2 p and C o n l y as proved i n Chapter . 5 . I f , then, the i n t e r a r r i v a l time d i s t r i b u t i o n i s . a r b i t r a r y and the s e r v i c e time d i s t r i b u t i o n i s a r b i t r a r y , i s the average l e n g t h o f the system, L , a 2 2 f u n c t i o n o f p , C& and C g only? Our purpose i s t o show, e m p i r i c a l l y , t h a t i t i s . The problem i s t o determine the v a l u e o f L f o r each 2 ? combination of C - cr and p . T h i s problem i s r e s o l v e d cl S through f o u r separate t e c h n i q u e s . These, r e l a t e d t o the v a l u e s 2 2 " o f C and C , are i l l u s t r a t e d i n f i g u r e 9 and e x p l a i n e d CL S below. i ) For P o i s s o n d i s t r i b u t i o n a r r i v a l r a t e s and p ? a r b i t r a r y s e r v i c e times (C^ = 1, C g _> 0) the P o l l a c z e k - K h i a t c h i n e approach of Chapter 4 can be used. Thus val u e s o f L may be taken d i r e c t l y from Table I I . i i ) For a r b i t r a r y a r r i v a l r a t e s and e x p o n e n t i a l d i s t r i b u t e d s e r v i c e times ( C 2 > 0, C 2 = 1) 58.. the work o f Chapter 5 i s p e r t i n e n t . Values o f L can be taken from Table I I I . i i i ) Approaches i and i i are i n d i c a t e d i n f i g u r e 5 by l i n e s 1 and 2 r e s p e c t i v e l y . F or a l l o t h e r areas o f f i g u r e 5, L must be determined by 2 2 s i m u l a t i o n , except f o r one p o i n t (C& = 0, 'CR = The sampling d i s t r i b u t i o n s must be h y p e r - e x p o n e n t i a l 2 2 f o r v a l u e s o f C or C > 1 : must be E r l a n g i a n f o r v a l u e s a s p p o f Ci" or C; < 1 : and must be constant f o r v a l u e s o f a s 3 2 2 C a o r C s = 0 * T h u s the a c t u a l s i m u l a t i o n technique depends 2 2 on the p a r t i c u l a r combination o f va l u e s o f C and C b e i n g c a s t e s t e d . The fl o w c h a r t s o f these techniques are d e t a i l e d i n Appendix I I I and are r e f e r e n c e d below t o the r e l e v a n t areas o f the c h a r t i n f i g u r e 9 . F i g . 9 A r e a c 2 a c l Flow Chart 3 C2>1 a 2 c|>i 1 4 0<C2<1 a cf>i 2 5 C2>1 a 0<C2<1 3 6 c? = o a C2>1 4 7 C2>1 a cf = o 5 8 0<C?<1 • a 0<C2<1 6 9 C 2 = 0 a 0<C2<1 7 10 0<C2<1 • a cf = o s 8 5 9 . -. o p F o r C = 0 and C = 0 , we know t h a t Lq = 0 . a s •» ^ Sin c e L = Lq + p , then L = p . T h i s a p p l i e s t o p o i n t 11 i n f i g u r e 9• Each s i m u l a t i o n was c a r r i e d through 10,000 t r a n s a c t i o n s and t o t a l w a i t i n g time i n the queue (TWT) was accumulated. Thus f o r each set of v a l u e s Wq , the average w a i t i n g time, c o u l d he determined. (Wq = TWT £ 10,000) The v a l u e o f L , average system l e n g t h , was de t e r -mined from the e q u a t i o n 1 L = Wq\ + p The r e s u l t s o f these c a l c u l a t i o n s and s i m u l a t i o n s are shown i n a s e t of graphs, f i g u r e s 1 0 - 2 9 . 2 F o r each of 10 v a l u e s o f C& , v a l u e s o f L are 2 p l o t t e d a g a i n s t t e n v a l u e s o f C g f o r va l u e s o f p from 0 . 1 2 t o 0 .9 . There are two graphs f o r each value o f C Q because the L s c a l e f o r v a l u e s o f p between 0 and 0 . 5 was inadequate f o r h i g h e r p v a l u e s . From the p l o t t e d p o i n t s , curves have been c o n s t r u c t e d by i n s p e c t i o n r e p r e s e n t i n g each v a l u e o f p . For va l u e s o f p below 0 . 6 there appears t o be a l i n e a r r e l a t i o n s h i p be-2 tween C and L . s i B u f f a , E.S., Models i n P r o d u c t i o n and Operations  Management, John Wil e y and Sons, Inc., New York, 1966. 6o. C a P r a c t i ° n a l C o e f f i c i e n t o f I n t e r a r r i v a l Time Square F i g u r e 9 I l l u s t r a t i o n o f D e r i v a t i o n s f o r A r b i t r a r y I n t e r a r r i v a l . and S e r v i c e Time D i s t r i b u t i o n s 64 o 68. KEUFFEL & ESSER CO. 71. 7 2 . 74. 75< 76. 78 o m mm - Trrt ffi m mm ttlH :SSS ±n±3 rffT 1 ttttt -co am. :t:nd± 79 8 0 . 8 1 . T e s t i n g o f Graphs The v a l i d i t y o f the graphs w i l l now be t e s t e d w i t h v a r i o u s a r b i t r a r y i n t e r a r r i v a l time d i s t r i b u t i o n s and a r b i t r a r y s e r v i c e time d i s t r i b u t i o n s w i t h the a i d of the U n i v e r s i t y o f B r i t i s h Columbia I.B.M. 7040 computer. A p o f 0.8 were used f o r the d i f f e r e n t combinations o f a r b i t r a r y i n t e r a r r i v a l time d i s t r i b u t i o n and s e r v i c e time d i s t r i b u t i o n . A second s e t v/ith p = 0.5 was a l s o t e s t e d . The program G.P.S.S. I l l v/as used t o simulate the queue. A run o f 10,000 t r a n s a c t i o n s v/ere simu-l a t e d f o r each o f the combination. The flow c h a r t f o r the program i s as shov/n i n Appendix I I I . The d i s t r i b u t i o n s used and'the r e s u l t s from G.P.S.S. I l l and from the graphs ( f i g u r e s 10-29) are shown i n Table 5. Appendix I/x : The d i s t r i b u t i o n used as g i v e n i n Appendix I, s e c t i o n " x . e.g. Appendix i / c i s the d i s t r i b u t i o n i n s e c t i o n c of Appendix I. p : The u t i l i z a t i o n f a c t o r used. pc : The u t i l i z a t i o n o f the f a c i l i t y as o b t a i n e d from G.P.S.'S. I I I . Lqc : The average l e n g t h o f queue as o b t a i n e d from G.P.S.S. I I I . Lc : The average w a i t i n g l e n g t h as obtained from G.P.S.S. I l l = Lqc + pc . Lp : The average w a i t i n g l e n g t h i n the system as o b t a i n e d from graphs w i t h the u t i l i z a t i o n f a c t o r used. Lpc : The average w a i t i n g l e n g t h i n the system as ob t a i n e d from the graphs w i t h u t i l i z a t i o n f a c t o r Pc . : I n t e r a r r i v a l time d i s t r i b u t i o n C 2 a S e r v i c e time d i s t r i b u t i o n C 2 °s P Lqc Lc. Lp LPc |Lpc-Lc ( 1 ) Appendix I/d 1 . 6 Appendix I/c 5 . 8 0.8 0 . 8 3 1 9 1 4 . 4 3 1 5 . 2 6 1 2 . 2 0 1 6 . 0 0 .74 ( 2 ) Appendix I/a 0.4046 Appendix I/b 1 0 . 8 0 . 8 8 7 0 4 . 8 9 3 . 7 5 2.9O 5 . 8 0 .04 ( 3 ) Appendix 1 / b 1 Appendix I/c 5 . 8 0 . 8 0 . 8 1 5 3 9 . 8 5 . , 1 0 . 6 7 1 1 . 6 7 1 2 . 8 0 . 1 3 ( 4 ) Appendix I/d 1 . 6 Appendix I/a 0.4046 0 . 8 0 . 7 4 5 0 , 2.84 .. 3.59 4 . 0 6 3 . 6 1 0 . 0 2 ( 5 ) Appendix I/c 5 . 8 Appendix I/e 0 . 3 3 3 3 0 . 8 O . 7 4 3 2 8 . 7 1 ' 9 . 4 5 1 0 . 5 0 i 9 . 0 1 0 . 4 4 ( 6 ) Appendix 1 / e 0 . 3 3 3 3 Appendix I/d 1 . 6 0 . 8 0 . 8 6 2 4 . 3 . 3 3 6 . 1 9 4 . 0 5 6 . 2 5 0 . 0 6 ( 7 ) Appendix I/b 1 Appendix I/c 5 . 8 0 . 5 0 . 5 2 0 1 < 1 . 8 8 2.40 2 . 2 0 2.50 0 . 1 0 (3) Appendix I/d 1 . 6 Appendix I/c 5 . 8 0 . 5 O . 5 0 1 7 1 . 9 5 2 . 4 5 '2 . 5 O . 2 . 5 2 0 . 0 7 ( 9 ) Appendix I/a 0.4046 Appendix I/a 0.4046 0 . 5 0.4722 0 . 2 5 • 0 . 7 2 0 . 6 V 0 . 6 1 0 . 0 9 . ( 1 0 ) Appendix I/d 1 . 6 Appendix I/b • 1 0 . 5 0 . 5 0 1 1 1 . 1 1 I . 6 I ' l . l 3 1 . 2 0 0 . 4 1 ( I D Appendix I/e 0 . 3 3 3 3 Appendix I/e 0 . 3 3 3 3 0 . 5 0 . 4 5 4 7 0 . 1 6 0 . 6 1 0 . 6 2 0 . 5 5 0 . 0 6 ( 1 2 ) Appendix I/c 5 . 8 Appendix I/d 1 . 6 0 . 5 0 . 4 9 0 1 3 . 3 5 3.84 2 . 1 8 2 . 1 5 1 . 6 9 -TABLE V T e s t R e s u l t s f o r A r b i t r a r y I n t e r a r r i v a l Time D i s t r i b u t i o n s and A r b i t r a r y S e r v i c e Time D i s t r i b u t i o n s 00 ro o The f o l l o w i n g f a c t o r s w i l l account f o r the d i f f e r e n c e between Lc and Lp- : a) The C a of the a c t u a l number of t r a n s a c t i o n s o f . p 10,000 i s not equal to the C o f an i n f i n i t e * ^ a number of t r a n s a c t i o n s , so e r r o r i s i n c u r r e d i n both, The G.P.S.S. I l l program and the c o n s t r u c t i o n o f the graphs. T h i s i n c r e a s e s the maximum p o s s i b l e e r r o r . b) pc and p. are not equ a l . c) The C of the a c t u a l number o f t r a n s a c t i o n s o f s p 10,000 i s not equal t o the C o f an i n f i n i t e a number o f t r a n s a c t i o n s , so e r r o r i s i n c u r r e d . i n both, the G.P.S.S. I l l program and the c o n s t r u c t i o n o f the graphs. T h i s i n c r e a s e s the maximum p o s s i b l e e r r o r . d) The accurac y o f the random number ge n e r a t o r . Average pc = (7.7950/12) = 0.6625 * Average p = (7.8/12) = O.65 S i n c e average pc > P , i t i s p o s s i b l e t h a t Lpc > Lc and . Lc > Lp i f the assumptions i n the usage of the graphs are c o r r e c t . Since t h i s i s borne out t o the f a c t s : Lc = 60.55 Lp = 57.65 and LpC = 63.02 , then L p c > Lc > Lp . Furthermore we s h a l l examine the d i f f e r e n c e s between Lc and L p c . We use Lpc i n s t e a d o f Lp f o r comparison be-cause pc . i s the a c t u a l u t i l i z a t i o n f a c t o r o f the s i m u l a t i o n The d i f f e r e n c e s between Lc and Lpc f o r the twelve s e t s o f s i m u l a t i o n are shown i n the f i n a l column of Tab l e V. The , 84. average d i f f e r e n c e i s 0 . 4 9 . Where p r e v i o u s l y we used exponent-i a l i n t e r a r r i v a l and s e r v i c e time d i s t r i b u t i o n s and where we were sure of Lpc because i t v/as c a l c u l a t e d from the C l a s s i c a l E r l a n g ' s formula, (Table IV) t h e r e was a d i f f e r e n c e between Lc and Lpc of 0 . 3 1 . Hence the value o f 0 . 4 9 average d i f f e r -ence i s net d i s t u r b i n g . I t i s n e c e s s a r y f o r f u r t h e r works t o be done on the t e s t i n g o f the graphs ( f i g u r e s 1 0 - 2 9 ) . T h i s , u n f o r t u n a t e l y , i s o u t s i d e the scope o f the l i m i t e d time a v a i l a b l e i n the pre-s e n t a t i o n o f t h i s t h e s i s . • 8 5. CHAPTER 7 CONCLUSION The usage o f the graphs w i l l enable one to compute the r e q u i r e d c h a r a c t e r i s t i c s t o make t h e . n e c e s s a r y d e c i s i o n s without going through the process of s i m u l a t i o n or curve f i t t i n g f o r each p a r t i c u l a r case of s i n g l e channel queue w i t h custom-er s coming from an i n f i n i t e p o p u l a t i o n . I t i s not n e c e s s a r y t o know the forms of d i s t r i b u t i o n o f i n t e r a r r i v a l time and s e r v i c e time t o o b t a i n the r e q u i r e d c h a r a c t e r i s t i c s . S i nce the average w a i t i n g l e n g t h o f the system i s dependent o n l y on u t i l i z a t i o n f a c t o r , p , square o f the ... p f r a c t i o n a l c o e f f i c i e n t o f i n t e r a r r i v a l time, C;~ and square ' '' 2 o f the f r a c t i o n a l c o e f f i c i e n t o f s e r v i c e time C . The f o l l o w -s i n g p r o c e s s w i l l enable one t o compute the n e c e s s a r y c h a r a c t -e r i s t i c s . The p o s s i b l e e r r o r i s r e l a t i v e l y s m a l l and as shown by comparison"'with s i m u l a t i o n o f 10,000 t r a n s a c t i o n s v i a G.P.S.S. I l l f o r twelve combinations o f a r b i t r a r y d i s t r i b u t i o n s , the average d i f f e r e n c e s i n w a i t i n g l e n g t h o f the system i s o n l y 0.49 customer. T h i s g i v e s a percentage d i f f e r e n c e o f 10°/o. a) C a l c u l a t e the average i n t e r a r r i v a l time. Average i n t e r a r r i v a l time = = ( T o t a l time o f A, o b s e r v a t i o n / T o t a l number of a r r i v a l s ) = ~ . b) C a l c u l a t e the v a r i a n c e f o r i n t e r a r r i v a l time. n 2 V a r i a n c e = V a r ( t ) = £ ( t . ™ ) / T o t a l number 1=1 2 o f a r r i v a l = £ f . f t .-^) 1 a i V . . . . 86.. c) C a l c u l a t e the f r a c t i o n a l c o e f f i c i e n t o f v a r i a n c e p squared f o r i n t e r a r r i v a l time d i s t r i b u t i o n , C . . d) C a l c u l a t e the average s e r v i c e time. Average s e r v i c e time = — = ( T o t a l time f a c i l i v y H i s i n o p e r a t i o n / T o t a l number s e r v i c e d ) = — ' m e) C a l c u l a t e the v a r i a n c e f o r s e r v i c e times. m , 2 V a r i a n c e = V a r ( t ) = £ ( t .—) / T o t a l number s j = l S J U s e r v i c e d = £ f 4 (t„.-— ) 2 • • 1 S I (Jl f ) C a l c u l a t e the f r a c t i o n a l c o e f f i c i e n t o f v a r i a n c e p squared f o r s e r v i c e time d i s t r i b u t i o n , C g . g) U t i l i z a t i o n f a c t o r , P = (Average s e r v i c e t i m e / average i n t e r a r r i v a l time) o P h) With the v a l u e s p , Cf^ , and C g , re a d from the graphs ( f i g u r e s 10 - 29 ) the v e r t i c l e a x i s , L . i ) Compute; The average queue l e n g t h Lq = (L - p ) The average queueing time, Wq = ( L - p ) / \ The average w a i t i n g time i n the system, W = L / x The busy p e r i o d = p . For an example o f the usage o f the graphs ( f i g u r e s 1 0 - 2 9 ) , vie s h a l l assume t h a t a saw r e c e i v e s l o g s from two s o u r c e s . D u r i n g the e n t i r e p e r i o d o f o b s e r v a t i o n , i t i s 87. ' noted that a t o t a l of 1000 logs a r r i v e at the saw. The t o t a l p e r i o d of ob s e r v a t i o n i s 5000 minutes. I t i s noted that 50°/o of t h e . i n t e r a r r i v a l time i s 3 minutes and 50°/o i s 7 minutes. • I t i s ' a l s o known that 30°/o of "".he logs r e q u i r e 3 minutes t o saw each l o g , 4o°/o r e q u i r e s 4 minutes and 30°/o r e q u i r e s 5 minutes. I t i s r e q u i r e d to compute the average w a i t i n g l e n g t h , the average queue le n g t h , the average w a i t i n g time, the aver-age queue time and the u t i l i z a t i o n f a c t o r a) Average i n t e r a r r i v a l time = = 5 minutes/log. b) Variance of I n t e r a r r i v a l time = [ ( 0 . 5 0 ) ( 3 - 5 ' ] 2 + ( 0 . 5 0 ) ( 7 - 5 ) 2 ] =4 d) Average s e r v i c e time =.(.30x3 +.40x4 + .30x5) • = 4 minutes/log - e) Variance of s e r v i c e time = [ 0 . 3 0 ( 3 - 4 ) 2 + 0.40(4-4) 2 + 0.50(5-4) 2] = 0.80 f ) C 2 = % ^ = 0.05 g) . > .= I = 0.8 h) From f i g u r e s . 1 0 - 2 9 , and us i n g i n t e r p o l a t i o n : From f i g u r e 11 w i t h C 2 = 0 , p = 0.8 and C? = 0.05 cl S L = O.85 P From f i g u r e 13 w i t h = 0.25 , P. = 0.8 and c l * c 2 = 0.05 By i n t e r p o l a t i o n : 8 8 . L = "o 3 5 + 0 . 1 5 > ^ 1 6 . ^ 0 . 2 5 = 0 . 9 5 l o g s 1 ) Lq = (L - p ) = ( 0 . 9 5 - 0 . 8 0 ) = 0 . 1 5 l o g s . W = ± = 0 . 9 5 x 5 = 4 . 7 5 minutes/log. • Wq = Lq = O.I5X5 = 0 . 7 5 minutes/log. \ APPENDIX I DISTRIBUTIONS FOR TESTING OF GRAPHS 89, a ) f ( t ) p ( t ) t . f ( t ) ( t - 1 ) ( t - 1 ) 2 2 V a r i a n c e ( t ) C 0 . 0 0 . 0 5 0 . 0 5 0 . 0 0 0 - 1 . 0 1 . 0 0 0 . 0 5 0 0 0 . 1 0 . 0 7 0 . 1 2 0 . 0 0 7 - 0 . 9 0 . 8 1 0 . 0 5 6 7 0 . 2 0 . 0 6 0 . 1 8 0 . 0 1 2 - 0 . 8 0 . 6 4 0 . 0 3 8 4 0 . 3 0 . 0 5 0 . 2 3 0 . 0 1 5 - 0 . 7 0 . 4 9 0 . 0 2 4 5 0 . 4 0 . 0 6 0 . 2 9 0 . 0 2 4 - 0 . 6 O . 3 6 , 0 . 0 2 1 6 0 . 5 0 . 0 5 0 . 3 4 0 . 0 2 5 - 0 . 5 0 . 2 5 0 . 0 1 2 5 0 . 6 0 . 0 4 0 . 3 8 0 . 0 2 4 - 0 . 4 O.I6 0 . 0 0 6 4 0 . 7 0 . 0 3 0 . 4 1 0 . 0 2 1 . - 0 . 3 0 . 0 9 0 . 0 0 2 7 0 . 8 0 . 0 2 0 . 4 3 0 . 0 1 6 - 0 . 2 0 . 0 4 0 . 0 0 0 8 0 . 9 0 . 0 1 0 . 4 4 0 . 0 0 9 - 0 . 1 0 . 0 1 0 . 0 0 0 1 1 . 0 0 . 0 3 0 . 4 7 0 . 0 3 0 0 . 0 , 0 . 0 0 0 . 0 0 0 0 1 . 1 0 . 0 3 0 . 5 0 0 . 0 3 3 0 . 1 0 . 0 1 0 . 0 0 0 3 1 . 2 0 . 0 5 0 . 5 5 0 . 0 6 0 0 . 2 0 . 0 4 0 . 0 0 2 0 1 . 3 0 . 0 6 0 . 6 1 0 . 0 7 8 0.3 0 . 0 9 0 . 0 0 5 4 1 . 4 o . o s O . 6 9 0 . 1 1 2 . 0 . 4 0 . 1 6 0 . 0 1 2 8 ' 1 . 5 0 . 0 7 O . 7 6 0 . 1 0 5 0 . 5 0 . 2 5 0 . 0 1 7 5 1 . 6 0 . 0 3 0 . 7 9 0 . 0 4 8 0 . 6 0 . 3 6 0 . 0 1 0 8 1 . 7 0 . 0 8 0 . 8 7 0 . 1 3 6 0 . 7 0 . 4 9 0 . 0 3 9 2 . 1 . 8 0 . 0 7 0 . 9 4 O .I26 0 . 8 0 . 6 4 0 . 0 4 4 8 1 . 9 0 . 0 1 0 . 9 5 0 . 0 1 9 0 . 9 0 . 8 1 0 . 0 0 8 1 2 . 0 0 . 0 5 1 . 0 0 0 . 1 0 0 1 . 0 1 . 0 0 0 . 0 5 0 0 1 . 0 0 0 0 . 4 0 4 6 0 . 4 0 4 6 • X •. ''••'/( ••• • • • - • • 0.05 f ( t ) • • • • 0 1 2 9 0 . b) t f ( t ) F ( t ) t . f ( t ) ( t - 1 ) ( t - 1 ) 2 V a r i a n c e ( t ) C 2 0 . 0 0 . 5 0 . 5 0 . 0 - 1 . 0 1 .00 0 . 5 0 2 . 0 0 . 5 1 .0 i . o . 1 .0 1 . 0 0 0 . 5 0  . 1 .0 1 . 0 0 1 . 0 0 0/ f(t) I I f(t)i 1 t 0 1 2 • t 9 1 . c) t f ( t ) . F ( t ) t . f ( t ) ( t - 1 ) ( t - 1 ) 2 V a r i a n c e ( t ) C 2 C O 0 . 3 0 . 8 O . O - 1 . 0 1 . 0 0 0 . 8 0 2 . 0 0 . 1 0 . 9 0 . 2 . 1 . 0 1 . 0 0 0 . 1 0 8 . 0 0 . 1 1 . 0 " 0 , 8 7 . 0 4 9 . 0 0 _ 4 . 9 0 5-f ( t ) F ( t ) 1 . 0 5 . 8 0 5 . 8 O 92. t f ( t ) F ( t ) t . f ( t ) ( t - 1 ) ( t - 1 ) 2 V a r i a n c e ( t ) C 2 0.0 0.6 0.6 0.0 -1.0 1.00 0.6 2.0 0.2 0.8 0.4 . ' 1.0 1.00 0.2 3.0 0.2 1.0 0.6 2.0 4.00 0.8 1.0 1.6 1.5 e) f ( t ) = 0.5 when 0 _< t < 2 =0.0 when 0 > t > 2 F ( t ) = f 0.5 dt = 1 d 0 2 2 Average ( t ) = f 0.5 t . d t = [ ° ' 5 t .]. = 1 J 0 d 2 I -0 Va r i a n c e ( t ) = f ( t - l ) 2 0 . 5 d t = f t 2 0 . 5 d t - 2 f t.0.5dt + J0 J 0 J 0 J 0.5dt = 0.5[-5-] - 2 p 0 + 1 APPENDIX I I Flow Chart; G.P.S.S. I l l : S i n g l e Channel Queue ADVANCE 8,FN1 RELEASE V TABULATE APPENDIX I I I Flow Chart 1 DATA _ 1 , 1 1  g a ~ 2 " ' v ? " 2(1+ j a ) LAMBDA 1,2,3.,. J f l 2,3,'... ,6 J s 2,3,.. .,6 2~.JT£' 2(1+3,) INITIAL CONDITIONS. MU = 10 AT = 0 TWT WT = 0 TIDT IDT = 0 AT = - l o g ( A) /( 2(T LAMBDA) AT = - log(A)/(.2(l-oJLAMBDA) -E<tr GENERATE S ST = - log(S:)/(2a sMU) . I E > G 8 GENERAT] 3 S > ST = - l o g ( S ) / ( 2 ( l - f f s ) M U ) 9.6. :DT=O WT=ST-AT • TwT=TWT+WT ST>AT ST<AT• WT=0 H)T=AT-ST T IDT=T IDT+IDT Plow Chart 2 9 7 . 1A 10,000 DATA = 1 1 . 1 -CTs 2 " ^ . ¥ " 2 ( 1 + 0 " ) t r a n s a c t i o n s INITIAL cor© TIONS GENERATE A l GENERATE A2 LAMBDA - 1 , 2 , 3 , . . . , 9 . O q = 2 ,3 S . • • , 6 . MU = 10 AT = 0 WT =. 0 IDT = 0 TWT = 0 TIDT = 0 i GENERATE Ak AT•= - (logAl+logA2+...+logAk)/(k.LAMBDA) GENERATE S I ST=-logS/(2c7_MU) ( IDT=0 WT=ST-AT TWT=TWT+WT GENERATE S ST = - l o g S / ( 2 ( ; l - 0 _ ) M J -1 WT= 0 IDT=AT-ST TIDT=TIDT+IDT Flow Chart 3 Cf - 3 a • c? = 1 / t DATA T a „ 1 . 1 1 2 " V Tf " 2 ( 1 + T T INITIAL CONDITIONS LAMBDA = 1 , 2 , 3 , . . . . , 9 . J a = 2 , 3 , • • •, 6". MU = 1 0 AT .=. 0 T¥T = 0 WT = 0 TIDT = 0 IDT = 0 GENERATE A A T = - l o g A / ( 2 a LAMBDA) a L AT=-logA/( 2 ( 1 - a )LAMBDA) 9 9 . Flow Chart 4 C 2 = 0 a DATA T Z 1 ,1 1 2 ~ V ¥ " 2 ( 1 + 3 - ) INITIAL CONDITIONS A \ 10,000 t r a n s a c t i o n s AT=1/LAMBDA AT=AT-WT GENERATE E -E<ac GENERATE S ST=-logS/(2 0 oMU) -ST>AT- J3T :AT, IDT=0 ST=AT 1 WT=0 LAMBDA = 1 , 2 , 3 , . . . , 9 < 3" = 2 , 3 , . . . , 6 . MU = 1 0 AT = 0 TWT = ( WT = 0 TIDT = ( IDT = 0 •E>ac GENERATE S ST=-logS/( 2 ( 1 - a Q)MU) -ST<ATi WT=0 WT=ST-AT ID1] IDT=AT-ST * TWT=TWT+WT T LDT=T IDT+ IDT 1 1 0 0 . Flow Chart 5 e 2 = 3 a "a C 2 = 0 s DATA LAMBDA = 1 , 2 , 3 , . . . . , 9 . J „ = 2 , 3 , . . . , 6 . = 1 . 1 1  a a -2 " * X ."' 2(1+3 J INITIAL CONDITIONS A , 1 0 , 0 0 0 t r a n s a c t i o n s MU* = 10 AT "= 0 TWT = 0 WT = 0 IDT = 0 GENERATE D a GENERATE A AT=-logA/(2a 0LAMBDA) a I - • GENERATE A •> AT: =-logA/( 2 ( l - o )LAMBDA) ST<AT — WT=0 IDT=AT-ST TIDT=TIDT+IDT Flew Chart 6 1A TO,OOP t r a n s a c t i o n s DATA INITIAL CONDITIONS 1 • •> • GENERATE A l t GENERATE A2 LAMBDA = 1,2,35...,< MU = 10 AT = 0 WT = 0 IDT = 0 • GENERATE Ak < • • •' AT = -(logAl+logA2+...+logAk)/(k.LAMBDA) . A AT=AT-WT GENERATE SI 'GENERATE S2 I I I JL. GENERATE St ST = -(logSl+logS2+...logS*)/(£.MJ) -ST>AT • ST<AT-IDT=0 WT=ST-AT WT=0 I D : D=0 WT=0 TWT=TWT+WT IDT==AT-ST TIDT=TIDT+IDT Flow Chart 7 0 1 0 , 0 0 0 t r a n s a c t i o n s DATA| B r a COND: ?IAL CTIONi.' GENERATE A l • GENERATE A2 LAMBDA = 1 , 2 , 3 , . . . , 9 . MU = 10 AT = 0 TWT WT = 0 TIDT IDT = 0 GENERATE Ak > AT=AT-WT t . ST=l/MU 103. Flow Chart 8 10,000 t r a n s a c t i o n s DATA INITIAL CONDITIONS AT=1/ LAMBDA • AT=AT-WT GENERATE SI > GENERATE S2 LAMBDA = 1,2,3, ...,9. MU = 10 AT = 0 WT = 0 IDT = 0 i _4_ GENERATE Si ST = -(logSl+logS2+... ..+logS-t)/(<t.MU) -ST>AT • •SKAT-IDT=0 L WT=ST-AT WT=0 • > HJ 1] ?=0 WT=0 TWT=TWT+WT H)T=AT-ST T LDT=T IDT+IDT TWT = C TIDT = C APPENDIX IV 104. The following tables contain the values of the average length of the system, L , with given coefficient of service p time variance squared, C g , coefficient of interarrival time 2 variance squared, C , and u t i l i z a t i o n factor, p, as obtained from the application of the various techniques l i s t e d in Chapter 6. The values are used to obtain the graphs (figures 10-29) for arbitrary service and interarrival time distributions. 0 0.25 0.333 0.5 o i 2 '. 3 . • .' • .4 5 6 ' o 0.25 0.333 0.5 0^25 : 1 2 3 ^ 4 . 5 6 0.3 0.4 0.1 0.10000 0.10000 0. 10000 01.'lOOOO 0.10000 0.10159. 0.10319 0.10682 0.11079 0.11261 0.10008 0.10016 0.10024 0.10021 0.10068 0.10313 0.10710 0.11014 0.11375 0.11875 0.2 0.20000 0.30000 0.20000 0.20018 0.20141 0.21481 0.23285 0.25517 O.26836 0.33056 0.20112 0.20190 0.20237 0.20385 0.20898 0.22119 C25373 0.27090 0.29479 0.33523 0.30000 0.40171 0.30397 0.30292 0.31279 O .3615O 0.43432 0.47471 0.56204 0.56176 3.30217 0.30769 0.31090 0.31793 .0.33645 0.39443 0.44913 0.52167 0.57640 0.65790 0.40000 0.50964 0.40401 0.41179 0.44813 0.54731 0.70498-O.76565 0.96452 I.08654 0.41327 0.42557 0.43293 0.44485' 0.49771 O.60567 O.70385 0.92615 1.00238 I.03617. APPENDIX IV P 0.5 0.50000 0.63715 0.51843. O.53650 0.62757 0.97988 1.06953 1.46049 1.63314 1.87173 0.53172 O.56871 0.57821 0.61133 O.71626 O.98002 1.12102 1.29407 1.53755 1.85463 0.6 0.60000 0.63715 0.64974 O.70786 0.88784 1.37234 2.01297 2.71278 3.04609 , 2.79277 0.67211 0.74431 O.78070 0.81837 1.03739 1.47787 I.88185 2.25240 3.18722 3.28373 0.7. 0.70000 0.79283 0.86288 0.96359 1.31307 2.30510 3.34897 .4.06366 4.21116 5.22927 0.85272 1.04190 .1.06483 1.21688 I.56569 2.4 2293 4.07435 4.72930 5.22968 6.19331 0.8 0.80000 1.05906 1.22588 1.38123 2.15343 4.72062 5.40883 8.22504 6.23288 10.6126 .••..02341 1.52993 1.62454 1.82870 2.61391 4.61372 5.20162 8 . 6 1 7 3 I 6.91972 9.47219 0.9 0.90000 1.63273 I . 99542 2.59995 4.65116 I I . 0258 10.7971 13.7601 20.2304 20.2308 I . 87272 3.12692 2.95744 3.37091 5.74475 I I . 7321 18.9732 21.3472 20.5841 22.7431 (Continued) H o c 2 c 2 a s o 0.25 0.333 0.5 0.333 l 2 • •• 3 5-6 . o 0.25 0.333 V 0.5 0.5 1 2 3 4 5 6 . 0 , 1 0 . 1 0 0 1 0 0 . 1 0 0 3 0 0 . 1 0 0 3 0 0 . 1 0 0 7 4 0 . 1 0 1 2 8 0 . 1 0 5 1 7 0 . 1 0 7 6 8 0.11424 0 . 1 1 6 4 6 0 . 1 2 5 1 7 0 . 1 0 0 6 6 0 . 1 0 1 2 0 0 . 1 0 1 5 5 0 . 1 0 1 5 2 0 . 1 0 3 0 1 0 . 1 0 6 1 0 0.11146 0 . 1 1 4 8 9 O . 1 2 5 0 9 0 . 1 3 0 7 9 0.2 0.20143 0 . 2 0 3 7 6 0.20453 0 . 2 0 5 7 3 0 . 2 1 2 5 5 0 . 2 3 1 5 3 0 . 2 4 2 5 6 0 . 2 7 4 5 7 O . 2 8 9 8 1 0 . 3 1 0 5 2 0 . 2 0 5 0 5 0 . 2 0 9 0 6 0 . 2 0 9 3 5 0 . 2 1 3 2 7 0 . 2 2 0 7 0 0 . 2 4 1 1 8 0 . 2 6 5 7 2 0 . 2 8 5 4 7 0 . 2 9 8 8 9 0 . 3 2 2 1 6 : 0 . 3 o i 3 0 6 0 7 0 . 3 1 4 6 0 O O I 6 6 3 O J 3 2 5 2 1 0 . 3 4 5 5 8 0 , 4 0 1 9 1 0 . 4 4 8 6 1 0 . 5 2 3 3 3 O . 6 0 9 6 0 0 . 6 2 1 6 6 0 . 3 1 6 4 3 0 . 3 2 7 2 5 0 . 3 3 6 0 5 0.33541 0 . 3 6 4 9 8 0 . 4 2 5 1 0 0 . 4 7 0 9 4 0 . 5 5 9 4 5 0 . 6 8 1 0 1 0 . 6 0 4 9 7 0 . 4 1 9 3 2 0 . 4 4 0 3 4 O . 4 4 7 7 6 0 . 4 , 6 2 3 1 0.51543 0 . 6 4 4 1 3 0 . 7 6 6 0 8 0 . 8 2 2 6 6 0 . 9 7 2 6 3 I . 2 1 8 6 0 0.44242 0 . 4 7 1 4 7 0.47274 0 . 5 0 2 8 8 0 . 5 5 1 9 6 0 . 6 8 3 9 7 0 . 7 8 1 4 0 1 . 0 2 0 7 1 I . . I 6 2 3 8 I . l 6 0 l 6 . APPENDIX IV 0.5 0.54524 0 . 5 8 6 4 1 O . 5 9 8 6 5 0.64446 0.74686 1 . 0 2 1 1 5 1 . 2 1 2 9 8 1 . 3 5 0 4 2 I . 6 7 2 8 5 1 . 7 6 4 4 1 0 . 5 9 1 2 6 0.64001 O . 6 6 7 5 6 0 . 7 0 5 1 9 0 . 8 0 9 0 2 I . 0 8 0 1 3 I . 2 5 2 6 7 1 . 5 9 7 1 8 1 . 6 2 4 8 7 2 . 2 6 7 6 0 0 . 6 0 . 6 9 3 4 3 O . 7 9 2 9 6 0 . 8 0 3 3 9 0 . 8 9 0 4 2 1 . 0 8 8 0 4 1 . 5 8 7 3 4 1 . 8 9 2 2 4 2 . 5 0 7 2 4 2 . 6 7 4 0 3 2.68410 0 . 7 8 4 1 9 0 . 8 6 3 9 4 0 . 9 1 1 9 5 1 . 0 3 0 3 1 1 . 1 9 0 1 0 1 . 7 0 6 5 4 2 . 2 4 0 1 9 2 . 4 7 4 9 6 2 . 5 7 0 6 7 3 . 2 0 6 9 2 0 . 7 0 . 9 0 2 4 5 1 . 0 8 4 8 6 1 . 1 1 8 ^ 7 1 . 2 7 2 8 5 1 . 6 5 0 4 1 2 . 4 3 8 9 0 2 . 9 8 2 5 0 4 . 6 8 5 9 2 5 . 2 3 7 1 2 5 . 7 4 8 5 2 1 . 0 4 2 6 2 1 . 2 1 5 0 0 . 1 . 2 8 5 5 2 1 . 4 0 8 7 4 1 . 8 2 0 4 4 3 . 0 2 8 0 3 . 3 . 7 3 8 6 4 3 . 8 7 3 2 2 5 . 2 3 5 6 6 . 4 . 6 4 3 1 5 6.8 1 . 1 9 6 5 6 1 . 6 2 2 7 1 I . 6 5 8 0 3 1 . 8 3 5 2 8 2 . 7 6 7 5 4 . 4 . 3 ^ 8 2 8 4 . 8 2 7 6 5 8 . 8 9 5 1 1 7 . 0 7 7 0 2 9 . 1 3 2 4 6 1 . 5 3 8 2 9 1 . 8 0 9 4 9 2 . 0 6 1 4 5 2 . 2 8 2 0 5 3 . 0 7 5 1 8 5 . 3 2 4 6 5 5 . 8 3 0 9 1 8 . 8 6 9 6 6 9 . 8 8 2 8 7 1 2 , ^ 5 2 0 ' 0 . 9 2.33076 3 . 0 5 3 3 0 3.46527 4.10961 6.10625 10,1476 12.4942 1 9 . 5 8 0 2 2.17321 2.29763 2.98837 3.60442 4 . 3 5 9 8 8 5 . 0 2 2 6 8 . 6.82946 1 0 . 0 6 4 4 11.9469 2 6 . 9 7 2 9 15.2311 23.0245 (Continued) 0 . 1 ' . / " ' 0 . 1 0 5 5 6 0 . 2 5 0 . 1 0 6 9 5 0 . 3 3 3 0 . 1 0 7 4 1 0 . 5 0 . 1 0 8 3 3 1 0 . 1 1 1 1 1 2 0 . 1 1 6 6 7 3 0 . 1 2 2 2 2 4 0 . 1 2 7 7 8 5 0 . 1 3 3 3 3 6 0 . 1 3 8 8 9 0 0 . 1 0 7 5 7 0 . 2 5 0 . 1 0 8 9 5 0 . 3 3 3 0 . 1 1 1 2 7 0 . 5 0.11049 1 0 . 1 1 5 0 7 2 0.12146 3 0 . 1 2 9 4 8 4 0 . 1 3 3 1 2 5 0 . 1 4 1 7 4 6 0 . 1 5 4 0 7 0 . 2 0 . 3 0 . 2 2 5 0 0 0 . 3 6 4 2 9 0 . 2 3 1 2 5 O . 3 8 O 3 6 0.23333 0 . 3 8 5 7 2 0 . 2 3 7 5 0 0 . 3 9 6 4 3 0 , 2 5 0 0 0 0 . 4 2 8 5 7 0 . 2 7 5 0 0 0 . 4 9 2 8 6 0 . 3 0 0 0 0 0 . 5 5 7 1 4 0 . 3 2 5 0 0 0.02143 . 0 . 3 5 0 0 0 0 . 6 8 5 7 2 0 . 3 7 5 0 0 0 . 7 5 0 0 0 0 . 2 3 6 4 8 0 . 3 9 0 2 9 0 . 2 4 1 8 1 0.42221 0 . 2 4 8 3 7 0 . 4 2 0 8 4 0 . 2 5 4 1 7 0 . 4 3 5 3 5 O . 2 6 9 0 1 0 . 4 8 0 5 1 0 . 3 0 1 9 5 0 . 5 6 3 9 5 0 . 3 5 4 8 3 0 . 6 0 1 7 9 0 . 3 5 2 0 8 0 . 7 3 6 9 4 O . 3 6 3 5 8 0.77948 0 . 3 9 6 1 4 0 . 9 2 3 4 8 0.4 0 . 5 3 3 3 3 O . 5 6 6 6 7 O . 5 7 7 7 8 0 . 6 0 0 0 0 O . 6 6 6 6 7 0 . 8 0 0 0 0 0 . 9 3 3 3 3 1 . 0 6 6 6 7 1 . 2 0 0 0 0 1 . 3 3 3 3 3 O . 6 I O 8 3 0 . 6 3 9 2 5 0 . 6 5 1 7 2 0 . 6 1 8 4 2 0 . 7 8 0 7 8 0 . 9 0 0 4 9 1 . 0 9 1 5 4 1 . 1 4 8 0 7 1 . 3 3 5 0 3 1 . 4 8 9 5 7 APPENDIX IV 0 . 5 0 . 7 5 0 0 0 0 . 8 1 2 5 0 0 . 8 3 3 3 3 O . 8 7 5 0 0 1 . 0 0 0 0 0 1 . 2 5 0 0 0 1 . 5 0 0 0 0 1 . 7 5 0 0 0 2 . 0 0 0 0 0 2 . 2 5 0 0 0 0 . 8 9 8 8 4 0 . 9 6 7 6 4 0 . 9 8 3 1 3 1 . 1 1 0 1 9 1.22247 I . 6 1 3 8 5 1 . 7 0 1 7 8 1 . 9 7 2 3 1 2 . 4 2 6 1 4 2 . 8 2 5 1 0 0 . 6 I . 0 5 0 0 0 I . 1 6 2 5 0 1 . 2 0 0 0 0 1 . 2 7 5 0 0 1 . 5 0 0 0 0 1 . 9 5 0 0 0 2 . 4 6 0 0 0 2 . 8 5 0 0 0 3 . 3 0 0 0 0 3 . 7 5 0 0 0 1 . 3 2 6 8 2 1 . 4 2 8 1 9 1 . 5 6 8 5 2 1 . 5 6 7 8 7 1 . 9 2 1 1 7 2.11494 3 . 3 1 5 5 1 3 . 7 3 9 6 6 3 . 3 2 6 7 9 4 . 2 0 1 5 1 0 . 7 1 . 5 1 6 6 7 1 . 7 2 0 8 3 1 . 7 8 8 8 9 1 . 9 2 5 0 0 2 . 3 3 3 3 3 3 . 1 5 0 0 0 3 . 9 6 6 6 7 4 . 7 8 3 3 3 5.6OOOO 6 . 4 1 6 6 7 2 . 2 1 7 6 6 2 . 4 5 8 9 6 2 . 5 5 7 0 2 2 . 7 0 7 8 4 3 . 1 2 1 3 2 ' 3 . 5 4 5 5 9 5 . 5 2 6 9 6 5 . 4 0 1 7 3 6 . 5 3 7 8 8 7 . 3 7 2 5 9 0.8 2.40000 2 . 8 0 0 0 0 ' 2 . 9 3 3 3 3 3 . 2 0 0 0 4 . 0 0 0 0 0 5 . 6 0 0 0 0 7 . 2 0 0 0 0 8 . 8 0 0 0 0 1 0 . 4 0 0 0 1 2 . 0 0 0 0 3 - 3 7 3 9 9 4 . 2 7 2 4 5 9 . 2 0 9 9 1 4 . 8 7 8 5 5 5 . 5 7 6 0 4 7 . 6 4 9 3 0 6 . 0 5 8 7 6 9 . 8 7 4 3 0 1 0 . 4 9 6 4 9 . 6 6 2 5 3 0.9 4 . 9 5 0 0 0 5 . 9 6 2 5 0 6 . 3 0 0 0 0 6 . 9 7 5 0 0 9 . 0 0 0 0 0 1 3 . 0 5 0 0 1 7 . 1 0 0 0 2 1 . 1 5 0 0 2 5 . 2 0 0 0 2 9 . 2 5 0 0 1 0 . 2 1 4 7 9 . 6 0 6 3 1 1 1 . 0 3 0 9 9 . 6 5 1 5 3 1 3 . 0 3 5 6 1 9 . 0 2 2 6 2 0 . 1 1 6 0 2 3 . 2 0 6 6 2 3 . 2 6 7 5 4 4 . 0 1 7 4 (Continued) 0 . 1 0 0 . 1 0 8 5 5 0 . 2 5 0 . 1 1 1 2 8 0.333 0.11087 0 . 5 0.11325 .1 0.11727 2 .0.12453 3 0.13425 4 0.13857 5 0.14537 6 0.15013 0 0 . 1 0 8 6 3 0.25 0 . 1 1 1 6 5 0.333 0.11438 0.5 0.11357 1 0 . 1 1 8 6 6 2 0.12743 3 0.13452 4 0.14352 5 0.16333 6 0 . 1 5 0 5 0 0.2 0 . 3 0.24247 0.41111 0 . 2 5 3 7 1 0 . 4 4 5 1 7 0 . 2 5 3 9 4 0 . 4 5 4 1 6 0 . 2 6 4 3 7 0 . 4 6 6 7 1 0 . 2 8 0 7 8 0 . 5 1 6 . 5 4 0 . 3 2 6 1 9 0 . 6 2 0 0 3 0.34646 0 . 6 4 9 0 2 0 . 3 3 1 9 5 0 . 8 9 8 7 2 0 . 4 2 1 3 1 0 . 8 1 5 4 1 0 . 4 0 3 9 0 0 . 8 1 6 2 7 0 . 2 4 7 1 5 0 . 4 2 9 9 6 0 . 2 5 2 5 9 0 . 4 5 5 5 7 0 . 2 5 7 3 6 0 . 4 7 4 3 6 0 . 2 6 9 5 5 0 . 4 9 5 1 0 0 . 2 8 8 8 4 0 . 5 4 3 3 4 0 . 3 2 2 9 4 0 . 6 4 8 2 5 0 . 3 5 7 0 4 0 . 7 6 6 7 7 0 . 4 1 2 9 9 0 . 8 2 3 2 8 0 . 4 1 5 2 7 0 . 8 7 2 8 7 0 . 4 8 7 3 8 1 . 0 4 9 1 5 0 . 4 0 . 6 7 6 4 6 0 . 6 9 2 2 7 0 . 7 1 2 2 7 0 . 6 7 1 3 3 0 . 8 6 8 5 2 0 . 9 7 6 4 5 1 . 2 6 6 6 9 . 1 . 4 7 6 7 4 1 . 4 6 7 8 7 1 . 6 8 9 9 2 0 . 7 2 6 0 3 0 . 7 4 6 5 5 0 . 7 7 5 4 4 0 . 8 2 4 8 9 0 . 9 3 9 9 0 1 . 0 9 6 9 4 1.24433 1 . 3 9 3 3 9 1 . 6 7 8 0 7 1 . 6 1 5 0 2 APPENDIX 0.6 0.7 I . 6 1 5 8 2 2 . 8 1 6 0 6 1 . 9 0 4 5 5 3.03737 1.97946 3.17864 2.09178 3.15166 2 . 3 0 2 7 7 3.87192 2.82192 5 . 0 7 1 8 7 3.41783 5.46346 3 . 6 7 1 5 6 6 . 9 8 8 3 5 4.88489 6.44592 4.24825 8.77335 0 . 5 1 . 0 3 7 3 7 1 . 1 4 7 3 8 1 . 1 9 1 2 4 1 . 1 8 9 5 3 1.41421 1 . 7 6 8 2 4 2.10237 2 . 2 6 9 1 1 3 . 0 0 5 0 9 3 . 2 8 5 4 7 1.12429 1 . 3 4 3 7 4 1 . 2 8 2 4 1 1 . 3 5 8 8 4 1 . 5 8 1 1 4 1 . 8 6 5 0 1 2 . 5 4 8 1 4 2.42111 2 . 4 9 2 0 0 2 . 8 8 7 7 9 2 . 0 5 3 6 3 2 . 0 7 0 0 6 2 . 2 2 1 6 8 2 . 2 2 7 1 6 2 . 6 0 5 9 9 2 . 9 6 8 7 1 3 . 2 3 8 4 2 3 . 6 6 7 8 8 4 . 5 9 0 6 1 5 . 0 5 6 7 6 3 . 3 3 2 5 2 3 . 8 3 7 9 0 3 . 6 6 6 8 9 4 . ~ 3 8 9 0 1 4 . 6 0 1 1 4 5 . 3 5 5 9 7 6 . 1 1 9 5 0 6.66^66 7 . 5 3 2 5 5 8 . 6 9 0 1 1 0 . 8 4 . 9 5 3 9 6 6 . 0 3 6 1 6 4 . 9 7 1 9 3 5 . 7 8 7 8 6 7 . 1 2 3 1 1 9 . 7 0 3 6 4 1 2 . 2 2 2 0 1 0 . 7 0 4 7 1 6 . 6 0 3 0 1 0 . 3 9 7 3 6 . 1 2 2 7 2 7 . I 8 5 3 6 7 . 3 7 6 0 6 1 0 . 7 3 9 9 8 . 6 5 5 3 5 1 1 . 2 4 1 6 9 . 1 6 5 0 4 1 3 . 3 9 8 0 1 0 . 3 1 0 2 1 6 . 1 0 9 6 0 . 9 1 4 . 1 6 7 2 14.2494 1 4 . 9 0 1 9 0 I I . 0 1 5 6 1 7 . 0 5 5 4 2 2 . 7 8 2 2 2 4 . 8 3 7 3 2 1 . 1 8 3 2 2 6 . 3 2 7 8 ' 5 0 . 1 9 8 2 1 3 . 6 4 6 7 1 5 . 5 8 6 2 1 5 . 3 6 5 5 1 4 . 2 9 1 3 2 1 . 0 6 8 0 24.6432 3 0 . 9 0 2 6 39•5400 2 8 . 0 ^ 3 1 2 4 . 2 3 4 8 (Continued) a c2 S 0 . 1 0 0 . 1 0 9 7 4 0 . 2 5 0 . 1 1 2 1 3 0 . 3 3 3 0 . 1 1 2 5 9 0 . 5 0 . 1 1 6 0 2 1 0 . 1 1 9 6 3 2 0 . 1 2 7 7 8 3 0 . 1 4 0 5 9 4 0.14772 5 0 . 1 5 7 1 0 6 0 . 1 5 6 2 1 0 . 2 0 . 2 4 9 7 6 0 . 2 5 9 8 4 0 . 2 6 8 0 7 0 . 2 7 4 7 1 O . 2 9 4 7 2 7 O . 3 1 8 6 0 0 . 3 6 3 1 5 0 . 4 2 5 3 8 0 . 4 2 1 8 4 0 . 4 5 8 4 8 0 . 3 0 . 4 5 4 6 9 O . 4 7 3 5 0 0 . 4 7 8 6 1 . 0 . 5 1 0 1 0 0 . 5 6 4 2 7 O . 6 6 3 9 8 0 . 7 3 8 9 4 0 . 8 1 7 5 9 ' O . 8 3 8 7 1 1 . 0 0 9 2 7 0 . 4 0 . 7 4 1 9 8 0 . 8 4 7 6 5 ' 0 . 8 2 7 4 1 O . 8 9 6 7 5 1 . 0 0 0 0 0 1 . 2 7 5 6 6 1 . 3 4 6 9 5 1.64341 I . 6 7 0 6 5 2 . 1 2 4 2 9 0 . 5 1 . 1 8 9 5 9 1 . 3 9 5 2 9 1 . 4 9 7 0 4 1.45440 1 . 7 3 2 0 5 2 . 0 5 4 5 0 2 . 4 4 1 6 8 2 . 3 9 5 3 4 3 . 0 7 5 5 2 3 . 7 7 1 7 8 0 . 6 2 . 3 4 0 4 7 2 . 2 8 0 6 1 2 . 4 2 4 6 1 2 . 3 8 9 5 2 3 . 0 0 0 0 3 . 3 6 5 3 8 3 . 9 0 6 7 2 4 . 2 9 7 7 5 4 . 1 6 7 4 2 5 . 0 5 7 3 6 0 . 7 4 . 1 5 9 8 8 5 . 2 5 8 2 8 4 . 3 0 6 0 5 3 4 . 8 8 1 1 0 5 . 3 1 6 6 2 5 . 7 2 1 0 2 6 . 9 2 7 4 7 9.47411 1 2 . 4 9 5 8 9 . 7 9 0 2 0 0 . 8 9 . 9 5 1 2 9 9 . 4 2 3 7 2 1 0 . 1 0 9 9 11.411 I O . 1 7 8 9 1 0 . 7 0 5 1 1 1 . 9 5 3 8 1 2 . 6 6 3 9 1 2 . 5 8 2 1 2 6 . 0 7 9 9 0 . 9 1 7 . 7 0 3 8 1 4 . 6 5 0 8 1 9 . 8 9 4 8 2 3 . 1 1 9 5 2 5 . 0 7 6 7 21.0272 30.1044 2 7 . 8 2 3 6 3 4 . 4 9 6 6 3 1 . 3 9 1 1 0 0 . 1 1 0 0 6 0 . 2 5 0 . 1 1 3 3 7 0 . 3 3 3 0.11484 0 . 5 0 . 1 1 6 1 1 1 0 . 1 2 0 3 5 2 0 . 1 3 0 3 5 3 O . 1 3 5 9 2 4 0.14544 5 '. 0 . 1 5 3 1 2 6 0 . 1 6 9 5 3 0 . 2 5 2 1 2 0 . 4 5 9 1 5 0 . 2 6 4 0 9 0 . 4 7 7 1 7 0 . 2 6 8 2 5 0 . 4 9 9 7 9 0 . 2 7 6 1 6 0 . 5 0 7 4 0 0 . 2 9 9 2 3 0 . 5 8 1 1 4 0 . 3 4 0 1 5 0 . 6 9 2 2 1 0 . 3 3 4 9 1 0 . 8 2 1 5 7 0 . 4 4 4 8 1 0 . 9 9 0 5 4 0 . 4 3 6 3 3 O . 9 8 6 9 6 0 . 4 8 8 0 6 0 . 9 1 9 2 4 0 . 7 6 3 7 3 1 . 3 8 4 5 3 0 . 8 9 5 2 7 1 . 4 6 1 6 1 0 . 8 4 8 7 1 1 . 5 8 2 2 0 0 . 8 8 3 5 9 I . 6 9 4 3 9 q 1 . 0 5 1 7 9 I . 8 7 0 8 3 1 . 3 4 7 5 7 2 . 0 3 0 0 4 1 . 4 1 9 4 2 3 . 1 7 1 3 8 I . 7 0 7 0 6 2 . 4 0 0 2 9 1 . 6 3 3 9 1 2 . 8 4 0 3 3 I . 8 0 8 2 5 3 . 9 1 6 1 4 2 . 5 1 3 3 8 4 . 9 5 0 8 2 2 . 8 7 6 5 7 5 . 4 7 0 0 0 2 . 7 5 6 9 5 4 . 7 7 6 2 0 2 . 3 8 9 5 2 5 . 5 2 5 2 8 3 . 3 2 7 6 8 6 . 0 0 2 2 7 3.44372 6 . 8 0 7 5 0 4 . 5 5 6 0 9 6 . 2 1 8 3 3 4 . 7 9 8 1 1 1 3 . 3 4 4 5 5 . 3 8 7 0 0 1 1 . 0 1 0 3 5 . 3 3 4 4 3 1 0 . 6 4 6 9 1 0 . 9 4 9 4 2 8 . 3 9 1 4 7 . 5 6 9 2 2 2 6 . 1 5 6 1 9 . 6 3 0 9 2 5 0 . 0 9 2 3 1 2 . 1 3 8 2 2 6 . 2 8 1 7 I I . 6 9 6 9 2 9 . 0 8 3 1 12.4843 3 6 . 0 8 4 4 1 4 . 0 3 7 7 3 1 . 4 3 2 3 1 3 . 4 0 9 9 2 7 . 5 8 5 2 1 6 . 7 4 4 3 6 3 . 1 3 3 0 1 9 . 6 6 1 8 2 9 . 4 3 5 5 APPENDIX IV (Continued) o 110. BIBLIOGRAPHY [ I ] /Arrow, K.J., S. Karlin, and H. Scarf, Studies In Applied Probability and Management Science, Stanford University Press, Stanford, California, 1952. [2] Benes, V.E., General Stochastic Processes in the Tneory  of Queues, Addison-Wesley Publishing Co. Inc., Reading. 1963. [3] Bierman, H., L.E. Fouraker, and R.K. Jaedicke, Quantitative  Analysis for Business Decisions, Richard D. Irwin, Inc., Homewood, I l l i n o i s , 1 9 6 1 . [4] Bowman, E.H. and R.B. Fetter, Analysis for Production Management, Richard D. Irwin, Inc.,: Homewood I l l i n o i s , 1 9 6 l . [ 5 ] Buffa, E.S. Model for Production and Operations Management, John Wiley and Sons, Inc., New York, 1966. Edited. ' [6] Buffa, E.S. ' Readings in Production and Operations Manage- ment, John Wiley and Sons, Inc., New York, 1966. [7] Clelland, R.C. et a l , Basic Statistics with Business Applications, John Wiley and Sons, Inc., New York, 1 9 6 6 . [8] Cox, D.R. and ¥.L. Smith, Queues, Metheun and Co. Ltd., London, 1 9 6 5 . [9]. Cruon, R., Edited, Queuing Theory, Recent Developments  and Applicatons, The English Universities Press Ltd., London, 1965. [10] Keeping, E.S. Introduction to S t a t i s t i c a l Inference, D. Van Nostrand Company, Inc., Princeton, N.J., 1962.' [ I I ] Morse, P.M., Queues, Inventories and Maintenance, John Wiley and Sons, Inc., New York, 1 9 6 3 . "~ ~ [12] Naylor, T.H., et a l , Computer Simulation Techniques, John Wiley and Sons, Inc., New York, 1967. 111. [ 1 3 ] Peck, L.G. and R.N. Hazelwood, Finite Queueing Tables, John Wiley and Sons, Inc., New York, 195b". [14] Probhu, N.U., Queues aid Inventories. John Wiley and Sons, Inc., Nexv York, 1965. [ 1 5 ] Saaty, T.L., Elements of Queueing Theory with Applications McGraw H i l l Book Co., Inc., New York, 1961. ~~~~ [ 1 6 ] Takacs, LaJos, Introduction to the Theory of Queues, Oxford University Press, New York, 1962. • 

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