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Effects of capture, power loss factor, and variable transmit power level in multiple-access systems Wong, Victor J. K. (Jack Keung) 1991

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EFFECTS OF CAPTURE, POWER LOSS FACTOR, AND VARIABLE TRANSMIT POWER L E V E L IN MULTIPLE-ACCESS SYSTEMS by VICTOR J . K. WONG B.Sc.(EE), University of Manitoba, 1989 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF ELECTRICAL ENGINEERING  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA August 1991 © Victor J. K. Wong, 1991  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department, or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall riot be allowed without my written permission.  Department of E l e c t r i c a l  Engineering  The University of British Columbia Vancouver, Canada Date  DE-6 (2/88)  Aug 26, 1991  Abstract The problem of data transmission in a slotted ALOHA system with one central base station and a number of mobiles is addressed. Of particular interest is the probability qi of successful packet reception ("capture") when i mobile users transmit in a given time slot. This probability depends on the propagation law, as well as fading, transmit power variations, spatial distribution and capture models. The capture probability in a noisy system is studied and the values of <?•, for large i appear to be independent of the noise level as long as the noise is reasonable (i.e. the normalized noise power is less than one). Conditions under which q-. decreases monotonically with i are derived. Examples are also given to illustrate cases in which qi may not decrease monotonically with i. The effect of the power loss factor, 3, on different spatial distribution and capture models is studied. It is shown that in a noiseless system, q-. increases with 0 for the two capture models considered. A new spatial distribution, referred to as the inverse-distance spatial distribution, is proposed which allows an exact expression for qi to be obtained. A transmit power selection scheme, which decreases with distance over a selected range of distance, is proposed. Simulation results are used to show that under certain conditions, the throughput performance can be greatly improved with such a transmit power law.  ii  Table of Contents Abstract  ii  List of Figures  vi  Acknowledgement  ix  1 Introduction  1  2 Preliminaries  3  2.1  Channel Model  , . 3  2.2  Spatial Distribution of Mobiles  3  2.3  Capture Models  7  2.4  Markovian Model  8  3 Analysis, Numerical and Simulation Results of Capture Effect 3.1  10  Analysis of Capture Probability  10  3.1.1  Capture model 1  10  3.1.2  Capture model 2  15  3.2 Properties of Capture Probability 3.2.1  18  Condition of monotonically decreasing qi for capture model 1 with no noise  3.2.2  3.2.3  19  Cases of monotonically decreasing qi for capture model 2 with no noise  20  Some examples of non-monotonically decreasing qi  21  iii  3.3  Capture Probability and System Throughput  21  3.3.1  Capture model 1  23  3.3.2  Capture model 2  25  3.3.2.1 Uniform and bell-shaped spatial distributions  25  3.3.2.2 Inverse-distance spatial distribution  28  3.3.2.3 Uniform received power distribution  28  4 Effect of Power Loss Factor on Capture Environment  32  4.1  A Condition for an Improved Capture Effect  32  4.2  Capture Model 1  34  4.3  Capture Model 2  36  4.3.1  Uniform and bell-shaped spatial distributions  36  4.3.2  Inverse-distance spatial distribution  39  5 Enhancing Capture Effect with Transmit Power Level Selection  43  5.1  A Condition for an Improved Capture Effect  43  5.2  The Proposed Transmit Power Law  46  5.3  Effect of Varying the Parameters on the Proposed Transmit Power Law . . . 47 5.3.1  Effect of varying A  48  5.3.2  Effect of varying d  49  6 Conclusions and Future Work  55  max  iv  Bibliography  57  Appendices  60  A Inverse-distance Spatial Distribution and Capture Model 2  60  A.l  Monotonic Decrease of qi with x  60  A. 2 Monotonic Decrease of qi with i  61  B Bell-shaped Spatial Distribution and Capture Model 2 B. l  Derivation of qi  62 62  B.2 Monotonic Decrease of qi with i  63  C The Proposed Transmit Power Law  65  V  List  o f Figures  2.1  Spatial distributions: the uniform spatial density for p = 0 and r = 1, the bell-shaped spatial density for p = 0 and the inverse-distance spatial density for p= \ and v = 2 6  2.2  Corresponding received power p.d.f. curves of Figure 2.1 with 0 = 4 and the uniform received power p.d.f. for 7min 0 &nd 7 m a i 10  6  2.3  State transition of a mobile  8  3.1  Capture probabilities, q\, q2 and £73, as functions of normalized noise power 77 for the uniform spatial distribution and capture model 1 with 8 = 4., c = 4 , p = 0 and r =l . 22  max  =  =  m a i  3.2  Capture probabilities, qi, 92 and 93, as functions of normalized noise power 77 for the bell-shaped spatial distribution and capture model 2 with 0 = 4, c=4 and p = 0 22  3.3  Capture probability for capture model 1: the uniform and bell spatial distributions with 8 = 4, c = 4 and p = 0; the inverse-distance spatial distribution with 0 = 4, c = 4 u = 2 and v = 1.1; the uniform received power distribution with c = 4 and 7 , „ = 0 24 m  3.4  Corresponding throughput curves of Figure 3.3  24  3.5  Capture probability for the uniform spatial distribution and capture model 2 with 8 = 4, c=4, p = 0 and r = 1 at different noise levels 26 max  3.6  Corresponding throughput curves of Figure 3.5  3.7  Capture probability for the bell-shaped spatial distribution and capture model 2 with 8 = 4, c = 4 and p = 0 at different noise levels 27  3.8  Corresponding throughput curves of Figure 3.7  27  3.9  Capture probability for the inverse-distance spatial distribution and capture model 2 with 0 = 4, c=4, n = 0 and different values of v  29  3.10 Corresponding throughput curves of Figure 3.9  vi  26  29  3.11 Capture probability for the inverse-distance spatial distribution and capture model 2 with 8 = 4, c=4, /> = 0.1 and v = l.l at different noise levels  30  3.12 Capture probability for the inverse-distance spatial distribution and capture model 2 with 8 = 4, c=4,77 = 0.001,1/= 1.1 and different values of p  30  3.13 Capture probability for the uniform received power distribution and capture model 2 with c=4, 7mm = 0 and 77 = 0  31  3.14 Corresponding throughput curve of Figure 3.13  31  4.1  . 4.2  4.3  4.4  4.5  Capture probability for the uniform spatial distribution and capture model 1 with c = 4, p = 0 and different values of 8  36  Capture probability for the bell-shaped spatial distribution and capture model 1 with c = 4, p = 0 and different values of 8 36 Capture probability for the inverse-distance spatial distribution and capture model 1 with c=4, i/ = 2 and different values of 8  37  Capture probability for the inverse-distance spatial distribution and capture model 1 with c = 4, u = l,l and different values of 8  37  Capture probability for the uniform spatial distribution and capture model 2 with c—4, p = 0, r = 1 and different values of 8  38  max  4.6  Capture probability for the bell-shaped spatial distribution and capture model 2 with c = 4, p = 0 and different values of 8 38  4.7  Capture probability for the inverse-distance spatial distribution and capture model 2 with c=4, u = 2 and different values of 8  4.8  5.1  Capture probability for the inverse-distance spatial distribution and capture model 2 with c=4, i/ = l . l and different values of 8  39  The proposed transmit power law as function of distance, r, for P -from = 0.01, d = 1, d i = 0 and different values of A  46  max  max  5.2  m  n  = 1»  Capture probability for the bell-shaped spatial distribution and capture model 2 with 8 = 4, c=4, /9 = 0, r/ = 0.001, d = 0, d = 1 and different values of A. 47 mtn  5.3  39  Corresponding throughput curves of Figure 5.2 vii  max  47  5.4  Capture probability for the bell-shaped spatial distribution and capture model 2 with 8 = 4, c = 4, p = 0,77 = 0.1, d i = 0, d = 1 and different values of A.. 48 m  n  max  5.5  Corresponding throughput curves of Figure 5.4  5.6  Capture probability for the bell-shaped spatial distribution and capture model 2 with 0 = 4, c = 4, p = 0,77 = 0.001, A = 5, d ,„ = 0 and different values of d . 50 m  48  max  5.7  Corresponding throughput curves of Figure 5.6  5.8  Capture probability for the bell-shaped spatial distribution and capture model 2 with 0 = 4, c = 4, p = 0,77 = 1, A = 5, <i , = 0 and different values of d . . . 51 m n  50  max  5.9  Corresponding throughput curves of Figure 5.8  51  B.l  Illustration of (B.8, B.9)  61  viii  Acknowledgement I would like to express sincere gratitude to my research supervisor, Dr. Cyril Leung, who has provided me with the research topic as well as many helpful suggestions and constant supervision. I would also like to thank my parents for their encouragement and Mr. William Cheung for his help in using Publisher™  to prepare this thesis.  This research was partially supported by a University of British Columbia graduate fellowship and NSERC grant OGP0001731.  ix  Chapter 1 Introduction Since the ALOHA random access concept wasfirstproposed by Abramson [1], it has been widely studied [2-23]. It can be applied to the problem encountered by a number of mobile users attempting to transmit data packets over a commonly shared (slotted) channel to a central base station. For bursty data traffic, random access schemes are more efficient than schemes in which the channel is split into dedicated sub-channels, e.g. TDMA or FDMA. However, random access schemes give rise to the possibility of a collision when several packets are transmitted at about the same time. If the packets are received with more or less the same powers, it is reasonable to assume that they will all be destroyed [1-6]. However, this assumption is somewhat pessimistic in a mobile radio environment. The performance improvement brought about by the fact that the mobiles may be at different distances from the base station (and hence their respective packets received with different power levels due to attenuation) has been discussed in [7-10]. This has been referred to as the "near-far" effect. The effects of fading and transmit power variations in the ALOHA system has also been studied [11-14]. Of particular interest is the probability q\ of successful packet reception ("capture") when i mobile users transmit in a given time slot. This probability depends on the attenuation of the transmitted signals with distance, i.e. the propagation law, as well as fading, transmit power variations, capture and spatial distribution models. This thesis is mainly concerned with the effect of capture in slotted ALOHA systems although the ide&S and techniques used could also be applied to other random access systems. It is organized as follows: in Chapter 2, the propagation model, mobile spatial and received power distributions and capture models to be used in subsequent chapters  l  Chapter 1. Introduction  2  are described. The Markovian model for evaluating the throughput of a finite population slotted ALOHA system is also discussed. In Chapter 3, expressions for the capture probability, qi, are derived for a number of different spatial distribution and capture models. Conditions under which q. decreases monotonically with i are obtained and some examples of non-monotonically decreasing <7i are given. Finally, plots of capture probability and system throughput for different spatial distribution and capture models are shown. Chapter 4 examines the effect of the power loss factor on capture effect for the uniform, the bell-shaped and the inverse-distance spatial densities under two capture models. In general, when noise is negligible, a larger power loss factor results in an increased capture probability. In Chapter 5, a scheme in which the transmit power level of a mobile is chosen according to its distance from the base station is proposed. The proposed transmit power law, which is a decreasing function of distance over a selected range of distance, has an effect similar to that of increasing the power loss factor. Several properties of the proposed transmit power law are studied. The main results of the thesis are summarized in Chapter 6. A number of topics suitable for further study are also outlined.  Chapter 2 Preliminaries In this chapter, a number of models and assumptions which will be used in subsequent chapters are discussed. First, the channel model offlatterrestrial propagation is described briefly. Next, three spatial distributions and one received power distribution of mobiles as well as two capture models are presented. Finally, the Markovian model for evaluating the throughput of a finite population slotted ALOHA system is discussed.  2.1 Channel Model We use a propagation model [24] in which the normalized received signal power T varies with the distance r between the transmitter and the receiver as (2.1) where 8 is the power loss factor. For mobile radio systems, a typical value of 8 is 4. The received signal may contain a noise component with normalized power 77 so that the received signal-to-noise ratio corresponding to a mobile at unit distance from the base station is ^.  2.2 Spatial Distribution of Mobiles Let the spatial density of data traffic from mobiles around the base station be denoted by G(r), 0 < r < 0 0 . The total channel traffic, measured in packets per time slot, is then given by OO  (2.2) 0  3  Chapter 2. Preliminaries  4  It is assumed that the locations of different mobiles are statistically independent and the spatial traffic distributions of the mobiles are identical. Three examples of G(r) will be considered. The first example of G(r) is a uniform spatial traffic density G (r) [9] over a u  punctured circle centered at the base station, i.e., (2.3) otherwise. In (2.3), the traffic on the channel has been normalized to unity. The corresponding probability density function (p.df.) of the distance R between a mobile and the base station is  {  ,i  2  _  r  D  P< r < r  2 ,  max  (2.4) 0, otherwise. The p.d.f. of the received power T can be derived from (2.1) and (2.4) and is given by  -fi  2  Mt)  = {  m a X  0,  fi  <  ~  1  ~  (2.5)  P  otherwise.  The second example of G(r) is a bell-shaped spatial traffic density Gi(r) [11] centered at the base station, i.e.,  0,  otherwise  oo  J t~ dt is the complementary error function. In this case, the  where erfc (z) =  x  z  corresponding p.d.f. of the distance R between a mobile and the base station is I f (r) R  =  {  2  r  ^  0,  / R  1  i ,  V  ?  p < r <  "  oo  otherwise.  .  (2.7)  5  Chapter 2. Preliminaries  The p.d.f. of the received power T can be derived from (2.1) and (2.7) and is given by / ( ) r  7  =  {  /Jeric(^)  '  - 1  0,  -  I*  (  2  g  )  otherwise.  The third example of G(r) is an inverse-distance spatial traffic density Gi(r) centered at the base station, i.e.,  Gi(r)  =I  (2.9)  „ 0, otherwise where v > 1. The corresponding p.d.f. of the distance R between a mobile and the base station is  ^ -  p—i  iv— 1)  , p < r < oo /ii(r) = < „ ' /? (2.10) 0, otherwise. The p.d.f. of the received power T can be derived from (2.1) and (2.10) and is given by N  r  l i ^ l i u-i p  7  ^ - i  o<  ?  7  < p-9  (2.11)  otherwise.  Another received power p.d.f. of interest is the uniform one given by Kmax  If min  Kmin  5:  7  —  Imax  (2.12) = \ ^ 0, otherwise. As an example, such a p.d.f. results if the mobiles move on a thin ring centered at the Mi)  base station and each mobile transmits with a power chosen randomly according to a uniform distribution. Figure 2.1 shows three spatial distributions: the uniform spatial density for p = 0 and r x = 1, the bell-shaped spatial density for p = 0 and the inverse-distance spatial density ma  for p = \ and v = 2. Figure 2.2 shows the three corresponding received power p.d.f. curves for 8 = 4. and the uniform received power p.d.f. for imin 0 =  i x = 10. ma  Chapter 2. Preliminaries  6  Uniform spatial Bell-shaped Inverse-distance (v=2)  Distance Figure 2.1 Spatial distributions: the uniform spatial density for p=0 and r = 1, the bell-shaped spatial density for p = 0 and the inverse-distance spatial density for p= \ and i/ = 2. mac  0.5  Uniform spatial —  I  -•  1 •1 \1  -a  1  d o  1  0.3  1 -11  a, -a  •  r. N  i \  >  •l\ 0.1  \  \  \ \  S \  \  \  \  \  Ti 1 I  10  15  Distance Figure 2.2 Corresponding received power p.d.f. curves of Figure 2.1 with /?=4 and the uniform received power p.d.f. for 7 „ = 0 and y = 10. m i  max  1  Chapter 2. Preliminaries  2.3 Capture Models Various conditions under which a data packet is assumed to be successfully decoded ("captured") have appeared in the literature [7-21]. The following two models have been commonly used. Let the probability of capture given a set of i contending packets with received powers { 71, 72, • • • , 7i } and normalized noise power 77 be denoted by  Pc (71, 72, • • • , 7«, rj). It is assumed that the received signal powers are more or less constant over a packet duration and that only the packet with the largest received power can be captured [7-23]. Let 7  m  be the largest received signal power.  Then the capture model in [7, 8, 14, 15] corresponds to  {  1,  if 7  0,  otherwise  m  > c max{ 71, • • • , 7 _ i , 7 +i, • • • ,7.} m  m  (2.13) and the capture model in [9, 11, 13] corresponds to  {  1,  if 7  m  > c  lj j=i,&m  0,  (2.14)  otherwise.  In (2.13) and (2.14), the parameter c> 1 is referred to as the capture ratio. In the case where 77 >0, the capture conditions in (2.13) and (2.14) can be rewritten more generally as 7  m  > c max{ 71, • • • , 7 _ i , 7 m  m +  i , • • • , 7,-, 77 }  (2.15)  and  For convenience, we will refer to conditions (2.15) and (2.16) as capture models 1 and 2, respectively. Even though capture model 1 is somewhat unrealistic and leads to optimistic results, its use often allows for simpler analytic derivations.  8  Chapter 2. Preliminaries  2.4 Markovian Model For the different capture and spatial distribution models in Sections 2.2 and 2.3, the Markovian Model described in [9] will be used to evaluate the throughput for a finite population slotted ALOHA system. Consider a slotted ALOHA system with N mobiles. As shown in Figure 2.3, a mobile can be in one of three states, namely, the origination state (O), the transmission state (T) and the retransmission state (R). At the beginning of each time slot, a mobile is in either 0-state or J?-state and the transitional probabilities to T-state are p and p , respectively. 0  r  At the end of a time slot, if the packet sent by a mobile (in the T-state) is successfully received, the mobile returns to O- state; otherwise, it goes to R-state. Such a system can be described by a homogeneous Markov chain with N+l states corresponding to the number of mobiles in the retransmission state at the end of a time slot. The throughput of a slotted ALOHA system, defined as the average number of  *- time  Figure 2.3 State transition of a mobile.  Chapter 2. Preliminaries  9  successful transmissions per time slot, can be evaluated as in [9]. To evaluate the system throughput for given values of p , p and qi, 1 ^ i £ N, the state transition probabilities 0  r  are calculated first. From the state transition probabilities, the steady state probabilities are derived and then the expected throughput of the system in each state can be obtained. The system throughput is simply the overall mean of the state throughputs.  Chapter 3 Analysis, Numerical and Simulation Results of Capture Effect This chapter analyzes the capture probability, qu for a number of different spatial distribution and capture models. Conditions under which qi decreases monotonically with i are obtained and examples of non-monotonically decreasing qi are provided. Finally, plots of capture probability and system throughput for different spatial distributions and capture models are shown.  3.1 Analysis of Capture Probability In this section, the capture probabilities, qi, for a number of different spatial distributions on two capture models are analyzed. In some cases, only numerical or simulation results will be provided because of the difficulty in obtaining a closed-form expression for qi. Let the p.d.f. and the cumulative density function (c.d.f.) of the received signal power, T, be denoted by /r(7) and FY{~I), respectively. Assume the corresponding smallest and largest possible values of T are 7  m m  and 7  m a i  . Suppose the i contending  packets have received signal powers {71, 72, • • • , 7, }. For convenience, it is assumed that 71 is the largest received signal power.  3.1.1 Capture model 1 From (2.15), q\ can be expressed as  = 1 10  F (cq). T  (3.1)  11  Chapter 3. Analysis, Numerical and Simulation Results of Capture Effect  From (2.15), q,, i>2, can be expressed as 21.  qi  =i  j  J  frAti)  /r (72) ••• J  '2 dy fri(li)d"fi ••• dy.  2  x  7m i n  cmax(-y,„,t/) m  i-l  2L  Kmax  dyi  = I fmin  emax(7i„,7/) m  = i  7moi  J  (3.2)  /r (7i)[Fr (^-)] " d7i. ,  1  1  J  cmax(7 ,»7) m n t  Note that if n > Zf*, qi = 0. For the uniform spatial distribution,  (0,  7 — max r  (3.3)  rmax < 1 < p  7 >p-e.  If T ? <  = 1. F o r ^ < r ? <  it can be shown from (3.1) and (3.3) that _  91  (ci/)~* -  ?  —2  2  _  2  (3.4) '  f  ''max  If 77 < rmax, qi, i > 2, can be obtained from (2.5), (3.2) and (3.3) as <*7i  cr" -  » - i  *"* max2*  2 mar  - *  2  v  dy.  -(7)  y  *'* max  Letting x = 7 0 in (3.5), we have  cr p  Qi =  2(1-1)  i c  0  /"  i - l  -2. 2  maI  c  r  ^ max ~ r  x  (3.5)  Chapter 3. Analysis, Numerical and Simulation Results of Capture Effect  — •* 2  2(«-l)  V max  r  <T* f  =  C  "  U  /  max  f  i-lj x  Q  7 ~_ ^ c  max  ax  | .  2  (3.6)  P  r  For any given 8 and c,  2  l*T -p  H )  12  depends on the ratio of r , ^ to p. Let < = —  ,  P  (3.7)  then  _ 2.  It follows that qi = c " for any r a  If  r ax < m  —0  V < -^-,  Qi, i >  £  m  a  i  > 0 and p = 0.  2, can be obtained from (2.5), (3.2) and (3.3) as  qi = i J hAli) [ r ( ^ ) ] ' ^ 7 i -  (3-9)  i ?  J  Using a similar derivation to the case of rj < _ 2. C ^  9» =  I'mai  it can be shown that  r a , m  X  (^L,-cV) '-(^L,-^) •]. ,  f  7  ,  (3-10)  J  For the bell-shaped spatial distribution and 0 < 7 < p  the c.d.f. of the received  power is given by 7  F() r  7  =  ] _  N  / x-(  1 +  ^)e-T  I _  ^x.  (3.11)  /3erfc(^) 7 _ 2_  Letting ?/ = x " in (3.11) yields 00  v  =  ' 1p  \ . erfc(^ry> } J  2  (3.12)  13  Chapter 3. Analysis, Numerical and Simulation Results of Capture Effect  For 7 > p-P, F (7) = 1. r  Note that for r)>^-,  q = 0. If T? <  we can use (2.8), (3.2), (3.1) and (3.12)  {  to obtain for i > 1, »-i  2i q i  =  7  >  r f  <^ )]"' 2  /  erfc  ^  CT/  V  2 */ .  J . 7  (3.13)  7  Substituting a; = 7 0 in (3.13) yields i-l  e «  9: = *  erfc  x  (ir *) ci  da:  (3.14)  If T) = 0 and /o = 0, (3.14) reduces to the result given in [20, 21], i.e., t-i  * - '7  da:.  erfc  (3.15)  0  In this case, }™ qi = c~0. But when p > 0, then ,11™ 9, = 0 [20, 21]. 0  For the inverse-distance spatial distribution and 0 < 7 < p~^, the c.d.f.  of the  received power is given by F (r)  (3.16)  = (pi*)"' . 1  T  Note that for 77 > ^ - , 9, = 0. If 77 <  we can use (2.11), (3.2) and (3.16) to obtain  e  for i > 1, <li  J  i{v-\)p^  1_  C  C7J  t ( t / - l)/?'^-/) 7 1  («-D(i-i)  e  v "  c  ^  07  c»?  ,-«("-!) C  0*7  3  '(«-») _  (crj)  0  (3.17)  Chapter 3. Analysis, Numerical and Simulation Results of Capture Effect  If 77 = 0 , qi is independent of p and equals c  14  "  For the uniform received power distribution,  If  77 <  2iEia.,  0 )  7  <  V. 1 ?  7  -  t7i = 1. For  < 77  q  i  Imax-CTJ  =  ^  7max  If  V<lmin>  Qi,  Umax •  >  it can be shown from (3.1) and (3.18) that  2SM=.,  <  Imin  (  3  1  9  )  Imin  i>2, can be obtained from (2.12), (3.2) and (3.18) as  q i  =  _  _  /  (7max  7m«'n)  7  J  ,  n  J  f f y  Chimin Imar  . —Tmtn  /  c  f  I C  (7max  m  C  7min)  •_!  7  , _ 1  J7  0  =  \  If 7mm < n <  2^!L ) .  ( _s  c  Umax  7min  3.20)  (  /  then for i > 2, Tmax  -1 q i  =  :  ?  /  ( * - « ) "  U (7mox  7min)  V  J  (7max  7min)  L  7 - ^ 7  ymin  / 7mox  7 \y  <*7  ^  \*  "imin J  /  \V ~ Imin )  (3.21)  15  Chapter 3. Analysis, Numerical and Simulation Results of Capture Effect  3.1.2 Capture model 2 From (2.16) we can write OO  qi = i  J  j c  hAli)  ft (^-v)dind 1)  (3.22)  11  tmin+V  where / £ ' ^{jn — v), the p.d.f.  of the total interference power, is the (i-l)-fold  convolution of /r(7) and then translated by rj. For the uniform spatial distribution, only simulation results will be provided because of the difficulty in obtaining a closed-form expression for the convolution of frit)For  the bell-shaped spatial distribution, only the case of 8 = 4 has been attempted  analytically. The p.d.f of the interference power, with r/ = 0, is shown in [20, 21] as  erfc for  0 < 7„ < p~ . If 0 < 77 <  (3.23)  i-l  the p.d.f. of the total interference power can be  4  expressed as i-l  erfc for rj<j <p- .  From (2.7), (3.22) and (3.24), q i > 1, can be obtained as  4  n  it  2L  -4  i(i-l)erfc I Letting 7 = (  7 n  - 7/)  -^r—p'  2T_  _3  /  c»7  7I  2  E  4 7 1  /"  (in-rj)  _3 2  tt-(i-l)  2  e *(•»>-'>) dj d-fi . (3.25) n  2 in (3.25) yields 7T(.-1) -, 2  erfc  /  Ct]  7i  -f  e  A - 1 eric I —-—  TT C  71 - CQ  2  dj dji  di 7  (3.26)  Chapter 3. Analysis, Numerical and Simulation Results of Capture Effect  Using x = 7j  2  in (3.26), we obtain  erfc  qi = i  16  m\ i  _2L,2  ,  — \  eric I •  e *  \ 2  7T  c  y a ; — cr;  dx.  (3.27)  - 2  It is shown in Appendix B.l that for p = 0 and 77 = 0, (3.27) reduces to the result given in [20, 21], i.e. for t > 1,  2t  qi = — tan  ,  (3.28)  L(«-i)VcJ  7T  For the inverse-distance spatial distribution, let the Laplace transform of /r(7) be denoted by #(s).  Then  =  (v - DP P "  1  J  Or )  - 1  r  1  («-o  5  /»  (3.29)  ,  where T(z) = / i e tfi,i?e(2r)>0, is the Gamma function. The (j-l)-fold convolution 2_1  °f  fr(l)>  -t  i / r ^(7n)» is simply the inverse Laplace transform of [ff(s) ]'t -. 1  A  7i  4i" (7 ) = 1}  i _ 1 ) 1  " , 1  0 < n < 7  n  (3.30)  otherwise  0, where x = v  —1  (3.31)  The p.d.f. of the total interference power, with n > 0, is  r[(i-i)*1 k  0,  otherwise.  (3.32)  Chapter 3. Analysis, Numerical and Simulation Results of Capture Effect  17  Assuming 77 < -^-, we can use (2.11), (3.22) and (3.32) to obtain for i > 1, £  Qi = ikx J  J  (in-vf-V-idlnd-n  n  cn  ct? where (3.34) If 77 = 0, (3.33) reduces to  t- 1  0  i-l (3.35)  (t-i)zr[(t-i)x]'  In this case, qi is independent of p. It is shown in Appendix A . l that qi is a monotonically decreasing function of x, x > 0, and hence, qi is also a monotonically decreasing function of v, v >  1,  and lim . _ lim n  V-y\  H\ — x—0 Hi  lim  _  lim  c-^ix^r  1  (i-l)a?r[(i-l)x] [xT^)]'"  1  («-l)ir[(i-l)i] l  i  m  ^ ° = 1.  [ r ^ + i)]'-  1  r[(i-i)x +i ] (3.36)  18  Chapter 3. Analysis, Numerical and Simulation Results of Capture Effect  For the uniform received power distribution, the p.d.f. of the total interference power is the (i — l)-fold convolution of the p.d.f. of (2.12) and then translated by 77. Using the Laplace transform of the unit step function with a delay of jmin, it can be shown that  /(-^(^ _ ) v  ~ * '~  [ 7 w  7  =  \imax  for i > 2 and  7  Z  < 7„ < 7  m  a  x  ]  (3.37)  2  immJ  ^«  ^) •  where  •J = ( i - l ) 7mm + tj.  (3.38)  x  For  7„ < 7  X  or 7 „ > (i — 1) 7  necessary to know / r  n  - 1 )  m a  (7n -  a;+r/,  / f ' ~ ^ ( 7 n — TJ) = n  v) for 7 m a i  <7n <  0. In the calculation of qu it is not  (i - 1) 7 m a * + 7 ? . If  7max >  cj , using x  (2.12), (3.22) and (3.37), qi, i > 1, can be expressed as 2L  "(max  •„  c  *' (T».-W / y =  Tmax  (imax  ~c~ "fx  7 /  /  7mm) ^  «J 'max  (7mox — 7mtn) (*  (i-l)!  /max C  Note that for a given value of c and 77 =  TT^Hnd-n ) '•  «-l  1) !  —  -  7max  ^  2  0  CTx  ^  (.--2)i  7x  »  (3.39)  7n 0,  qi only depends on the ratio of  7 ^  to  7 ,„. OT  3.2 Properties of Capture Probability In this section, it is shown that for a number of commonly used models, qi decreases monotonically with i. A number of examples in which qi does not decrease monotonically with i are also given.  Chapter 3. Analysis, Numerical and Simulation Results of Capture Effect  19  3.2.1 Condition of monotonically decreasing g- for capture model 1 with no noise t  A sufficient condition to ensure that qi decreases monotonically with / for capture model 1 is as follows:  Theorem 3.1: For capture model  1 with n = 0, a sufficient condition for qi to decrease  monotonically with i is that M l) /r(7) c  A  9(1)  =  (3.40)  decreases monotonically with 7 over the range of 7 for which the received power p.d.f., /r(7), is non-zero.  Proof:  From (2.15) we have i-l  2L  qi = i J Mn) 0  J /r (7i)^7> 0  dyi  J  00  = i J Ml) 0  [ r(£)] ~ <*7 i r  ,  1  00  Mci)[F (l)r dy.  = icj  (3.41)  l  T  0 Let 70 be such that ^(70) = 7^5-. Then from (3.41) we can write 00  q i  -  q  i  +  1  = c J  M n )  [t -  (* + 1 ) ^ ( 7 ) ] [ J F H T ) ! ' ' -  0  J / r ( c ) [t 7o  = c  7  (i + l ) F ( ) ] r  7  0  [Fr( )r 7  1  i  d  1 dy  00  + cJ  / ( c ) [t - (i + l)F (y)} [FrCT)]'"" dy. 1  r  7  T  (3.42)  7o  For 7 G [0, 70], the term [z-(i-f l)Fr(7)] is non-negative and fr(cy) > g{io) fr(l)For ye [70, 00), the term  + 1) JFT(7)] is non-positive and fr{cy) < g(ya)Ml)-  20  Chapter 3. Analysis, Numerical and Simulation Results of Capture Effect  Therefore, from (3.42) we have oo  qi-qi+i  > cg( ) 10  J  / ( ) [t - (t + 1) F ( ) ] [F (l) r  7  r  7  oo  J  oo  / r ( 7 ) [ ^ r ( 7 ) r ^ 7 - ( i + l)  0 oo  = cg( ) 10  i  /  <*7  T  J Mi)[ /T(7)NT  0 oo  I [[FTWrUFrM-ii + l) I [/r(7)N^r(7) ^ r ( 7 ) r ^ r ( 7 ) - ( i + l) J [FrWUFrb) _ 1  -(3-43)  L 0 0 Since each of the two terms in the square bracket in (3.43) is equal to 1, it follows that qi > qi+i-  Q.E.D.  It can be easily shown that all four received power p.d.f.'s defined in Section 2.2 satisfy the sufficient condition of Theorem 3.1. Hence in these cases qi decreases monotonically with /.  3.2.2 Cases of monotonically decreasing qi for capture model 2 with no noise Here, we consider three cases for capture model 2 with TJ = 0 for which qi decreases monotonically with i. For the bell-shaped spatial distribution with B = 4, it is proved in Appendix B.2 that qi, as given by (3.28), decreases monotonically with /. For the inverse-distance spatial distribution, it is proven in Appendix A.2 that q, given by (3.35) decreases monotonically with i. For the uniform received power distribution, qi given by (3.39) reduces to qi =  (i-l)!  [  ¥  " Imax  for 7? = 0. Clearly, q decreases with / for ^ t  (»-l)7«  max  >(i-  (3.44)  7m in  l)c7  qi; = 0. Therefore, qi decreases monotonically with i.  mm  . If ^  max  < (i - l ) c 7  mm  ,  21  Chapter 3. Analysis, Numerical and Simulation Results of Capture Effect  3.2.3 Some examples of non-monotonically decreasing qi In this section, three examples are given in which qi does not decrease monotonically with i. For capture model 1 and the uniform spatial distribution, as shown in (3.4) and (3.10), the normalized noise power rj affects q\ only if r\ > f^ and qi, i>2, only if rj > la  ra. m  X  For rj > r ax, p = 0 and i > 1, we have m  (3.45) Hence,  is an increasing function of i for rj > r  p  m ax  . Figure 3.1 shows a plot of q\, q  2  and q-i as functions of rj with 8 = 4, c = 4, p = 0 and r  max  = 1.  For capture model 2, the bell-shaped spatial distribution with 8 = 4, the expression for qi is given by (3.27). Figure 3.2 shows q\, q and q-i as functions of rj with 8 = 4, 2  c = 4 and p = 0. It can be seen that for rj > 2.1, q\ < qi < 03. For our last example, consider a system in which the mobiles move on one of two thin rings centered at the base station with radii r\ and r2. Each mobile is on the ri-ring with probability p and the r2-ring with probability (1 — p). Let the corresponding received powers be T i and T2, respectively. If j^. > 2c, then for both capture models 1 and 2, qi = 1. 92 = 2p(l - p) and 03 = 3p(l - p) . Clearly, q > q for p<\. 2  3  2  3.3 Capture Probability and System Throughput In this section, the capture probability and system throughput for capture models 1 and 2 are plotted. The throughput is evaluated using the Markovian Model described in Section 2.4. The number of mobiles considered in the Markovian Model is 50 and p = 10p where p is the probability of packet origination and p is the probability of r  o  0  packet retransmission.  r  Chapter 3. Analysis, Numerical and Simulation Results of Capture Effect  oj I 1  1  •  1  • *• ' • • i • • • • 2 3 4  • • • • 5  1  1  1  • • • i • • • • 6 7  1  • • • • 8  1  • • • • * 9  1  1  1  • 10  Normalized noise power Figure 3.1 Capture probabilities, c/i, q and c/ , as functions of normalized noise power 77 for the uniform spatial distribution and capture model 1 with /? = 4,c = 4,p=0 and r = 1. 2  3  max  0  1  2  3  4  5  6  7  8  9  10  Normalized noise power Figure 3.2 Capture probabilities, 91, q and 93, as functions of normalized noise power r? for the bell-shaped spatial distribution and capture model 2 with /? = 4, c = 4 and p=0. 2  Chapter 3. Analysis, Numerical and Simulation Results of Capture Effect  23  3.3.1 Capture model 1 Here, we provide plots for capture model 1 with different spatial distributions. For simplicity, we assume that noise is negligible in this section. For the uniform spatial distribution, qi = 1 and qi, i > 2, is given by (3.6). If p = 0, _ 2  then qi depends only on 0 and c and qi = c e, i > 2. For the bell-shaped spatial distribution, q-. is given by (3.15) for p = 0.  If the  integration in (3.15) is performed over [0,6], the truncation error, c, can be bounded as shown in [20, 21], namely  6  (3.46) The truncation error is negligible for 6 > 2. For the inverse-distance spatial distribution, qi is obtained by using (3.17). In this case, qi depends only on 0, c and v and q. is independent of p. For the uniform received power distribution, q is given by (3.20). If 7 m t n = 0, qi is x  independent of j x ma  and depends only on c.  Figure 3.3 shows the capture probability for different spatial distributions. For the uniform and bell spatial distributions, the following parameters are used: 0 = 4,0=4 and p = 0. Note that the asymptotic value of qi is the same for the uniform and bell-shaped spatial distributions. Two curves with 0 = 4 and c = 4 are plotted for the inverse-distance spatial distribution corresponding to v = 2 and v = 1.1. For the uniform received power distribution, c=4 and 7 m t n = 0. Figure 3.4 shows the corresponding throughput curves.  Chapter 3. Analysis, Numerical and Simulation Results of Capture Effect  Uniform spatial Bell-shaped Inverse-distance (v=2) Inverse-distance (v=l.l) Uniform received power JO  2  I  0 6  '  iU  10  15  20  25  Number of contending transmitters Figure 3.3 Capture probability for capture model 1: the uniform and bell spatial distributions with /? = 4, c = 4 and p = 0; the inverse-distance spatial distribution with /? = 4,c = 4 i / = 2 and v = 1.1; the uniform received power distribution with c = 4 and 7 , „ = 0. m  Probability of packet origination Figure 3.4 Corresponding throughput curves of Figure 3.3.  Chapter 3. Analysis, Numerical and Simulation Results of Capture Effect  25  3.3.2 Capture model 2 In this section, we provide plots of capture probability and throughput for different spatial distributions with capture model 2.  3.3.2.1 Uniform and bell-shaped spatial distributions For the uniform and bell-shaped spatial distributions, the effects of puncturing and different capture ratios on the capture probability and throughput have been studied [20, 21]. Here, we examine the effect of increasing the noise level. Figure 3.5 shows the Monte Carlo simulation results of capture probability for the uniform spatial distribution and capture model 2 with 8 = 4, c=4, p = Q and r  max  =1  at different noise levels. The 99 percent confidence intervals are so narrow as to be indiscernible on the graph. Figure 3.6 shows the corresponding throughput curves. For moderate noise levels (77 < 0.2), the throughput is only slightly degraded. When the traffic is heavy, the throughput is quite insensitive to increased noise. The asymptotic value of qi appears to be the same at any reasonable noise level (77 < 1). The reason is that when the number of colliding packets increases, noise becomes a less significant part of the interference. Figure 3.7 shows the results, obtained using (3.27), of capture probability for the bell-shaped spatial distribution and capture model 2 with 8 = 4, c = 4 and p = 0 at different noise levels. The asymptotic value of q; for the bell-shaped spatial distribution appears to be the same as that of the uniform spatial distribution. Figure 3.8 shows the corresponding throughput curves. The results are similar to those of the uniform spatial distribution.  Chapter 3. Analysis, Numerical and Simulation Results of Capture Effect  Number of contending transmitters Figure 3.5 Capture probability for the uniform spatial distribution and capture model 2 with /? = c = 4, p = 0 and r = 1 at different noise levels. max  Chapter 3. Analysis, Numerical and Simulation Results of Capture Effect  n  =o 0.1  1  Tl =  1  in = 0.2 11=1  i 0.4  0  1  5  10  15  20  25  30  Number of contending transmitters Figure 3.7 Capture probability for the bell-shaped spatial distribution and capture model 2 with /3 = 4, c = 4 and p = 0 at different noise levels.  28  Chapter 3. Analysis, Numerical and Simulation Results of Capture Effect  3.3.2.2 Inverse-distance spatial distribution For the inverse-distance spatial distribution and r? = 0, qi, given by (3.35), is independent of p. Figure 3.9 shows the capture probability for Q = 4, c = 4, rj = 0 and different values of v. It is proved in Appendix A . l that for 77 = 0, q is a monotonically t  decreasing function of v. Figure 3.10 shows the corresponding throughput curves. It can be seen that in heavy traffic, the throughput is very sensitive to v and degrades rapidly. If p is fixed and 77 > 0, qi can be obtained by (3.33). For v close to 1 and small values of i, qi is extremely sensitive to noise and decreases rapidly as illustrated in Figure 3.11. At any noise level, 77 > 0, qt decreases with puncturing radius, p, as depicted in Figure 3.12.  3.3.2.3 Uniform received power distribution For the uniform received power distribution, q is given by (3.39). For any given x  c, qi is maximized when y in m  = 0 and 77 = 0. Figure 3.13 shows the maximized qi for c = 4. However, the maximized <?•, still decreases rapidly with i. The corresponding throughput, shown in Figure 3.14, also decreases rapidly under heavy traffic. In general, the throughput of the uniform received power distribution is much worse than those of the uniform, bell-shaped and inverse-distance spatial distributions except for the inversedistance spatial distribution with v near 1 and 77 > 0. The reason for the poor performance of the uniform received power distribution is that there is little "near-far" effect. The performance of the uniform received power distribution is comparable to that of the standard slotted ALOHA system with no capture.  Chapter 3. Analysis, Numerical and Simulation Results of Capture Effect  1  5  10  15  20  25  30  Number of contending transmitters Figure 3.9 Capture probability for the inverse-distance spatial distribution and capture model 2 with (3 = 4, c = 4, r? = 0 and different values of v.  to'  3  to"  2  Probability of packet origination Figure 3.10 Corresponding throughput curves of Figure 3.9.  to'  1  30  Chapter 3. Analysis, Numerical and Simulation Results of Capture Effect  0.8  I B  0.6  0.4  S 0.2  1  i  5  i  10  '  1  -L 25  1  15  20  30  Number of contending transmitters Figure 3.11 Capture probability for the inverse-distance spatial distribution and capture model 2 with /? = 4, c = 4, p = 0.1 and v = 1.1 at different noise levels.  0.8  •8 1 8 S  & u  p = 0.001 • p = 0.01 • p = 0.1 • p = 1.0 •  -  0.6  0.4  0.2  •  i  i  i  I  10  .  I  15  _L 20  •  1  •  1  25  •  •  30  Number of contending transmitters Figure 3.12 Capture probability for the inverse-distance spatial distribution and capture model 2 with B = 4, c = 4, rj = 0.001, u= 1.1 and different values of p.  Chapter 3. Analysis, Numerical and Simulation Results of Capture Effect  Number of contending transmitters Figure 3.13 Capture probability for the uniform received power distribution and capture model 2 with c = 4, y in = 0 and r? = 0. m  Probability of packet origination Figure 3.14 Corresponding throughput curve of Figure 3.13.  Chapter 4 Effect of Power Loss Factor on Capture Environment A typical value for the power loss factor, 8, is 4 but 8 can vary between 2 and 5 depending on the terrain. In this chapter, the effect of different values of 8 on the capture probability and throughput for various spatial distribution and capture models are examined. For simplicity, we assume that noise is negligible throughout this chapter.  4.1 A Condition for an Improved Capture Effect Suppose the i contending packets originate at distances { r i , r , • • • , r, } from the 2  base station.  Let r  m  and r denote the smallest and the second smallest distances, s  respectively. Assume the maximum possible distance between the base station and a mobile is rmax and let the noise power be zero. Then for capture model 1, q can be t  expressed as (4.1) and for capture model 2, qi can be expressed as (4.2) If 8 increases to 8+A8 where A8 > 0, then the resulting capture probability, denoted by q*, can be written for capture model 1 as  32  Chapter 4. Effect of Power Loss Factor on Capture Environment  3 3  For capture model 2, q* can be written as  (4.4) r  r  m  Lemma 4.1: For capture model 1 with no noise, q*>qi, i > 1. Proof: For capture model 1 with no noise, q$ = q\ = 1. Since 1 fr \"  p  A/3  n  . 1  < 4,  rf V« /  ( -5) 4  rf  r  it follows from (4.1) and (4.3) that q* > q for i > 2. {  Q.E.D.  {  Lemma 4.2: For capture model 2 with no noise, q*>qi, « > 1. Proof: For capture model 2 with no noise, q^ = q = l. Since 1  '  i  /  \ A/9  «'  -,  it follows from (4.2) and (4.4) that 9; > q i > 2.  Q.E.D.  it  4.2 Capture Model 1 In this section, we show the effect of varying 0 on capture probability and throughput for three spatial distributions on capture model 1. For the uniform spatial distribution, q\ = 1 and qi, i > 2, is given by (3.6). If 0 increases to 0+A0, the improvement of q* over qi, i>2,  is  34  Chapter 4. Effect of Power Loss Factor on Capture Environment  It can be easily seen that the ratio in (4.7a) is a monotonically increasing function of i. Figure 4.1 shows the capture probability for c=4, p = 0 and different values of 8. Since p — 0, q is independent of r x  and the improvement of q* over q i > 2, is given by  max  t>  the expression in (4.7b). For the bell-shaped spatial distribution,  is given by (3.15) for p = 0. Figure 4.2  shows the capture probability for c = 4, p = 0 and different values of 8. The results are similar to those of the uniform spatial distribution. For the inverse-distance spatial distribution, qi can be obtained by using (3.17). If 8 increases to 8+A8, the improvement of q* over qi is Si  =  c  ("-i)('-i)(*-*k?).  (4.8)  Qi  The ratio in (4.8) increases monotonically with i. Figures 4.3 and 4.4 show the capture probability with c = 4 for v = 2 and f = 1.1, respectively. The ratio in (4.8) increases with v, but the absolute difference in the capture probability may not increase with v. This is because a smaller value of v results in a larger qi if noise is negligible.  4.3 Capture Model 2 In this section, we examine the effect of varying 8 on capture probability and throughput for three spatial distributions on capture model 2.  4.3.1 Uniform and bell-shaped spatial distributions We only provide the simulation results for the uniform and bell-shaped spatial distributions because of the difficulty of obtaining a closed-form expression for q . t  Figure 4.5 shows the capture probability for the uniform spatial distribution with c = 4, p = 0 and r  m a i  = 1. From Figures 4.1 and 4.5, the main difference between  35  Chapter 4. Effect of Power Loss Factor on Capture Environment  the two capture models is that the asymptotic value of the capture probability curve for capture model 2 is lower than that of the corresponding curve for capture model 1. The asymptotic value increases with 8. Figure 4.6 shows the capture probability for the bell-shaped spatial distribution with c = 4 and p = 0. The results are similar to those of the uniform spatial distribution.  4.3.2 Inverse-distance spatial distribution The capture probability for the inverse-distance spatial distribution always approaches zero as i increases, i.e.  jll^g; = 0, except  for v = 1. The capture probability can be  obtained by using (3.35). If 0 increases to 0+A0, the improvement of q* over qi is  £  c-^[x*T{x*)r xT[{i-l)x] l  =  qi  eH'- ) 1  1  [x T(x)  x* T[ (i - l)x* }  > (i-i)(*-0 c  where x* =  and x —  (4 ) -9b  The result of Appendix A . l is applied to obtain (4.9b)  from (4.9a). Figure 4.7 shows the capture probability for c = 4 and v = 2. Figure 4.8 shows the corresponding curve for v = 1.1. The effects of varying 0 on both capture models 1 and 2 are similar for the inverse-distance spatial distribution.  Chapter 4. Effect of Power Loss Factor on Capture Environment  0.2  36  -  0 '  1  1  10  5  15  _L 20  25  30  Number of contending transmitters Figure 4.1 Capture probability for the uniform spatial distribution and capture model 1 with c = 4, p = 0 and different values of /?.  l  0 \ 1  i  i  i  I i 5  i  i  i  I i 10  i_i  i L_i i i i I i i i i 15  20  I i 25  i  i  i  30  Number of contending transmitters Figure 4.2 Capture probability for the bell-shaped spatial distribution and capture model 1 with c = 4, p = 0 and different values of /?.  Chapter 4. Effect of Power Loss Factor on Capture Environment  1  10  5  15  20  25  30  Number of contending transmitters Figure 4.3 Capture probability for the inverse-distance spatial distribution and capture model 1 with c = 4, v = 2 and different values of /?.  0 I 1  I  I  I  I 5  I  I  I  I—I  10  L _ l  I  I  L_l  15  I  I  I  I 20  I 25  I  •  30  Number of contending transmitters Figure 4.4 Capture probability for the inverse-distance spatial distribution and capture model 1 with c = 4, v = 1.1 and different values of /?.  Chapter 4. Effect of Power Loss Factor on Capture Environment  P=5 P=4 P=3 P=2  0.8  S  0.6  0.4  0.2  •  10  i — i -  15  20  25  i  i  i  i  30  Number of contending transmitters Figure 4.5 Capture probability for the uniform spatial distribution and capture model 2 with c=4, p — 0, r  max  = 1 and different values of (3.  p  i  u  Number of contending transmitters Figure 4.6 Capture probability for the bell-shaped spatial distribution and capture model 2 with c = 4, p = 0 and different values of 0.  39  Chapter 4. Effect of Power Loss Factor on Capture Environment  B=5  P=4 P=3 P=2  1  0.8  S  •1 1 -  M\\  0.6  • • w\ '  \\  \\\\  I 0.2  :\\\ 1  5  10  15  20  25  30  Number of contending transmitters Figure 4.7 Capture probability for the inverse-distance spatial distribution and capture model 2 with c = 4, v = 2 and different values of /?.  0.8  «  -  0.6  s 0.4  -  0.2  J  1  5  10  15  I  I  20  L.  I  25  I  I  I  30  Number of contending transmitters Figure 4.8 Capture probability for the inverse-distance spatial distribution and capture model 2 with c = 4, v = 1.1 and different values of /?.  Chapter 5 Enhancing Capture Effect with Transmit Power Level Selection If noise is negligible in a packet radio system, a larger value of the power loss factor, 0, can increase the capture probability as shown in Chapter 4. The reason is that a larger value of 0 can result in a larger signal-to-interference ratio. In the previous chapters, it has been assumed that the transmit power level of a mobile is constant. A class of transmit power level selection schemes which assign a higher transmit power level to a mobile closer to the base station is referred to as a decreasing transmit power law. In this chapter, we give some conditions on a decreasing transmit power law which can improve the capture effect. A particular decreasing transmit power law is proposed. It is shown that the proposed transmit power law has an effect similar to that of increasing 0. Several properties of the proposed transmit power law when used in conjunction with the bell-shaped spatial distribution and capture model 2 are studied.  5.1 A Condition for an Improved Capture Effect Suppose the / contending packets originate at distances { n , r2, • • • , n } from the base station.  Let r  m  and r denote the smallest and the second smallest distances, s  respectively. Assume the minimum and maximum possible distances between the base station and a mobile are fmin &nd r^^, respectively, and let the normalized noise power be n. Let P(r) be the transmitted power level of a mobile at distance, r, from the base station. In the previous chapters, it has been assumed that every mobile in the system transmits with constant power, i.e. P(r) = P where P is a constant, then for capture 0  40  0  41  Chapter 5. Enhancing Capture Effect with Transmit Power Level Selection  model 1, Qi can be expressed as „ = P r { ^ > c  m  a x ( - | , , ) }  (5.1)  and for capture model 2, qi can be expressed as  qi  = Pr <  > c  ±  7  (5.2)  + r,  Now assume that P(r) decreases with distance r, i.e.  P(rj) >  (5.3)  P(r ) k  for r  • Condition (5.3) is referred to as a decreasing transmit power  min^ j^ k^ max r  r  r  law. We assume that P ( r capture model 1 as qi  = Pr  ) = P.  m i n  The capture probability, qi, can be written for  0  <— „0  > c max 7*  = Pr  <  —5-  }  •n  1 P(r )  rj  rS P( mV  P{r )  > c max  s  r  For capture model 2,  qi  r  3—  P  1  > c  m  4" > c  Lemma 5.1: For capture model 1, <7* > 9t  for z >  2.  (5.4)  can be written as  = .rr <  = Pr{  J  m  }  . , .,  gi  r  1 "ffa) ? P(r )  = ft = 1 if  ,  + P(r  TO  r; <  fl  f  ( y*). r  m  (5.5)  )  If  77  <  J >  ^ ° ^ , then m  1  Chapter 5. Enhancing Capture Effect with Transmit Power Level Selection  42  Proof: From (4.1) and (5.4), it can be seen that q = q = l if n < ( y"). p  1  r  1  C '"max  If j] < ^ P(  mai  ) , then n < -k^ also. Therefore, such a noise level does not affect p  '"max  **a  9i, i > 2. Since  -L-ffeUl it follows from (4.1) and (5.4) that  q  > qi for i > 2.  i  (5.6) Q.E.D.  Lemma 5.2: For capture model 2 and n = 0, qi > qi, i > 1. Proof: For capture model 2 and 77 = 0,  From (4.2) and (5.5), it follows that ?,•>?., t > 1.  Q.E.D.  Lemma 5.3: If noise is negligible and the transmit power level at distance r^v, is constant, a decreasing transmit power law with a smaller ratio of the transmit power level at that of at rj, where r , „ < rj < r* < r m  to  , can further increase the capture probability.  max  Proof: Let Pt(r) be a transmit power law where Pt(n) P(rt) Pt(rj) ~ P(rj) for r i m  n  <rj<r < k  r  max  (  58  ,  and Pt(r in) = P(r ). m  min  (5.9)  Let the capture probability with the transmit power law Pt{r) be denoted by qi. It follows from Lemma 5.1 and 5.2 that q~i > qi for the two capture models if noise is negligible.  Q.E.D.  The probability of a mobile located at distance r to capture the receiver does not decrease by using a decreasing transmit power law if noise is negligible. This is because  43  Chapter 5. Enhancing Capture Effect with Transmit Power Level Selection  only the strongest signal can capture the receiver and the capture probability depends only on the signal-to-interference ratio. If the signal sent by the mobile is the strongest, the probability for the mobile to capture the receiver increases because the interference power relative to the signal power decreases. If the signal sent by the mobile is not the strongest, its contribution to the interference decreases relative to the strongest signal.  5.2 The Proposed Transmit Power Law We have shown in the previous section that the capture effect, in a noiseless environment, can be improved by using a decreasing transmit power law. In this section, we propose a transmit power law, P(r), defined as  p  «  = 7*T1  (  5  1  0  )  where g, A and 6 are constants and A > 0. Let the minimum and maximum transmit power levels be Pmin and Pmax, respectively.  Assume that a mobile is restricted to  transmit packets with non-zero power level within a circular ring centered at the base station. Let dmin and dmax be the inner and outer radii, respectively, of the circular ring. The corresponding transmit power levels at dmi and dmax are Pmax and Pmin- The n  transmit power level is assumed to be zero outside the ring. The capture probability, qi, can be interpreted as the probability of successful packet reception when i mobiles wish to transmit, with zero or non-zero transmit power level, in a given time slot. From (5.10), we have Pmin  and  =  ,  A  \  "max  ,  +0  (5.H)  44  Chapter 5. Enhancing Capture Effect with Transmit Power Level Selection  From (5.11) and (5.12), we can express g in terms of 8 and A as g = P in{d m  +  max  s) = Pma {d X  S) .  +  min  (5.13)  From (5.13), 8 can be expressed in terms of A as £ _  Pmm r] max — P max r] • X  1  X  u  1  ^  u  mtn  Pmax  Pmin  Therefore, the proposed transmit power law can be defined by A for given Pmin, Pmax, dmin  and dmax- From (5.10), (5.13) and (5.14), we can obtain P(r) as p( \ _  for d i m  n  <r<d  .  max  PmaxPmin(d  —^ j )  max  m  ._ .  w  ' ~ JPm i n (d \ max -r )4-P " ) + max \( r - d min)}  {  a  x  x  r  r  x  a  X  It is shown in Appendix C that if Pmin decreases, the proposed  transmit power law, P(r) with constant Pmax, dmin and dmax, has a smaller ratio of the transmit power level at  to that of at rj, where rf ,„ < rj <rk< d . m  max  It follows from  Lemma 5.3 that a smaller value of Pmin can increase the capture probability if noise is negligible. Theoretically, we can set Pmin to be arbitrarily close to zero if n = 0. If 7?  > 0, the minimum value of Pmin should be cqd  max  to ensure that a packet originating  at distance dmax can capture the receiver in the absence of any other packet transmission. The received power, T, of a packet transmitted from distance r is T(7) =  (5.16)  The p.d.f. of the received power can be derived from (5.16) and the p.d.f. of the distance R between a mobile and the base station. If the spatial distribution is punctured and 8 approaches zero, we can easily see from (5.16) that the proposed transmit power law has an effect similar to that of increasing 0.  45  Chapter 5. Enhancing Capture Effect with Transmit Power Level Selection  5.3 Effect of Varying the Parameters on the Proposed Transmit Power Law In this section, we study the effect of varying the parameters A and dmax of the proposed transmit power law on the bell-shaped spatial distribution and capture model 2. The results are obtained by computer simulation and the throughput is evaluated using the Markovian Model described in Section 2.4. The number of mobiles considered in the Markovian Model is 50 and p = 10p where p is the probability of packet r  o  0  origination and p is the probability of packet retransmission. We assume that P r  max  =1  and Pmin = cr}d ax throughout this section. m  5.3.1 Effect of varying A From (5.15), we can write P(r) as (5.17)  for d i < r < d ax- If A increases, both terms of the denominator in (5.17) decrease m  n  m  and hence P(r) increases. If A is large and r is not close to dmax, the first term of the denominator in (5.17) approaches Pmin and the corresponding second term approaches zero. Therefore, P(r) approaches Pmax for large A and r not near dmax- If A is small and r is not close to dmin, the first term of the denominator in (5.17) approaches Pmin and the corresponding second term approaches Pmax- Therefore, P(r) approaches Pmin for small A and r not near dmin- Figure 5.1 shows the proposed transmit power law as function of distance, r, for P  max  = 1, P  m i n  = 0.01, d  max  = 1, d  max  = 0 and different values of A.  Chapter 5. Enhancing Capture Effect with Transmit Power Level Selection  0.2  0.4  0.6  0.8  1  46  1.2  1.4  Distance Figure 5.1 The proposed transmit power law as function of distance, r, for P  max  = 1, P ,„ = 0 . 0 1 , m  dmax = 1. d in = 0 and different values of A. m  Figures 5.2 and 5.3 show the simulation results of capture probability and the corresponding throughput curves for 0 =  4,  c=  4,  p = 0, n =  0.001,  di m  n  =  0,  d  max  = 1  and different values of A on the bell-shaped spatial distribution and capture model 2. The capture probability and throughput curves with constant transmit power level of 1, labelled as "constant", are also shown in Figures 5.2 and 5.3 for comparison. Figure 5.4 shows the results of capture probability under the same conditions as those of Figure 5.2 except for n = 0.1 and Figure 5.5 shows the corresponding throughput curves. It appears that the peak of the throughput curve increases and then decreases with A. In most cases, a A value of about 5 maximizes the peak for the throughput curve. When the traffic is light, the proposed transmit power law slightly degrades the throughput, especially for large values of n. However, the power law improves the throughput for moderate and heavy traffic.  Number of contending transmitters Figure 5.2 Capture probability for the bell-shaped spatial distribution and capture model 2 with /? = 4, c = 4, p = 0, T? = 0.001, d = 0, cf = 1 and different values of A. max  min  constant 51 = 7  i  to'  3  i  t  i  i  i  i  t  I  ,  t  i  n ,  io"  z  Probability of packet origination Figure 5.3 Corresponding throughput curves of Figure 5.2.  i  t  i  t  t  to"  1  Chapter 5. Enhancing Capture Effect with Transmit Power Level Selection  Number of contending transmitters Figure 5.4 Capture probability for the bell-shaped spatial distribution and capture model 2 with p = 4, c = 4, p = 0, rj = 0.1, d i„ = 0, d ax = 1 and different values of A. m  m  Chapter S. Enhancing Capture Effect with Transmit Power Level Selection  5.3.2 Effect of varying  d  49  max  The parameter dmax should be chosen such that every mobile within dmax has a chance to capture the receiver. Hence, a smaller dmax can reduce the interference and limit the traffic. Figures 5.6 and 5.7 show the results of capture probability and the corresponding throughput curves for 0 = 4, c = 4, p = 0, n = 0.001, A = 5, d i = 0 m  n  and different values of dmax- It can be seen that as dmax is reduced, the peak of the throughput curve shifts towards the heavy traffic side. If a system can estimate the traffic intensity and choose dmax accordingly, an improved throughput curve can be achieved. For example, for p = 10 , the throughputs with d -1  0  max  = 1.5 and d  max  = 0.3 are 0.33  and 0.70, respectively. Figures 5.8 and 5.9 show the results of capture probability and the corresponding throughput curves for a noisier system with 77 = 1.  Chapter 5. Enhancing Capture Effect with Transmit Power Level Selection  -i  1  I  5  i  i  i_  10  15  i  20  i  I  i  25  i  i  30  Number of contending transmitters Figure 5.6 Capture probability for the bell-shaped spatial distribution and capture model 2 with P = 4, c = 4, p = 0, 77 = 0.001, A = 5, d ,-„ = 0 and different values of d . m  max  51  Chapter 5. Enhancing Capture Effect with Transmit Power Level Selection  1  5  10  15  20  25  30  Number of contending transmitters Figure 5.8 Capture probability for the bell-shaped spatial distribution and capture model 2 with 3 = 4, c = 4, p= 0, 77 = 1, A = 5, d in = 0 and different values of d . m  max  Chapter 6 Conclusions and Future Work This thesis has extended the research on the performance improvement due to the existence of capture effects for different spatial distribution and capture models in a slotted ALOHA system. In addition to considering the uniform and bell-shaped spatial distributions and the uniform received power distribution, a new spatial distribution, the inverse-distance spatial distribution, was introduced for ease of analysis. The throughputs, evaluated using a finite population Markovian model, of the uniform and bell-shaped spatial distributions are quite insensitive to noise. The asymptotic values of the capture probability, qi, for both spatial distributions appear to be independent of the noise level (for 77 < 1). The <7,'s for the inverse-distance spatial distribution with v near 1 were shown to be very sensitive to noise even though the q?% are close to 1 in the absence of noise. In general, the throughput of the uniform received power distribution is much worse than those of the uniform, bell-shaped and inverse-distance spatial distributions. One exception is the inverse-distance spatial distribution with v near 1 and 77 > 0. The reason for the poor performance of the uniform received power distribution is that there is little "near-far" effect. The performance of the uniform received power distribution is comparable to that of the standard slotted ALOHA system with no capture. The capture probability for the three spatial distributions and the uniform received power distribution with capture model 1 was shown to decrease monotonically with i for 77 = 0. A sufficient condition for qi of capture model 1 to be a monotonically decreasing function of i was derived. The capture probability of the inverse-distance spatial distribution, the uniform received power distribution and the bell-shaped spatial distribution with 0=4 and capture model 2 was proved to decrease monotonically with i 52  53  Chapter 6. Conclusions and Future Work  for 77 = 0. Some examples of non-monotonically decreasing q for the two capture models t  with noise were provided. An example of a noiseless case in which qi does not decrease monotonically with i was also given. The effect of varying the power loss factor, 0, on qi for the three spatial distributions and the two capture models was studied. It was proved that qi increases with 0 for both capture models with 77 = 0. A scheme in which the transmit power level of a mobile is chosen according to its distance from the base station was proposed. The proposed transmit power law decreases with distance over a selected range of distance and has an effect similar to that of increasing 0. The effect of varying the parameters of the proposed transmit power law on the bell-shaped spatial distribution with capture model 2 was studied. The throughput can be greatly improved with a proper choice of parameters. Among the topics which could be further investigated are: •  To find more general sufficient conditions on a received power distribution under which qi decreases monotonically with 1 for capture model 2 and possibly other capture models.  •  To find a received power distribution which maximizes the throughput.  •  Tofindthe optimal parameter values of the proposed transmit power law for different spatial distribution and capture models.  •  To determine the transmit power law which maximizes the throughput.  Bibliography  [1] N. Abramson, "The ALOHA System-Another Alternative for Computer Communications," Proceedings of the AFIPS Fall Joint Computer Conference, Montvale, NJ,  USA.,  vol. 37, pp. 281-285, 1970.  [2] L. Kleinrock and S. Lam, "Packet Switching in a Multiaccess Broadcast Channel: Performance Evaluation," IEEE Transactions on Communications, vol. COM-23, pp. 410-423, Apr. 1975. [3] L . Kleinrock and F. Tobagi, "Packet Switching in Radio Channels:Part I-Carrier Sense Multiple-Access Modes and Their Throughput-Delay Characteristics," IEEE Transactions on Communications, vol. COM-23, pp. 1400-1416, Dec. 1975. [4] F. Tobagi and L. Kleinrock, "Packet Switching in Radio Channels:Part II-The Hidden Terminal Problem in Carrier Sense Multiple-Access and the Busy-Tone Solution," vol. COM-23, pp. 1417-1433, Dec. 1975.  IEEE Transactions on Communications,  [5] R. Metcalfe and D. Boggs, "Ethernet: Distributed Packet Switching for Local Computer Networks," Communications of the ACM, vol. 19, pp. 395-404, July 1976. [6] J. Capetanakis, "Tree Algorithms for Packet Broadcast Channels," IEEE on Information Theory, vol. IT-25, pp. 505-515, Sept. 1979.  Transactions  [7] L. Roberts, "ALOHA Packet System with and without Slots and Capture," Computer Communications  Review, pp. 28-42, Apr. 1975.  [8] D. Goodman and A. Saleh, "The Near/Far Effect in Local ALOHA Radio Communications," IEEE Transactions on Vehicular Technology, vol. VT-36, pp. 19-27, Feb.  1987. [9] C. Namislo, "Analysis of Mobile Radio Slotted ALOHA Networks," IEEE Transactions on Vehicular Technology, vol. VT-33, pp. 199-204, Aug. 1984.  [10] J-P. Linnartz and R. Prasad, "Near-Far Effect on Slotted ALOHA Channel with Shadowing and Capture," Proceedings  of the 39th IEEE  Vehicular  Conference, San Francisco, California, U.SA., pp. 809-813, Apr. 1989. 54  Technology  Bibliography  55  [11] J. Arnbak and W. Blitterswijk, "Capacity of Slotted ALOHA in Rayleigh-Fading Channels," IEEE Journal on Selected Areas in Communications, vol. SAC-5, pp. 261269, Feb. 1987. [12] I. Habbab, M. Kavehrad and C-E. Sundberg, "ALOHA with Capture Over Slow and Fast Fading Radio Channels with Coding and Diversity," IEEE Journal on Selected Areas in Communications, vol. SAC-7, pp. 79-88, Jan. 1989. [13] F. Kuperus and J. Arnbak, "Packet Radio in a Rayleigh Channel," IEE Electronics Letters, vol. 18, pp. 506-507, June 1982. [14] E. Zainal and R. Garcia, "The Effects of Rayleigh Fading on Capture Phenomenon," IEEE Infocom 1987, pp. 888-893, Mar. 1987. [15] N. Abramson, "The Throughput of Packet Broadcasting Channels," IEEE Transactions on Communications, vol. COM-25, pp. 117-128, Jan. 1977. [16] I. Gdon, H. Kodesh and M . Sidi, "Erasure, Capture, and Random Power Level Selection in Multiple-Access Systems," IEEE Transactions on Communications, vol. COM-36, pp. 263-271, Mar. 1988. [17] Y. Onozato, J. Liu and S. Noguchi, "Stability of a Slotted ALOHA System with Capture Effect," IEEE Transactions on Vehicular Technology, vol. VT-38, pp. 31-38, Feb. 1989. [18] D. Lyons and P. Papantoni-Kazakos, "A Window Random Access Algorithm for Environments with Capture," IEEE Transactions on Communications, vol. COM-37, pp. 766-770, July 1989. [19] A. Shwartz and M. Sidi, "Erasure, Capture, and Noise Errors in Controlled MultipleAccess Networks," IEEE Transactions on Communications, vol. COM-37, pp. 12281231, Nov. 1989. [20] C. Lau and C. Leung, "Capture Models for Mobile Packet Radio Networks." To Appear, IEEE Transactions on Communications. [21] C. Lau, Multi-Antenna and Receiver Slotted ALOHA Packet Radio Systems with Capture. PhD thesis, Dept. Elec. Eng., UBC, Vancouver, B.C., Canada, May 1990. [22] J. Metzner, "On Improving Utilization in ALOHA Networks," IEEE Transactions on Communications, vol. COM-24, pp. 447-448, Apr. 1976.  Bibliography  56  [23] C. Lau and C. Leung, "Performance of a Power Group Division Scheme for ALOHA System in a Finite Capture Environment," IEE Electronics Letters, vol. 24, pp. 915— 916, July 1988. [24] K. Bullington, "Radio Propagation for Vehicular Communications," IEEE Transactions on Vehicular Technology, vol. VT-26, pp. 295-308, Nov. [25]  1977.  M . Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas,  Graphs, and Mathematical Tables. New York: Dover Publications, Inc., 1970. [26]  I. Gradshteyn and I. Ryzhik, Table of Integrals, Series, and Products. New York:  Academic Press, 1980. [27] L. Couch II, Digital and Analog Communication. New York: MacMillian, 2nd ed., 1987. [28] A. Tanenbaum, Computer Networks. Englewood Cliffs, New Jersey: Prentice Hall, 2nd ed., 1989.  Appendix A Inverse-distance Spatial Distribution and Capture Model 2 In this appendix, it is shown that qi as given by (3.35) decreases monotonically with x, x>0, and i, i > 1. Since q\ = 1, we only need to prove that q decreases monotonically t  with x, x > 0, and i, i > 2. It is sufficient to prove that h(x, y) decreases with X, x > 0, and y, y > 1 where  [*r(*)]»  h(x,y) =  (A.1)  and T(-) is the Gamma function.  A.l Monotonic Decrease of qi with x In this section, we prove that h(x, y) as given by (A.l) decreases monotonically with x, x > 0 and y > 1. From (6.1.3) of [25], the Gamma function, T(-), can be expressed as  w-{""fi[(i + J)-]}  r  1  (A.2)  where a is Euler's constant. From (A.l) and (A.2), h(x,y) can be expanded to OO  e " " U  n „  =1  r [(l + ?)e-V  12/  (A.3)  La+s'J"  Let the n-th product term of h(x,y) be denoted by h (x,y). n  To prove that h(x,y)  decreases with x, x > 0, it is sufficient to prove that h (x,y) decreases with x, x > 0. n  57  Appendix A. Inverse-distance Spatial Distribution and Capture Model 2  The derivative of h (x,y)  58  w.r.t. x is  n  £ (1 + J)» _  dhn(x,y) dx  y  (i  < I ~  1  +  V J-  ^ n) ( 1+  (  B  +  (  1  +  x^y n)  (A.4a)  (  A  4  b  )  r »  < 0  (A.4c)  for a; > 0 and y > 1. Hence, <7, decreases monotonically with x, x > 0 and i > 1.  A.2 Monotonic Decrease of qi with ? In this section, we prove that h(x,y) as given by (A.l) decreases with y, y > 1 and x > 0. Following from the previous section, it is sufficient to prove that h (x,y) n  decreases with y, y > 1 and x > 0. The derivative of h (x,y) w.r.t. ;y is n  dh (x,y) n  =  =  * ( ! + * ) ' - (l + K ) (l + * ) » l n ( l + *) ^ - (l + f ) l n ( l + | )  (A.5)  (! + *)' From (4.1.33) of [25], we have -  for y > 1. Therefore, we have  < ( l + - ) Infl + - )  (A.6a)  < ( l + ^ ) ln(l + | )  (A.6b)  dhn  ^' ) < rj for a; > 0 and y > 1. Hence, <fr decreases y  monotonically with i, i > 1 and x > 0.  Appendix B Bell-shaped Spatial Distribution and Capture Model 2 In this appendix, we derive the expression for qi in (3.28) and show that qi decreases monotonically with i.  B.l Derivation of qi In this section, we derive (3.28). If p = 0 and r/ = 0, (3.27) reduces to oo  0  oo  oo  (B.l) 0  where  u = (i — l)y/c.  Using  y = &x  in (B.l), we obtain oo oo  (B.2) 0 uy  Letting r = y/y +z 2  2  and 0 = t a n  - 1  (f) in (B.2) yields oo  oo  2i = — tan-1 7T  59  1  (B.3)  60  Appendix B. Bell-shaped Spatial Distribution and Capture Model 2  B.2 Monotonic Decrease of qi with i In this section, it is shown that qi as given by (3.28) decreases monotonically with i, i > 1. Let w{x) = {x + 1) t a n "  .  1  (B.4)  Since q\ = l, it suffices to prove that w(x) decreases monotonically with x for x > 1 and ft < 1. For this, it will be shown that  = tan"' ( -)  ^±4  -  k  dx  \x J  ft  + x  2  2  (B.5)  is non-positive for x > 1 andft< 1. We can write 1) x  (x +  >  ft  +  2  x  (B.6)  2  or equivalently X \/ft + X 2  Vft  + x  2  ft  +  x  x +  1  2  2 >  -  2  2  Referring to Figure B.l and using (B.7), we have \/ft  + x  2  2  cos  9 >  +  x  (B.8)  X  -f  1  or equivalently ^  x  ft  2  +  +  1  ) >  x  2  -  fe  Vft  2  Vft  2  + x  2  + x2  > , ~ yjh + x 2  (B.9c)  1  . X  < 0 for x > 1 andft< 1.  (B.9b)  2  = tan" ( . This proves that  (B.9a)  cos 0  (B.9d)  Appendix B. Bell-shaped Spatial Distribution and Capture Model 2  Figure B.l Illustration of (B.8, B.9).  61  Appendix C The Proposed Transmit Power Law In this appendix, we show that if the proposed transmit power law, P(r), given by (5.15) has variable Pmin and constant Pmax, d i„,  and dmax, then the ratio of the  transmit power level at r to that of at r\, where  < n < r% < d  m  d in  2  m  Pmin- Let Pi(r),  i = l , 2, be a proposed transmit power law with P  P2min  From (5.15), we have  > Pimin-  Pl(r)  Plr  P2min  P mm  P2(r)  P\min(d  2  (dmax  — r*) + Pmax  (r*  — r )  (f  A  max  + Pmax  X  increases with  ,  max  = P ,„ where  m i n  ITO  ^row) —  (Cl)  <^ ) mm  We would like to show that Pl(r )  W  A(ri)  " P (ri)  2  for dmin  r  r  1  2  2  .  *'  < H < T2 < d m a r -  From ( C l ) and (C.2), it suffices to show that Pimin Plmin(d  (dmax  ~  r  l ) + Pmax  — r) X  max  (i  ~ d )  r  P2min(d  min  + Pmax ( 2 r  ^mtn)  —  P2min(d  max  —  + Pmax{Tl  max  — r$) + Pmaxiji  ~ ~  d jn) m  ^  m  ,„)  (C.3) or equivalently, PmaxP2min  <PmaxP2min  ~ ^ m m ) {^max ~ 2^ r  (dmax  ~ l r  ) ( 2 r  —  + PmaxPlmih  ^ m t n ) + PmaxPlmin  (d  ~  max  — d  m  r  m  l )( 2 r  ^ (d  max  —  ^m»n)  — r£ ^  (C.4) which can be simplified to Pmax(P2min  5; Pmax{P2min  ~ Pimin)  —  (^1 ~ ^ m i n )  P l m t n ) (d ax m  62  {^max ~~ 2^ r  ~ l^j {^2 ~ d inj r  m  •  (C.5)  Appendix C. The Proposed Transmit Power Law  We can further reduce (C.5) to rULx  + rd X  min  ( ' 2 - r )di X  Since d i m  n  <d  in  , (C.6) is clearly valid.  max  < r d in X  m  < (r£ -  +  rd X  max  r )d . X  max  

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