PERFORMANCE EVALUATION OF HIGHWAY MOBILE INFOSTATION NETWORKS Wing Ho Yuen WINLAB Rutgers University Piscataway, NJ 8854 anyyuen@winlab.rutgers.eu Roy D. Yates WINLAB Rutgers University Piscataway, NJ 8854 ryates@winlab.rutgers.eu Chi Wan Sung Department of Computer Engineering an Information Technology City University of Hong Kong Hong Kong itcwsung@cityu.eu.hk Abstract A mobile infostation network stipulates all transmissions to occur when noes are in proximity. In this paper, we evaluate the effect of mobility on highway mobile infostation networks. Each noe enters a highway section at a Poisson rate with a constant spee rawn from a known but arbitrary istribution. Both forwar an backwar traffic are consiere. For noe spee that is uniformly istribute, the fraction of connection time is inepenent of target noe spee for backwar traffic, while it increases with target noe spee for forwar traffic. We also exten our mobility moel such that each noe changes spee in each highway section. The long run fraction of connection time is epenent on the ratio of transmit range an connection time limit. Forwar traffic connections yiels better performance when the ratio is smaller than one an vice versa. We also compute the optimal transmit range an the corresponing ata rate for both traffic types. We conclue that forwar traffic connections yiels much higher ata rate in most scenarios. I. INTRODUCTION In a mobile infostation network, any two noes communicate only when they have a very goo channel. This occurs usually when noes are in proximity. Uner this transmission constraint, any pair of noes is intermittently connecte as mobility shuffles the noe locations. The network capacity of mobile infostation networks compares favorably to conventional multihop a hoc networks. In [3] Gupta an Kumar showe that the per noe throughput in a multihop network W rops to zero at a rate O( nln n in the limit of large n. Thus multihop networks o not scale with large network size. On the other han, Grossglauser an Tse showe in [2] that the per noe throughput of a mobile infostation network is O(1, inepenent of the number of noes. This capacity is achieve through a two hop relay strategy. Suppose each noe i has a packet to a estination noe (i. When a noe comes close to other noes k (i, it relays the packet to them, hoping that one of the relay noes reaches the estination (i an complete the secon relay on its behalf. At the steay state, each noe contains many packets aresse to various estinations. It is almost surely that each noe has a packet aresse to its nearest neighbor at any network snapshot. Nevertheless, the orer of magnitue improvement in network capacity comes at a cost. En to en transmissions incur a ranom elay that is at the same time scale of the mobility process. Motivate by the ramatic capacity improvement of mobile infostation networks, there are a number of recent papers that explore the mobile infostation paraigm. Whereas [2] focuse on unicast, [5], [8] aresse multicast in mobile infostation networks. [5] assumes noes cooperate with each other in the network. The issue of noncooperation between noes was explore in [8]. Transmissions between two proximate noes are allowe only when both noes benefit from a file exchange. Nevertheless, simple interference an mobility moels are use to facilitate analysis. In [7], a refine interference moel was use to evaluate the effect of transmit range on network capacity. In this paper, we examine the effect of mobility on mobile infostation networks. In [2], mobility provies a mechanism such that numerous instances of excellent channels between ifferent noes can be exploite. The realization of large network capacity comes from the translation of maximal spatial transmission concurrency in each network snapshot to the long run en-to-en network capacity. The physical implication of mobility in noe encounters has been glosse over. In reality, the total connection time of a noe over a specific interval epens on the noe encounter rate an the connection time in each encounter, both of which epen on the relative mobility of noes. Although a high noe spee results in more noe encounter, the connection time in each noe encounter also ecreases. It is not apparent whether high or low spee results in a larger connection time, an thus, ata rate. To this en we propose a general mobility moel for highway networks. The highway scenario proves to be interesting espite its mathematical simplicity. First, forwar traffic connection time is much larger than that of backwar traffic, but the noe encounter rate is also much smaller. It is not apparent which traffic type maximizes the fraction of connection time. Secon, the connection time in an encounter epens on the transmit range of the noes. For both forwar an backwar traffic, an optimal transmit range exists such that the long run ata rate of a noe is maximize. II. SYSTEM MODEL We consier a highway section as shown in Figure 1. Fixe infostations are eploye regularly on a highway at a istance from each other. An infinite number of highway sections are place on a line to mimic a highway network. Noes move at a constant spee V, a i.i.. ranom variable rawn from a known but arbitrary istribution G(V. Two types of traffic are consiere here. For forwar traffic, noes are injecte into the highway section at a Poisson rate of λ from infostation A. Similarly, noes are injecte into the highway at a Poisson rate λ from infostation B for backwar traffic. Since noes have ifferent spees, a noe may overtake other noes or being overtaken as it traverses the highway section. We assume there
Fig. 1. forwar traffic at rate λ A t Β 1 Β 2 Β 3 fixe infostation target noe v 4 5 Β Β backwar traffic at rate λ tt B N 1 (t B Illustration of the highway mobile infostation network moel. is no elay incurre in a noe encounter. This is plausible in a wie motorway with multiple lanes an moerate traffic, where all noe encounters occur in ifferent lanes. This is calle the wie motorway moel in [4]. More generally, a noe changes spee as time evolves. We assume each noe still moves at a constant spee in a highway section. Whenever a noe traverses a new highway section, we stipulate that each noe selects a new spee from the istribution G, inepenent of the previous spee. Consier the target noe moves at a spee V v on a highway section from infostation A to B. We enote the time for the noe to traverse a highway section as the cycle uration, given by T /V, an a corresponing istribution F. F an G are obviously relate, given by F(t G(/t, where the notation F(t 1 F(t is use throughout the paper. For forwar traffic, we enote N 1 (t as the number of noe encounters for the target noe in the uration t /v. The connection time Y 1 (t in each noe encounter is a ranom variable epenent on the relative spee of the noes an the common transmit range of all noes r. Two noes having a similar spee will therefore have an unboune connection time. In reality, however, each noe only has a finite amount of ata for issemination. We stipulate a connection time limit parameter c to limit the actual connection time B 1 (t in a noe encounter, given by B 1 (t min(y 1 (t, c. We also enote the total connection time of the target noe in a highway section as Z 1 (t. The expressions E[N 1 (t ]/t an η 1 (t E[Z 1 (t ]/t correspon to the expecte noe encounter rate an the expecte fraction of connection time as a function of the target noe cycle uration t /v. For backwar traffic, the corresponing expecte noe encounter rate an fraction of connection time are enote as E[N 2 (t /t ] an η 2 (t E[Z 2 (t /t ]. When spee change is incorporate to our moel, the long run fraction of connection time an ata rate are the appropriate metrics. It turns out that simple characterization of these metrics is possible by rawing results from renewal rewar theory [6]. At the n-th highway section, the target noe selects a ranom spee V n. Thus the corresponing cycle uration T n is an i.i.. ranom variable. We enote R n as the rewar earne at the time of the nth cycle uration, or renewal perio. If we let R(t N(t n1 R n, (1 then R(t is the total rewar earne by time t. Let E[R] E[R n ] an E[T n ], the renewal rewar theorem [6] t states that if E[R] < an <, then with probability 1, R(t lim E[R] (2 t t That is, the rate of earning rewar in the long run is just the ratio of the expecte rewar in a cycle an the expecte cycle uration. Accoringly, if we efine the rewar as the number of encounters N 1 (T in a highway section for forwar traffic, then the long run noe encounter rate of the target noe is simply N 1 E[N 1 (T]/. Similarly, when the rewar is efine as the total connection time Z 1 (T in a highway section, Z 1 E[Z 1 (T]/ correspons to the long run fraction of connection time of the target noe. Last, when the rewar is the total amount of elivere ata W 1 (T in a highway section, then W 1 E[W 1 (T]/ enotes the long run ata rate of the target noe. For backwar traffic the long run fraction of connection time Z 2 an ata rate W 2 are efine in a similar fashion, with Z 2 E[Z 2 (T]/ an W 2 E[W 2 (T]/. III. PERFORMANCE ANALYSIS Consier the forwar traffic case. Suppose the target noe arrives at infostation A at time s an infostation B at time s + t. We enote an event occurs at time t if a noe enters the highway section at infostation A. Since the noe travels with ranom spee V /T, this noe leaves the highway section at time t + T. Define p 1 (t as the probability that there is an encounter with the target noe for this event. It is straightforwar to show that p 1 (t F(s + t t t < s F(s + t t s < t < s + t (3 t > s + t The total number of noe encounters at the steay state is E[N 1 (t ] lim λ p(tt (4 s ( t λ F(tt + F(tt (5 It can be shown E[N 1 (t ] attains a global minimum when the target noe cycle uration t is the meian of the istribution F by twice ifferentiating (5. This coincies with the intuition that there are few noe encounters if the target noe goes along with the traffic flow. For backwar traffic, we assume an event occurs at time t if a noe enters the highway section from infostation B. For an event at time t, it can be shown the probability of an encounter with the target noe is given by p 2 (t t t > s + t 1 s < t < s + t F(s t t < s The total number of noe encounters at steay state is (6 E[N 2 (t ] lim λ p 2 (tt (7 s λ(t + (8
where is the expecte cycle uration given by F(tt (9 The long run noe encounter rate for both traffic types can be obtaine by averaging over the spee istribution. Thus E[N 1 (T] 2λ E[N 1 (t ]f(t t (1 F(tF(tt (11 E[N 2 (T] 2λ (12 (11 an (12 suggest that the expecte noe encounter rate for backwar traffic is always larger than the expecte noe encounter rate for forwar traffic, which is intuitively plausible. Moreover, (12 shows that the expecte noe encounter rate is completely characterize by the traffic intensity λ an the first moment of istribution F. To compute the expecte connection time in one encounter for forwar traffic E[B 1 (t ], we note that E[B 1 (t ] P[Y 1 (t > t]t (13 [ ] 2r P v V > t t (14 ( 2r G t + ( G 2r t (15 t t t Similarly, for backwar traffic we have E[B 2 (t ] P[Y 2 (t > t]t (16 ( 2r G t t t (17 Refer to Figure 1 again, the total connection time for forwar traffic is obtaine by summing all iniviual connection time B1 i(t, i [1, N 1 (t ] over the cycle. In the event that the connection time of the encounter N 1 (t overshoots the en of the cycle, the target noe unergoes a renewal an selects a new spee. This in turn moifies the connection time B N1(t 1. Nevertheless, the bounary effect of an overshoot connection time is minimal when either N 1 (t is large, or when B 1 (t c t /v. The former assumption is vali for mobile infostations when the traffic intensity of noes is moerate, such that N 1 (t 1. The latter assumption is vali when the istance between fixe infostations is large, which is likely in an initial eployment of a fixe infostation network. Ignoring the bounary effect of B N1(t 1 (t, we have N 1(t E[Z 1 (t ] E[ B1(t i ] (18 i1 E[N 1 (t ]E[B 1 (t ] (19 N 2(t E[Z 2 (t ] E[ B2 i (t ] (2 i1 E[N 2 (t ]E[B 2 (t ] (21 Since B i (t are i.i.. ranom variables an N(t is Poisson, (19 an (21 follow irectly from the efinition of a compoun Poisson process. The long run fraction of connection time of a target noe for both traffic types can be obtaine by conitioning on istribution F, given by, Z 1 E[Z 1(T] Z 2 E[Z 2(T] E[Z 1 (t ]f(t t E[Z 2 (t ]f(t t (22 (23 Finally, we are also intereste in the long run ata rate for both traffic types. Assuming non-aaptive raios are use, the ata rate is the Shannon rate at the transmit range bounary r, given by C(r ln(1 + 1/r 4 (24 where we have assume a path gain exponent of 4 an ignore the effect of mutual interference. We efine the long run ata rate as W 1 E[W 1 (T, r] C(rE[Z 1 (T, r] (25 W 2 E[W 2 (T, r] C(rE[Z 2 (T, r] (26 where we emphasize both connection time Z an the amount of elivere ata W are epenent on the transmit range r. Since W 1 (r an W 2 (r when the transmit range is zero or very large, an optimal transmit range r exists for both traffic types such that W 1 an W 2 are maximize respectively. IV. NUMERICAL STUDY We consier the case when noe spee is uniformly istribute accoring to (27. v v v G(v a v b v v b (27 1 v v b The corresponing istribution of the cycle uration T /V is t /v b v F(t b /t /v b t / (28 1 t / E[N 1 (t], E[N 1 (T] an E[B 1 (t ] can be reaily compute by evaluating (5,(11, (15 as [ ( ( ] 2 λ 1 E[N 1 (t ] ( + v b t + ln v b t e v b ( ( (29 2λ E[N 1 (T] (v b 2 ( + v b ln 2(v b (3 E[B 1 (t ] c( t v a+2r ln[(v b t ( 2r] ce t max(, + 2r c v b 2r c c(v b t +2r ln[( t v a( 2r] ce t min( +, 2r c v b 2r c 2r ln[( 2r ce 2 (v b t ( t v a] c v b 2r c + 2r c t t + 2r c v b 2r c (31
(a (b Fig. 3. Long run fraction of connection time Z versus transmit range r for connection time limit c.5, 1,2. (c Fig. 2. Fraction of connection time η(t versus noe mobility t /v for ifferent transmit range r an connection time limit c. (a r 1, c 1 (b r 2, c 1 (c r.5, c 1 ( r 1, c 1 Similarly by substituting (28 to (8,(12,(17 we have ( ln E[N 2 (t ] λ(t + (32 v b ( 2λ ln E[N 2 (T] (33 v b E[B 2 (t ] +/t 2r ln( va+/t /t +v 2r ln( b 2r ce c(/t + ( t max(,2r/c max(,2r/c v t a max(,2r/c v b c t max(,2r/c v b (34 We aopt the parameters 2,v b 1, 1 in our numerical stuy. With reference to Figure 2, the fraction of connection time η 1 (t an η 2 (t are plotte together versus t in the range /v b 1 to / 5. At mean spee v 6, the corresponing t is 166.67 unit. Consier scenario 1 for r 1, c 1. For forwar traffic, η 1 (t attains a global maximum of.6 when t is minimum. η 1 (t ecreases steaily as t increases an hits the minimum of.3 at t 267.73. Beyon that, there is a slight increase of η 1 (t when t is increase further. Similar trens are observe for other scenarios in Figure 2(b,(c,(. Nevertheless, a slight ip of η 1 (t occurs at low mobility (t 5 for Figure 2(. Although there are slightly more encounters at low mobility, there is a steeper ecrease in connection time. Thus η 1 (t is not convex in general. In fact, for large t, the target noe moves very slowly. The fraction of connection time for forwar an backwar traffic shoul come arbitrarily close. Thus, the ip in Figure 2( is consistent to our intuition. Whereas η 1 (t varies with t, the fraction of connection time for backwar traffic η 2 (t is almost constant at all target noe spee in the 4 scenarios. The relative value of η 1 (t an η 2 (t epens on the ratio of transmit range an connection time limit r/c. When r/c is large (Figure 2(b, it is likely that the connection time for forwar traffic is truncate. Thus η 1 (t is consistently smaller than η 2 (t except for very high target noe spee. When r/c is small (Figure 2(c,(, the connection time of each noe encounter is large. In fact, if there is no connection time limit, the expecte connection time for forwar traffic is unboune. The large connection time at large c stipulates that η 1 (t > η 2 (t at all noe spee. Incientally, when r/c 1 (Figure 2(a, η 1 (t an η 2 (t intersects at t 162.7, which is close to the cycle uration at mean spee /E[V ] 166.67. Thus, if a target noe moves at a constant spee v less than the mean spee E[V ], backwar traffic connections are more preferable. Similarly, forwar traffic connections are more preferable if a noe moves at a constant spee v E[V ]. When noes move with ranom spee in ifferent highway sections, the long run fraction of connection time Z 1 (r, c an Z 2 (r, c are relevant an epenent on the transmit range r an connection time limit c. In practice, c is typically long enough such that the connection time for for backwar traffic Y 2 is not truncate. This is satisfie when max(y 2 r/ c, or r/c 2. Thus Z 2 (r is inepenent of c for the cases of our interest. In Figure 3 the long run fraction of connection time for both forwar an backwar traffic is plotte for c.5, 1, 2. Both Z 1 an Z 2 are increasing functions of the transmit range. This is obvious since as c increases, the connection time B 1 an B 2 also increase. We also observe that when the long run fraction of connection time for both traffic types are the same, r c hols. For r/c > 1 the network noes have a large transmit range relative to c, backwar traffic connections are more preferable ue to the truncate connection time for forwar traffic. Similarly, forwar traffic connections are more preferable when r/c < 1. Whereas the fraction of connection time is an increasing function of the transmit range, there exists an optimal range such that the long run ata rate is maximize. With reference to Figure 4, the ata rate for both traffic types are plotte versus transmit range for the cases c.25,.5, 1, 2. The optimal transmit range corresponing to the cases c.25,.5, 1, 2 is r.15285,.1922,.22187,.24722 for forwar traffic. For backwar traffic, the optimal range is r.37713 inepenent of c. The smaller optimal range for forwar traffic connections is intuitively plausible. Forwar traffic enjoys
expecte connection time as warrante by the uniform spee istribution. (a (c Fig. 4. Long run ata rate W versus transmit range r for ifferent connection time limit c (a c.25, (b c.5, (c c1, ( c2. longer connection time. A short range is favore such that a high channel rate can be realize. When the connection time limit c is small (Figure 4(a, it is likely the connection time of forwar traffic is truncate. Backwar traffic enjoye much higher encounter rate that contributes to the total connection time. Thus backwar traffic outperforms forwar traffic at large transmit range. The long run ata rate for forwar an backwar traffic at optimal range are roughly the same. However, the iscrepancy of the optimize ata rate increases as c increases. At c 1, 2 (Figure 4(c(, the optimize ata rate of forwar traffic is more than twice the backwar traffic. Thus, forwar traffic connections yiels much higher ata rate. The result of Figure 4 can be compare to Figure 3. Figure 3 shows that for r/c > 1, backwar traffic connections are more preferable ue to the increase fraction of connection time. However, the achievable channel rate also ecreases rapily at large transmit range. Thus the extraneous ata rate ue to the increase fraction of connection time at r/c > 1 is negligible as shown in Figure 4(b,(c. In most scenarios, forwar traffic connections yiel much higher ata rate. The optimize range for forwar traffic is also consierably smaller than that in backwar traffic. Thus it is also energy efficient to maintain forwar traffic connections. Our results show that the ata rate of forwar traffic connections an backwar traffic connections is epenent on c. The value of c, in turn, is closely relate to the correlation of the contents between two noes. If noes have highly correlate contents, any two arbitrary noes may want to exchange only a few files with each other, effectively moele by a small c. It is more efficient to maintain backwar traffic connections an exchange files with more noes. In a content istribution application, this is an appropriate strategy when most noes get most of the files alreay. Similarly, when new content is isseminate, noes have few files in common an shoul maintain forwar traffic connections to exploit the long (b ( V. DISCUSSIONS In [2], it was shown that mobility increases the capacity of a mobile infostation network. Capacity gain arises from the realization of the maximal spatial transmission concurrency in each network snapshot. Mobility comes into the picture by shuffling noe locations, creating numerous instances when excellent channels between ifferent noes can be exploite (multiuser iversity. As a result of mobility, the sum capacity of each network snapshot translates to the long run en-toen network throughput. If there is no constraint on the elay requirement, en-to-en capacity oes not epen on noe mobility per se. On the other han, in this paper we focuse on the physical implications of mobility. The fraction of connection time of a target noe over an interval is etermine by the rate of encounters an the connection time of each encounter, both of which are obviously relate to noe mobility. It turns out that for backwar traffic, the fraction of connection time is really inepenent of noe mobility. For forwar traffic, however, the fraction of connection time (an thus the ata rate increases as mobility increases. Numerical results show that the fraction of connection time is almost twice the minimum value in all scenarios at high noe mobility. Thus, mobility not only provies a mechanism for the exploitation of multiuser iversity. The increase of the fraction of connection time an ata rate is a physical consequence of noe mobility. Incientally, this also provie an incentive for network noes to be mobile, which is crucial to ecrease the en-to-en elay. Whereas the performance of a mobile infostation network improves at high mobility, network performance usually egraes with mobility in other wireless paraigms. It is well known that mobility is etrimental to multihop networks. Extraneous overhea is neee for route maintenance to cope with link failures ue to noe mobility. On the other han, the fraction of connection time in a fixe infostation moel [1] is constant regarless of noe mobility. Thus the mobile infostation network paraigm is superior to multihop networks an fixe infostation networks in its robustness to noe mobility. REFERENCES [1] R. H. Frenkiel, B. R. Barinath, J. Borras, an R. Yates. The infostations challenge: Balancing cost an ubiquity in elivering wireless ata. IEEE Personal Communications, 7(2:66 71, April 2. [2] M. Grossglauser an D. Tse. Mobility increases the capacity of a-hoc wireless networks. In Proceeings of IEEE INFOCOM 1, volume 3, pages 136 1369, 21. [3] P. Gupta an P.R. Kumar. The capacity of wireless networks. IEEE Trans. on Info. Theo., 46(2:388 44, 2. [4] J.F.C. Kingman. Poisson Processes. Oxfor University Press, New York, 1993. [5] M. Papaopouli an H. Schulzrinne. Effects of power conservation, wireless coverage an cooperation on ata issemination among mobile sevices. In Proc. IEEE MobiHoc 1, 21. [6] S.M. Ross. Introuction to Probability Moels. Acaemic Press, Lonon, 2. [7] W.H. Yuen an R.D. Yates. Optimum transmit range an capacity of mobile infostation networks. submitte for publication. [8] W.H. Yuen, R.D. Yates, an S.-C. Mau. Exploiting ata iversity an multiuser iversity in mobile infostation networks. to appear, IEEE Infocom 23.