Optimal Spatial Reuse in Poisson Multi-hop Networks

Transcription

1 Otimal Satial Reuse in Poisson Multi-ho Networks Kostas Stamatiou an Martin Haenggi Deartment of Electrical Engineering University of Notre Dame, Notre Dame, IN Abstract We consier a wireless multi-ho network with sources that are Poisson istribute an relays which are lace on the source-estination line. Given a combine TDMA/ALOHA MAC rotocol, we exlore the following question of otimal satial reuse: Increasing the number of noes that are simultaneously scheule to transmit in a route allows noes to transmit more often. At the same time, it results in an increase of intraroute an inter-route interference, which has a negative imact on the en-to-en elay an throughut. In a regime of large source-estination istances R, we fin that it is elay-otimal for either only one noe, or a number of noes that increases linearly in R, to be scheule in each slot, eening on the ALOHA robability. If the transmission robability is also otimize, we fin that maximum satial reuse is elay-otimal. Scaling laws for the en-to-en elay an throughut are erive in all cases. I. INTRODUCTION The remise of multi-ho transmission in wireless networks is the eloyment of intermeiate noes to relay ackets from the source to the estination, in scenarios where irect communication is not ossible ue to ower or interference limitations. This aer aresses the esign issue ertinent to multi-ho networks, of jointly otimizing the number of hos an intra-route satial reuse in orer to minimize the en-toen elay. The traeoff involve is that allowing more noes to simultaneously access the channel er route leas to a higher satial reuse, but, at the same time, it increases interference both within the route an in the rest of the network. Our framework, which is an extension of [1], [2], encomasses ranom noe lacement, a channel moel with faing, ath-loss an interference, an queueing elays associate with multi-ho transmission. In articular, we consier a network of Poisson istribute sources, each with its own estination, an relays lace on the source-estination line. The MAC rotocol is a combination of ALOHA an TDMA: Within each route, a grou of noes with a given satial searation, is given a TDMA token which allows them to transmit with a certain robability. In the next slot, the token is asse to the next grou of noes an so on, until all grous have been given their turn an the TDMA cycle starts again. This rotocol is selecte in light of its relative simlicity an the fact that it allows us to tune the intra-route satial reuse in a straightforwar manner. We fin that the otimal reuse eens on the transmit robability. If the latter is also otimize, then maximum reuse is otimal. Scaling laws for the en-to-en elay an throughut are erive for large source-estination istances. Previous work on Poisson networks has mostly stuie the single-ho case, e.g., [3], [4] an multi-ho extensions have focuse on throughut but not elay [5]. In [6], the issue of otimal satial reuse was aresse for line networks in terms of the achievable en-to-en rates, but only a single route was consiere. A. Network setting II. SYSTEM MODEL The network consists of an infinite number of sources at locations {x i }, which form a homogeneous PPP Ψ = {x i } R 2 of ensity λ. Each source has a estination at istance R an a ranom orientation. Packets are relaye from the source to its estination by N 1 equiistantly lace relays, N Z + (if N = 1, we have single-ho transmission). The sources are backlogge, i.e., they always have ackets to transmit. Each relay has an infinite buffer, where ackets that are receive from the revious noe in the route can be store in a first-in, first-out fashion. Time is ivie into acket slots. Within a route of N hos, a TDMA/ALOHA rotocol is observe accoring to which, at any given time, noes at a istance of hos, = 1,..., N, are allowe to transmit with a certain robability. If the noe is a source, this robability is an if it is a relay, it is r. Let us label the relays with the numbers 1 to N 1. The rotocol oeration can be escribe as follows: At slot 1, the source is allowe to transmit with robability an the relays, 2,... are each allowe to transmit 1 ineenently with robability r. At slot 2, the relays + 1, 2 + 1,... are each allowe to transmit with robability r, an so on, until grous of noes have been given their turn an it is time for the first grou to be scheule again. Note that: The number of noes er grou in the tyical route may vary, but it is at most N an, on average, N. The case = 1 corresons to maximum satial reuse an the case = N to no intra-route satial reuse. If the signal-to-interference-an-noise ratio is above a target threshol θ, a acket is successfully receive. If it is not, the transmitting noe is informe via an ieal feeback channel 1 Note that a relay which is allowe to transmit is a otential transmitter. It may not transmit as its queue might be emty.

2 an the acket remains at the hea of its queue until the noe gets another oortunity to transmit. We also assume that the noes have access to a common clock (obtaine, e.g., by GPS), i.e., the network is synchronize at the slot level. However, the TDMA scheules nee not be aligne in any way, i.e., ifferent grous of noes across routes might be scheule in the same slot. Note that N,, are esign arameters, i.e., they are otimizable accoring to the esire metric(s) for given values of R, λ, r an θ. B. Physical layer The channel between two noes at istance r inclues Rayleigh faing (with a coherence time of one slot) an athloss accoring to the law r b, where b > 2 is the ath-loss exonent. For ease of exosition, we consier an interferencelimite setting, i.e., thermal noise is assume to be negligible an all noes have the same transmit ower, normalize to one. (The analysis can be extene to inclue thermal noise.) Consier the ho/slot in the tyical route which is subject to the largest number of intra-route interferers, an let the corresoning receiving relay (RX) be locate (without loss of generality) at the origin. The signal-to-interference-ratio (SIR) is a ranom variable (r.v.) efine as SIR = where A( R N ) b z Π e zt z B z z b + (1) N n=1,n N 2 r b n e n B n A is the faing coefficient between RX an its transmitting noe, exonentially istribute with unit mean. Π is the oint rocess of inter-route noes, scheule at the given slot. e z = 1, when the noe at location z transmits a acket. If the noe is a source, then P(e z = 1) = (the resective robability for a relay follows in the next section. B z is the faing coefficient between the noe locate at z an RX, an is exonentially istribute with unit mean. n is the inex of the intra-route noe scheule to transmit in the given slot. For = 1,...,N 1, the istance of that noe from RX is r n R N N 2 = n n + 1, n = 1,..., N 2 1 N 1, n = N ,..., N (The inex N 2 corresons to the esire transmitter.) {e n, B n } are efine similarly to {e z, B z }. We enote the total intra-route an inter-route interference as I i an I o, resectively. The resective SIRs are enote as SIR i an SIR o. C. Definition of metrics The mean en-to-en elay D corresoning to the tyical route is efine as the mean total time (in slots) that it takes a acket to travel from the hea of the source queue to its P(SIRo θ) Sim., N = 2k Poisson, ens. λk Sim., N = k Poisson, ens. λk Fig. 1. Success robability vs. k taking into account only inter-route interference (R = 5 m, λ = 1 4, =.5, θ = 6 B, b = 4). estination. D is the sum of the mean service time at the source, an the service times an waiting times along the relays of the route. The service time is measure from the moment a acket reaches the hea of the queue until it is successfully receive by the next noe. The waiting time is measure starting from the moment a acket arrives at a relay s queue until it becomes the hea-of-line acket, i.e., all ackets in front of it have been successfully transmitte to the next noe. The route throughut T is efine as the execte number of ackets successfully elivere to the estination er slot. By efinition, T > 1/D, i.e., the inverse of the elay rovies a lower boun on the throughut. As a result, by minimizing D, a lower boun on the throughut is also maximize. The next two sections focus on the evaluation an otimization of D. k III. DELAY ANALYSIS In orer to make the analysis tractable, assume that acket successes across all hos on all routes are ineenent events. The assumtion is base on the observation that if the robability that a noe is a transmitter is small, then, in combination with faing, a sufficient egree of ranomization is achieve in the network 2. Moreover, consier the worst-case scenario where acket success robabilities across hos of the tyical route are all equal to the smallest one, corresoning to the receiver(s) subject to the largest number of intra-route interferers. Denote this robability by s. It is then unerstoo that a necessary conition for the relay buffers to be stable is that < r, as then the acket arrival robability to the first relay, s, is smaller than the acket earture robability from the first an all subsequent relays, r s. As formally shown in [7], acket arrivals to all relays are ii geometric with arameter s, an the robability that a relay locate at z transmits a acket is simly P(e z = 1) =. 2 In the case = 1, a relay is either allowe to transmit or receive at a given time, which creates correlation between the success robabilities. However, such a scenario arises when successive noes actually have ackets to transmit, which, as we will see in Section IV, is unlikely for sufficiently small.

3 If = N, then, since Ψ is a PPP, it follows from the islacement theorem [8] that the oint rocesses of otential inter-route interferers Π, an actual inter-route interferers Π 2 = {z : e z = 1, t z = 1}, are PPPs with ensities λ an λ resectively. By Corollary 3.2 in [9], the ho success robability is therefore s = P(SIR o θ) = e λc(r/n)2, where c = Γ(1+2/b)Γ(1 2/b)πθ 2/b is the satial contention arameter [4] an Γ(x), x >, is the gamma function. For < N, Π 2 can be well aroximate as a PPP of ensity λ = λk, where k = N/ is the average number of scheule noes er route. This is shown in Fig. 1, where we have evaluate the success robability via simulation taking into account only inter-route interference, P(SIR o θ), an comare it to the exression e λkc(r/n)2, for ifferent k, an N = k, 2k (i.e, = 1, 2). Since P(A x) = e x, s = P(SIR θ), where the SIR is efine in (1), can be written as s = Φ Ii (γ)φ Io (γ), (2) where Φ X (s) = E[e Xs ], s >, enotes the Lalace transform of the f of the r.v. X an γ ( ) R b N θ. Base on the observation of the revious aragrah, we can aroximate Φ Io (γ) by [8] Φ Io (γ) e λkc(r/n)2 = e λcr2 N. (3) Since {e n, B n } are ineenent, it is also straightforwar to show that [4] Φ Ii (γ) = ( ) r b n n /γ, (4) where n takes values as in (1). Note that, for = 1, Φ Ii (γ) inclues the term 1, which is the robability that the receiver has no ackets in its queue. Moreover, Φ Ii (γ) oes not een on the hoing istance R/N. A. Delay an throughut exressions Following the analysis in [1], the service time for the heaof-line acket at the source is H = s + 1, an, similarly, the service time for the hea-of-line acket at a relay is H r = + 1. r s Moreover, the waiting time at the queue of a relay is Q r = r 1 r s ( r ) s. The en-to-en elay of the tyical route is therefore D = H + (N 1)(H r + Q r ) = + (N 1) 1 s N( 1). (5) s ( r ) s ΦIi N =.1 =.5 Fig. 2. Success robability taking into account only intra-route interference (b = 4, θ = 6 B). Since a acket is receive by the estination every slots with robability s, the first term is the inverse of the (stable) en-to-en throughut T = s /. The secon term is ominate by the value of ( r s s ) 1, i.e., the inverse of the ifference between the acket service an acket arrival rate at the buffer of each relay. Therefore, in orer to minimize D, the en-to-en throughut an the time sent in the relay queues must be otimally trae off. Given that for small values of, 1 s 1, the simler exression D = (N 1) + N( 1) (6) s ( r ) s rovies a tight uer boun to D. IV. DELAY OPTIMIZATION In this section, we exlore the eenence of D on the arameters N,,. For convenience, we let N [1, + ) an [1, N], an set Φ Ii (γ) 1, i.e., we temorarily ignore intra-route interference. As seen in Fig. 2, Φ Ii is insensitive to N for N 1 an a given, while, for a given N, it quickly aroaches unity as increases. In the first case, we thus exect that the aroximation will yiel a constant, albeit small, erformance ga for small, while, in the latter case, this ga will ecrease with increasing. These qualitative observations are confirme by numerical examles. First, we observe that the following scaling law hols. Proosition 1 As R, D = O(R 2 ) only if N/ = Θ(R 2 ). Proof: Due to the fact that s e λcr2 N, we can see from (6) that N/ = o(r 2 ) imlies that D = ω(e R 2 ). If N/ = ω(r 2 ), then, fixing an imlies that N = ω(r 2 ), hence D = ω(r 2 ). Accoring to Proosition 1, N/ Θ(R 2 ) results in a scaling of D which is worse than quaratic. Hence, in otimizing D over N,,, we constrain the arameter set to

4 1 5 r =.5 r =.1 r =.2 8 x , fixe q 5 D (slots) Max reuse, otimal Otimal reuse, fixe, otimal Fig. 3. q - efine in (7) - vs., for ifferent values of r. satisfy N/ = Θ(R 2 ). Uner this constraint, the success robability satisfies s = e Θ(1). A. Fixe source transmission robability Suose that is fixe. The following roosition characterizes the - jointly - elay-otimal N an. Proosition 2 Denote by N o an o the values of N an that jointly minimize (6) when R + an efine q = 1 1 r. (7) If q, then N o = o = Θ(R). If q > 1, then o < N o an N o = Θ(R), o = Θ(R). Proof: From Proosition 1 an (6) we have that D = qe Θ(1) + g ( r eθ(1) g 1 1 ), (8) where q is efine in (7) an g N. The erivative with resect to is D = qeθ(1) g 2. If q, the erivative is always negative an the minimum D is achieve for the maximum ossible, i.e., o = N o. Since N o o = Θ(R 2 ), we have o = N o = Θ(R). If q > 1, then (8) is minimize for o = g < g < N qe Θ(1) o. Since g = Θ(R 2 ), we have that N o = Θ(R) an o = Θ(R). Remarks: 1) The arameter q eens on the relation of to r. If r /2, i.e., q, then allowing only one noe to transmit er route minimizes the en-to-en elay. The intuition behin this rule is that, since the traffic is heavy in the relay queues (high-traffic regime), it is referable to kee interference low at the exense of satial reuse. On the other han, if the system is oerate in a low-traffic regime, i.e., q > 1, the elay is minimize if o an N o are linear functions of R, i.e., if a constant number of noes k o = N o / o (as a function of R) is scheule to transmit in each slot. As seen Fig. 4. D - eq. (5) - vs. R, otimize over, N, for =.4, an otimize over, N, jointly. For fixe, D = Θ(R 2 ). For otimize, = 1 is otimal an D = Θ(R). The resective curves with no satial reuse are shown for comarison. (λ = 1 4 routes/m 2, r =.1, b = 4 an θ = 6 B.) λt (ac./slot/m 2 ) 1 5 Otimal reuse, inter route interf. only Otimal reuse, total interf. 1 8 Fig. 5. λt vs. R, for =.4. All curves follow the tren R 1 but satial reuse rovies a throughut gain over no satial reuse. (λ = 1 4 routes/m 2, r =.1, b = 4 an θ = 6 B.) in Fig. 3, the transition of q from negative to ositive values is very stee aroun = r /2 (hence the range q (, 1) is inconsequential). We may seak of a hase transition from no satial reuse to satial reuse, as becomes smaller than r /2. 2) From (6), we can see that, in both high an low traffic regimes, D = Θ(R 2 ) an T = Θ(R 1 ). That no imrovement in elay is achieve by allowing satial reuse is exlaine by the fact that, on the one han, noes get an oortunity to transmit more often, on the other han the interference in the network increases. The benefit of satial reuse is manifeste in terms of a throughut gain, which is of the orer of qe Θ(1). Examle 1: Consier a network with arameters λ = 1 4 routes/m 2, r =.1, =.4, b = 4 an θ = 6 B. We numerically otimize (5) over N Z + an = 1,..., N, an lot the results in Figs We observe that the results of Proosition 2 are confirme an that taking into account intra-route interference results in a larger (or smaller N an T ) than the case where only inter-route interference is taken

5 Φ Ii Otimal reuse 2 15 Max reuse, otimal Otimal reuse, fixe, fixe s.6.5 No 1, otimal Fig. 6. s vs. R, for =.4. The asymtotic value of s is aroximately.37. Φ Ii is lotte for comarison; it is very close to one for all R. (λ = 1 4 routes/m 2, r =.1, b = 4 an θ = 6 B.) Total interf. Inter route interf. Fig. 8. Otimal number of hos vs. R, corresoning to Fig. 4. When is fixe, N is quaratic in R for each ste of - see Fig. 7 - but the overall tren is Θ(R). When is otimize, = 1 is otimal an N = Θ(R). The resective curves for no satial reuse are shown for comarison. (λ = 1 4 routes/m 2, r =.1, b = 4 an θ = 6 B.) λt (ac./slot/m 2 ) 1 5 Otimal reuse, inter route interf. only Otimal reuse, total interf. 2 Fig. 7. Otimal satial reuse vs. R, for =.4. Taking into account both inter an intra-route interference results in less aggressive reuse. (λ = 1 4 routes/m 2, r =.1, b = 4 an θ = 6 B.) into account. The corresoning elay curves, however, are inistinguishable. These results illustrate that, for sufficiently small, a throughut gain is achieve comare to the case of no satial reuse. However, as seen in Fig. 8, this comes at a cost in terms of the require number of hos. For a fair comarison between the two cases in the sense of require resources, i.e., relays, we constrain the number of hos to be no larger than 5, which is the maximum value of N with no satial reuse for R = 8 m. The network throughut er unit area is lotte vs. R in Figs. 9. At R 2 m, the constraint on N takes effect, so, by Proosition 1, starts to increase quaratically with R, hence the throughut ecreases as 1/R 2, until = N. The main message of Fig. 9 is that juicious (over no) satial reuse results in a throughut gain which eens on R, given the constraint lace on N. B. Variable source transmission robability We now consier the scenario where N, an are jointly otimize. We have the following result. 1 8 Fig. 9. λt vs. R for =.4 an N 5. In contrast to Fig. 5, the throughut gain achieve with satial reuse ecreases with R, since increases quaratically with R. (λ = 1 4 routes/m 2, r =.1, b = 4 an θ = 6 B.) Proosition 3 Denote by N o, o an o the values of N, an that jointly minimize (6) when R. Then o, N o o = Θ(1) an o N 2 o = Θ(R 2 ). Proof: D, given in (8), is strictly convex in (, r ) an lim + D = lim r D = +. As a result, the otimal, o, is obtaine by setting D = or =o,= o o 2 + o o ( r o ) 2 = g r ( r o ) 2 + g ( ) 1 1. (9) e Θ(1) o We have the following cases: 1) o = Θ(1): The scaling with R on both sies is the same iff o = Θ(R 2 ). However, the constraint N o o / o = Θ(R 2 ) woul then imly that N o = Θ(1), which violates the requirement o N o. 2) o : Due to the constraint 1, it is necessary that = ω(1). Since g = Θ(R 2 ), (9) imlies that o / 2 o = Θ(R2 ).

6 λt (ac./slot/m 2 ) 1 5 Max reuse Max reuse, 1/D Max reuse 1 9 Fig. 1. λt vs. R, for otimize. With maximum reuse, the throughut scales as R 1. The inverse of D is a lower boun to T. (λ = 1 4 routes/m 2, r =.1, b = 4 an θ = 6 B.) Fig. 11. Otimal vs. R. The scaling is = Θ(R 1 ); with no reuse, = Θ(R 2/3 ) [2]. (λ = 1 4 routes/m 2, r =.1, b = 4 an θ = 6 B.) Since N o o / o = Θ(R 2 ), we have that N o o = Θ(1) an o No 2 = Θ(R2 ). Remarks: 1) Eq. (6) can be rewritten as: D = ( 1 s + N 1 ) r N + N. Since o an N o o = Θ(1), = 1, i.e., maximum satial reuse minimizes D. As a result, N o = Θ(R) an o = Θ(R 1 ). The scaling o = Θ(R 1 ) also imlies that the robability of any two consecutive relays in the tyical route transmitting a acket is of the orer N 2 = Θ(R 1 ), i.e., it goes to zero as R grows large. 2) D = Θ(R) an T = Θ(R 1 ). Note that these scaling laws are erive with no constraint on N. If N (hence ) is constraine, then, by Proosition 1, = Θ(R 2 ). Examle 2: Consier a network with the same arameters as in Examle 1, with the only ifference that is allowe to vary in (,.1). The numerically otimize D is shown in Fig. 4 an the resective throughut an otimal in Figs Given the selecte arameters, we obtaine that = 1 was otimal for the whole range of R. Moreover, we verifie that the effect of intra-route comare to inter-route interference was negligible. Figs. 4, 1-11 confirm the scaling laws erive in Proosition 3. ACKNOWLEDGMENTS The artial suort of the DARPA/IPTO IT-MANET rogram (grant W911NF ) is gratefully acknowlege. REFERENCES [1] K. Stamatiou, F. Rossetto, M. Haenggi, T. Javii, J. R. Zeiler, an M. Zorzi, A elay-minimizing routing strategy for wireless multiho networks, in Worksho on Satial Stochastic Moels for Wireless Networks (SaSWiN), Seoul, Jun. 29. [2] K. Stamatiou an M. Haenggi, The elay-otimal number of hos in Poisson multi-ho networks, in IEEE Symosium on Information Theory (ISIT 1), Austin, TX, Jun. 21. [3] S. P. Weber, X. Yang, J. G. Anrews, an G. e Veciana, Transmission caacity of wireless a hoc networks with outage constraints, IEEE Trans. Inf. Theory, vol. 51, , Dec. 25. [4] M. Haenggi, Outage, local throughut, an caacity of ranom wireless networks, IEEE Transactions on Wireless Communications, vol. 8, no. 8, , Aug. 29. [5] J. G. Anrews, S. Weber, M. Kountouris, an M. Haenggi, Ranom access transort caacity, IEEE Transactions on Wireless Communications, vol. 9, no. 6, , Jun. 21. [6] M. Sikora, J. N. Laneman, M. Haenggi, D. J. Costello, an T. Fuja, Banwith- an ower-efficient routing in linear wireless networks, Joint Secial Issue of IEEE Transactions on Information Theory an IEEE Transactions on Networking, vol. 52, , Jun. 26. [7] J. Hsu an P. P. Burke, Behavior of tanem buffers with geometric inut an Markovian outut, IEEE Transactions on Communications, vol. 24, no. 3, , Mar [8] M. Haenggi, J. G. Anrews, F. Baccelli, O. Dousse, an M. Franceschetti, Stochastic geometry an ranom grahs for the analysis an esign of wireless networks, IEEE Journal on Selecte Areas in Communications, vol. 27, no. 7, , Se. 29. [9] F. Baccelli, B. Błaszczyszyn, an P. Mühlethaler, An Aloha rotocol for multi-ho mobile wireless networks, IEEE Trans. Inf. Theory, vol. 52, , Feb. 26. V. CONCLUSIONS We roose a framework to characterize the elay-otimal number of hos an intra-route satial reuse in Poisson multiho networks. The scaling of the elay an throughut as functions of R were characterize. Our results have alications in routing algorithms for multi-ho networks where the relays are also ranomly locate, e.g., a routing rotocol can select the relays which are foun closest to the ieal locations etermine by the analysis.