Throughput Optimal Control of Cooperative Relay Networks

Transcription

1 hroughput Optimal Control of Cooperative Relay Networks Emun Yeh Dept. of Electrical Engineering Yale University New Haven, C 06520, USA emun.yeh@yale.eu Ranall Berry Dept. of Electrical an Computer Engineering Northwestern University Evanston, IL 60208, USA rberry@ece.northwestern.eu Abstract We give a moel for cooperative communication in a parallel relay network that inclues the stochastic arrival of packets an queueing. For this moel we provie a throughput optimal network control policy which stabilizes the network for any arrival rate in its stability region. his policy is a generalization of imum ifferential backpressure policies which takes into account the potential cooperative gains in the network. I. INRODUCION In recent years, motivate chiefly by wireless networking applications, there has been interest in moels which jointly aress network layer issues such as the ranom generation of traffic, elay, an buffer occupancy, along with traitional physical layer issues such as moulation, coing, an channel moeling. In [1], [2], moels for multiaccess an broacast channels taking into account both queueing ynamics as well as information-theoretic capacity regions have been consiere. For these moels, the network stability region is characterize; this is the set of arrival rates for which all queues can be stabilize by a feasible rate an power allocation policy. Furthermore, simple throughput optimal resource allocation policies are specifie, which stabilize the system for any arrival rates in the stability region, without requiring any a priori knowlege of the arrival statistics. Results using similar techniques have been shown in [3], [4] for other non-information theoretic physical layer moels, where joint power/rate allocation an routing are performe for multi-hop transmission. A feature of all the above moels is that each packet follows a single route from the source to the estination. In particular, this oes not incorporate the potential gains from various cooperative relaying techniques e.g. [5], [6], [7], [8], [9], [10]. With such techniques, multiple noes may cooperate in relaying a packet, essentially forming a istribute antenna array. Cooperative communication has mainly been aresse from the physical layer viewpoint, i.e. by stuying the achievable rates or iversity gains of given cooperative schemes. A goal of this paper is to stuy a moel of cooperative communication which incorporates the stochastic arrival of traffic an queueing ynamics at the various noes in the 1 his research was supporte in part by NSF uner grants CCR an CCR , an by ARO uner grant DAAD Fig A four noe parallel relay network moel. network. In the next section, we escribe such a moel base on the parallel Gaussian relay channel stuie in [5], [6]. For this moel we characterize the network s stability region an give a throughput optimal network control policy. Compare to the moels in [1], [2], [3], [4], a new potential trae-off emerges: in orer to exploit cooperative gains, information has to be sent along multiple routes; this in turn temporarily increases the congestion within the network. II. NEWORK MODEL Consier a simple network with cooperative communication as shown in Figure 1. his network G consists of four noes V = {1, 2, 3, 4}. raffic originates at noes 1, 2, an 3, an the estination of all traffic in the network is noe 4. At the physical layer, we moel this as a moifie version of a Gaussian parallel relay network [5], [6]. Each noe i V has an average power constraint P. Noe 1 communicates with noes 2 an 3 over a Gaussian broacast channel with banwith W. If X 1 t is the transmitte signal by noe 1, then the receive signal at noe i = 2, 3, is given by Y i t = h 1i X 1 t + Z i t, 1 where Z i t is a white Gaussian noise process with noise ensity N 0 /2 an h 1i is the channel gain between noe 1 an i. 1 Noes 2 an 3 the communicate to noe 4 over a 2- user Gaussian multiaccess channel, also with banwith W. 1 Note all channel gains are fixe an known at the transmitters an receivers. Hence, using cooperative transmissions for iversity gains as in [9] is not relevant.

2 he receive signal at noe 4 is given by Y 4 t = hi4 X i t + Z 4 t, i=2,3 where Z 4 t is a white Gaussian noise process with noise ensity N 0 /2. All noise processes are mutually inepenent an inepenent of the channel inputs. For convenience, we normalize both the banwith an the noise power: W = 1, N 0 W = 1. We efine a symmetric network to be one where h 12 = h 13 = 1 an h 24 = h 34 = 1. o simplify our iscussion, we assume that at any time the network is either operating in the multi-access moe or the broacast moe, i.e., if X 1 t > 0 then X 2 t = 0 an X 3 t = 0, an likewise if X 2 t > 0 or X 3 t > 0 then X 1 t = 0. From the point of view of noes 2 an 3, this enforces a half-uplexing constraint [9], [10], i.e. these noes cannot transmit an receive simultaneously. his is consiere a realistic constraint in practical systems. However, we note that our assumption also prohibits scheules which o not violate the half-uplexing constraint, such as noe 1 transmitting to noe 2, while noe 3 transmits to noe 4. Also, as in [5], we o not consier irect transmissions from noe 1 to noe 4. his is reasonable when the istance between these noes in large. Both of the above-mentione possibilities coul in principle be inclue in our moel at the expense of more complicate notation. Cooperation in this network is achieve by having noes 2 an 3 cooperate to relay information from noe 1 to noe 4. We assume that this is accomplishe by having both noes use a ecoe an forwar strategy. Namely, they will both receive an ecoe the same packet from noe 1; eventually, they will simultaneously transmit this packet to noe 4 by coherently beamforming the receive signal at noe 4. For example, in a symmetric network, if noe 2 an 3 cooperatively transmit a packet using the imum power, P, then the receive signal power will be 4P. Of course, this requires that the noes must be perfectly synchronize. In the absence of perfect synchronization, other cooperative techniques coul be use. Also, we note that in general this is not the optimal cooperative strategy from the view of imizing capacity. 2 We o not require that all traffic from noe 1 to 4 is relaye using this cooperative moe; noe 1 can also sen irect traffic to either noe 2 or 3, which the receiving noe then iniviually relays to noe 4. Exogenous traffic arrives at noe i = 1, 2, 3 accoring to an ergoic counting process A i t, where A i t is the number of packet arrivals up to time t. he packet lengths Z i of exogenous traffic at noe i are i.i.. with E[Z i ] < an E[Zi 2 ] <. Each noe will store all arriving packets in an infinite capacity buffer until they are transmitte. Let U 1 t be the number of untransmitte bits unfinishe work at noe 1, an let U i t an U ic t respectively be the unfinishe work of irect an cooperative traffic at noe i = 2, 3. All exogenous arrivals at noes 2 2 Inee, the optimal strategy an the capacity of the parallel Gaussian relay channel is an open problem [6]. an 3 are inclue in the irect traffic. By assumption, all cooperative traffic is receive at both noes 2 an 3. Hence, for all t, U 2c t = U 3c t, an so we will enote both of these quantities by U c t. Let Ut = U 1 t, U c t, U 2 t, U 3 t enote the joint queue state at time t. We consier the case where given Ut at time t, a network controller specifies a rate allocation Rt = R1, c R12, R13, R4, c R24, R34, where Rij is the rate of irect traffic between noes i an j, Rc 1 is the rate noe 1 sens cooperative traffic to noes 2 an 3, an R4 c is the rate noes 2 an 3 cooperatively forwar traffic to noe 4. At times, it will be more convenient to enote the components of Rt as R 1 t, R 2 t,..., R 6 t, where for example R 6 R34. he rate allocation chosen for time t must respect the halfuplex constraint escribe above an given that the network operates in the broacast or multiaccess moe, the rates must lie in the corresponing capacity region. 3 We escribe these capacity regions next. First consier the broacast moe an without loss of generality assume that h 12 h 13. Let C BC be the capacity region of the two-user Gaussian broacast channel efine by 1. hen it follows that the rates R1, c R12, R13 must satisfy R12 + R1, c R13 C BC. Let the cooperative broacast region C CBC be the set of all such allowable R1, c R12, R13. For a symmetric network h 12 = h 13 C CBC reuces to the set of non-negative rates that lie in the simplex efine by R i log 1 + P. 2 In the multiaccess moe, if noes 2 an 3 only sen irect traffic R4 c = 0, then the transmission rates R24, R34 must lie in the corresponing multiaccess capacity region C MAC ; this is the set of non-negative rates that satisfy Ri4 log 1 + h i4 P S {2, 3}. If both noes sen only cooperative traffic R24 = R34 = 0 then the transmission rate R4 c must satisfy R4 c log 1 + h 24 + h 34 2 P. In aition, we allow the noes to transmit both cooperative an irect traffic simultaneously. One way to moel this is to allow time-sharing between the above two moes. More generally, we can view this as a type of 3-user multiaccess channel, with 2 users corresponing to the irect traffic for noes 2 an 3, respectively, an a thir user corresponing to the cooperative traffic. 4 he ifference here is that the power constraints of the users are couple. If both users 2 an 3, evote a fraction α [0, 1] of their power to 3 It is reasonable to assume that we can achieve any rate in these regions if the times at which we apply the controls are sufficiently separate as to allow the use of long coewors. 4 A key assumption here is that the encoing of the traffic by these three users is one only base on their own message an that the messages are inepenent.

3 cooperative traffic, then we assume that they can achieve any rates R c 4, R 24, R 34 R 4, R 5, R 6 which satisfy R i log 1 + P i α S {4, 5, 6}, 3 where P 4 α = h 24 + h 34 2 αp, P 5 α = h 24 1 αp, an P 6 α = h 34 1 αp. Let C CMAC α enote this set of feasible rates for a particular power splitting parameter α. hen we assume that the controller can choose any rates from the cooperative-mac capacity region given by C CMAC = α [0,1] C CMACα. It can be verifie that the resulting region is convex. Let C CBC an C CMAC be C CBC an C CMAC embee in R 6 +, respectively. hat is, R c 1, R 12, R 13 C CBC if an only if R c 1, R 12, R 13, 0, 0, 0 C CBC, an R c 4, R 24, R 34 C CMAC if an only if 0, 0, 0, R c 4, R 24, R 34 C CMAC. Uner the uplex constraint, the overall physical-layer capacity region is C = convc CBC, C CMAC, i.e. the convex hull of these two sets. III. NEWORK SABILIY REGION AND HROUGHPU OPIMAL RAE ALLOCAION Given the moel in Section II, we procee to characterize the network stability region an the throughput optimal rate allocation policy. Although the results we obtain here may be reminiscent of results for conventional networks [3], [4], we shall fin that the cooperative nature of the parallel relay network introuces some significantly new elements. A. Stability Region Let λ i = lim t A i t/t be the packet arrival rate of traffic to noe i. Let ρ i = λ i E[Z i ] be the corresponing bit arrival rate. We say that queue i is stable if 1 t lim sup t t 0 1 [U iτ>ξ]τ 0 as ξ, where 1 { } is the inicator function. he network stability region S is the closure of the set of all ρ 1, ρ 2, ρ 3 R 3 + for which there exists some feasible rate allocation policy R efine by R = Ru C, where u = u 1, u c, u 2, u 3, which can guarantee that all queues are stable. Note that R is a function of the queue state u an can assign any rate from the physicallayer capacity region C. Using arguments similar to those in [3], [4], we can establish the following. heorem 1: he stability region S of the parallel relay network of Figure 1 is set of all ρ 1, ρ 2, ρ 3 R 3 + for which there exist non-negative flow variables f c, f 12, f 13, f 24, f 34 which support ρ 1, ρ 2, ρ 3 relative to the weighte graph efine by the long-term rates given by C. hat is, the following flow conservation relations must be satisfie: ρ 1 = f c + f 12 + f 13, ρ 2 = f 24 f 12, ρ 2 = f 34 f 13. In aition, f c, f 12, f 13, f c, f 24, f 34 C. B. hroughput Optimal Rate Allocation heorem 1 states that if ρ = ρ 1, ρ 2, ρ 3 ints, then the queues can be stabilize. In general, however, this may require knowing the value of ρ. In reality, ρ can be learne only over time, an may be variable. One woul prefer to fin aaptive rate allocation policies which can stabilize the network without knowing ρ, as long as ρ ints. hese rate allocation policies are calle throughput optimal. As shown in [4], a throughput optimal resource allocation policy for stochastic networks with physical-layer capacity regions turns out to be a generalization of the imum ifferential backlog MDB policy first propose by assiulas [3]. Due to cooperative transmissions, however, the parallel relay network consiere here is somewhat ifferent from the networks consiere in [4]. Nevertheless, we show that the MDB policy can be aapte to prouce a throughput optimal rate allocation policy for the parallel relay network. We consier examine the evolution of the network at time instants separate by > 0 units of time, where is sufficiently large to allow for large coing lengths. We make the following assumptions for the packet arrival processes. Let A k = A 1k, A 2k, A 3k be the arrival vector for the kth -slot. We assume {A k : k Z + } are i.i.. accoring to istribution π A with mean E[A] = λ, where λ = λ 1, λ 2, λ 3 are the arrival rates. Furthermore, assume that for each i, E[A 2 i ] <, an Pr 3 {A i = 0} > 0. hese assumptions on the arrival process, for example, clearly hol for inepenent homogeneous Poisson arrival processes. he above assumptions can be relaxe to the Markov moulate case. heorem 2: A throughput optimal rate allocation policy R u for the parallel relay network of Figure 1 is given by the solution to the following optimization: R C u 1 2u c R c 1 + u 1 u 2 R 12 + u 1 u 3 R 13 +u 2 R 24 + u 3 R u c R c 4 4 Note that the policy in 4 is the not the same as the conventional MDB policy of [3], [4]. In particular, the coefficient terms u 1 2u c an 2u c reflect the queue coupling effect inuce by the cooperative transmission structure. We refer to the policy of 4 as the Cooperative Maximum Differential Backlog CMDB policy. Proof of heorem 2: o show that the CMDB policy stabilizes the network for any ρ = ρ 1, ρ 2, ρ 3 ints, it is convenient to consier a fictitious network G f which is the same as the network G, except that arrivals are allowe to enter the queues 2c an 3c. Let S f be the stability region of the network G f. It is clear that if the CMDB policy stabilizes G f for all ρ 1, ρ 2c, ρ 2, ρ 3c, ρ 3 ints f such that ρ 2c = ρ 3c = 0, then CMDB also stabilizes G for all ρ = ρ 1, ρ 2, ρ 3 ints. herefore, from now on, we concentrate on the artificial network G f.

4 o show that the CMDB policy stabilizes G f for all ρ 1, 0, ρ 2, 0, ρ 3 ints f, we use an extension of Foster s Criterion for the convergence of Markov chains [1], [2], [4]. Consier the Lyapunov function V u u i=2 u2 ic + u 2 i. We wish to show that there exists a compact subset Λ R 5 + such that uner the CMDB policy, E[V Ut + V Ut Ut = u] < ɛ for all u / Λ, where ɛ > 0. his, along with some other technical conitions [4], implies the existence of a steay state istribution for U. We have: U 2 1 t + = [U 1 t + B 1 t R c 1t + R 12t + R 13t + ] 2 U 1 t + B 1 t R1t c + R12t + R13t 2 U1 2 t 2 U 1 t R1t c + R12t + R13t B 1t +B1t 2 + R1t c + R12t + R13t 2 2 Here B 1 t is the number of bits arriving to queue 1 in the tth slot, an x + enotes x, 0. Similarly, for i = 2, 3, U 2 ict + U 2 ict 2 U ic tr c 4t R c 1t where β > 0 is an upper boun on a sum of terms involving the secon moments of the bit arrivals in the tth slot which are boune since the secon moments of the packet arrivals an the packet sizes are boune, an powers of transmission rates which are boune since C is boune. Let E u [X] enote E[X Ut = u]. Note that u 1 E u [R1t c + R12t + R13t] + 2u c E u [R4t c R1t] c + u i E u [Ri4t R1it] i=2 = u 1 2u c E u [R c 1t] + u 1 u 2 E u [R 12t] +u 1 u 3 E u [R 13t] + u 2 E u [R 24t] u 3 E u [R 34t] + 2u c E u [R c 4t] 6 For any ρ 1, 0, ρ 2, 0, ρ 3 ints f, there exists δ > 0 such that ρ 1 + δ, δ, ρ 2 + δ, δ, ρ 3 + δ S f. herefore, there exist non-negative flow variables f c 1, f 12, f 13, f c 4, f 24, f 34 C such that ρ 1 + δ = f 1 c + f 12 + f 13, ρ 2 + δ = f 24 f 12, ρ 3 + δ = f 34 f 13, an δ = f 4 c f 1 c. We therefore have u 1 ρ 1 + δ + u 2 ρ 2 + δ + u 3 ρ 3 + δ + 2u c δ = u 1 f 1 c + f 12 + f 13 + u 2 f 24 f 12 + u 3 f 34 f 13 +2u c f 4 c f 1 c = u 1 2u c f c 1 + u 1 u 2 f 12 + u 1 u 3 f 13 +u 2 f 24 + u 3 f u c f c 4 Let Rt = R c 1t, R 12t, R 13t, R c 4t, R 24t, R 34t be chosen accoring to the CMDB rule escribe in 4. hen, since f c 1, f 12, f 13, f c 4, f 24, f 34 C, u 1 ρ 1 + δ + u 2 ρ 2 + δ + u 3 ρ 3 + δ + 2u c δ is less than or equal to the RHS of 6. Since E[B 1 t/ ] = ρ 1 an E[B i t/ ] = ρ i for i = 1, 2, the RHS of 5 implies E[V Ut + V Ut Ut = u] β 2 δu 1 + u 2 + u 3 + 2u c. +R1t c 2 + R4t c 2 2 Let Λ = {u : u 1 + u 2 + u 3 + 2u c β+ɛ, 2 δ }. hen, for any ɛ > 0, an any u / Λ, E[V Ut + V Ut Ut = Uit 2 + Uit 2 2 U i tri4t R1it u] < ɛ. B i t/ + B i t 2 + 2B i tr1it We have shown that the CMDB policy stabilizes G f +R1it 2 + Ri4t 2 2 for all ρ 1, 0, ρ 2, 0, ρ 3 ints f. hus, we have also shown that the CMDB policy stabilizes G for all Note that since ρ 2c = ρ 3c = 0, there are no exogenous arrivals to queues 2c an 3c. aking conitional expecte value of both sies of the above inequalities given the event Ut = u, an re-arranging, we have ρ = ρ 1, ρ 2, ρ 3 ints. IV. CALCULAING HE CMBD POLICY We now focus on solving the optimization problem 4 E[V Ut + V Ut Ut = u] [ 2 u 1 E R1t c + R12t + R13t B ] require by the CMBD policy. We focus on a symmetric 1t Ut = u network h 12 = h 13 = h 24 = h 34 = 1 in this section. In this case, C CBC is given by 2 an the power variables 2 2u c E[R4t c R1t Ut c = u] use in the efinition of C CMAC α become P 4 α = 4αP, [ 2 u i E Ri4t R1it B ] P it 5 α = P 6 α = 1 αp. Ut = u For simplicity of notation, let w 1, w 2, w 3, w 4, w 5, w 6 enote u 1 2u c, u 1 u 2, u 1 u 3, 2u c, u 2, u 3, an let i=2 +β 5 R 1, R 2, R 3, R 4, R 5, R 6 enote R c 1,R 12,R 13,R c 4,R 24, R 34. he CMBD policy can now be expresse as R C w i R i, 7 Note that the solution R to 7 lies in convc CBC, C CMAC α for some α [0, 1], where C CMAC α is the appropriate embeing of C CMAC α in R 6 +. Since C CBC an C CMAC α are both convex polytopes, convc CBC, C CMAC α is also a convex polytope. Now since R also imizes the linear objective of 7 over convc CBC, C CMAC α, R is without loss of optimality an extreme point of convc CBC, C CMAC α. hus, R is an extreme point either of C CBC or an extreme point of C CMAC α. We now consier these cases separately.

5 Case 1: R is an extreme point of C CBC. In this case, R has the form R 1, R 2, R 3, 0, 0, 0. hus, R 1, R 2, R 3 solves R C CBC w i R i. 8 Since C CBC is a simplex, R1, R2, R3 is easily given as follows. Let w [1] w [2] w [3] be w 1, w 2, w 3 arrange in ecreasing orer. hen R[1] = log1+p an R [2] = R [3] = 0. hus, whenever the CMDB policy operates in the broacast moe, it allocates imum rate log1 + P to the traffic type with the largest weight w i. he optimal value of 8 is then L CBC w 1, w 2, w 3 =,2,3 w i log1 + P. Uner the assumption that R is an extreme point of C CBC, L CBC w 1, w 2, w 3 is equal to the optimal value of 7, L w 1,..., w 6. Case 2: R is an extreme point of C CMAC α. In this case, R has the form 0, 0, 0, R4, R5, R6. hus, R4, R5, R6 solves w i R i 9 R C CMAC α i=4 Since C CMAC α is a polymatroi [11], R4, R5, R6 can be explicitly given as follows. Let w [4], w [5], w [6] be the largest, secon largest, an smallest element of {w 4, w 5, w 6 }, respectively. hen R[i] 1 = log P [i] α j<i P [j]α, i = 4, 5, For instance, if w 4 w 5 w 6, then R 4 = log1 + 4α P, R 5 = log α P 1+4α P, R 6 = log α P 1+3α +1P. Next, to fin α, we can solve P [i] α w [i] log 1 + α [0,1] 1 + j<i P. 11 [j]α i=4 Let Lα be the objective in 11. For the case of w 4 w 5 w 6, it can be verifie that Lα is concave in α over [0, 1], an that L α 0 for all α [0, 1]. hus, α = 1, i.e. all power is allocate to cooperative transmission over the MAC. In other cases, Lα may not be concave, an one nees to solve for the stationary points of Lα an compare the value of Lα at the stationary points with its values on the bounaries. Let L CMAC w 4, w 5, w 6 be the optimal objective of 11. Uner the assumption that R is an extreme point of C CMAC, we have L CMAC w 4, w 5, w 6 = L w 1,..., w 6. log1 + P an R [2] = R [3] = 0. Otherwise, R = 0, 0, 0, R 4, R 5, R 6, where R [i] is given by 10 an α is given by 11. V. CONCLUSIONS In this paper we consiere a moel of a parallel relay network that incorporates the stochastic arrival of traffic within the network. For this moel we showe that a variation of a imum back pressure policy is throughput optimal, where this policy is moifie to incorporate the potential gains of cooperative communication. We only consiere one type of cooperation, namely ecoe an forwar combine with beamforming. Potential future irections inclue consiering other types of cooperation as well as other network topologies. REFERENCES [1] E. Yeh an A. Cohen, hroughput optimal power an rate control for queue multiaccess an broacast communications, in Proceeings of the International Symposium on Information heory, Chicago, IL, [2] E. Yeh an A. Cohen, Information theory, queueing, an resource allocation in multi-user faing communications, in Proceeings of the Conference on Information Sciences an Systems, Princeton, NJ, [3] L. assiulas an A. Ephremies, Stability properties of constraine queueing systems an scheuling policies for imum throughput in multihop raio networks, IEEE ransactions on Automatic Control, vol. 37, no. 12, pp , [4] M. J. Neely, E. Moiano, an C. E. Rohrs, Dynamic power allocation an routing for time varying wireless networks. Proceeings of Infocom, [5] B. Schein an R. G. Gallager, he gaussian parallel relay network, in Proc. IEEE Int. Symp. Information heory ISI, Sorrento, Italy, p. 22, June [6] B. Schein, Distribute Coorination in Network Information heory. PhD thesis, Massachusetts Institute of echnology, Cambrige, MA, [7] A. Senonaris, E. Erkip, an B. Aazhang, User cooperation iversity - part i: System escription, IEEE rans. On Communications, vol. 51, pp , November [8] A. Senonaris, E. Erkip, an B. Aazhang, User cooperation iversity - part ii: Implementation aspects an performance analysis, IEEE rans. On Communications, vol. 51, pp , November [9] J. N. Laneman, D. N. C. se, an G. W. Wornell, Cooperative iversity in wireless networks: Efficient protocols an outage behavior, IEEE rans. Inform. heory, vol. 50, pp , Dec [10] A. Hst-Masen an J. Zhang, Capacity bouns an power allocation for the wireless relay channel.. [11] D. se an S. Hanly, Multi-access faing channels: Part I: Polymatroi structure, optimal resource allocation an throughput capacities, IEEE ransactions on Information heory, vol. 44, no. 7, pp , he previous iscussion for Cases 1 an 2 operate uner the assumption that R is an extreme point of C CBC or that R is an extreme point of C CMAC. Using these arguments, on the other han, it is easy to see that the following is true: heorem 3: Consier the four-noe parallel relay network with ifferential queue backlogs w 1, w 2,..., w 6. If L CBC w 1, w 2, w 3 L CMAC w 4, w 5, w 6, then the optimal solution R to 7 is R 1, R 2, R 3, 0, 0, 0, where R [1] =