On the Energy-Delay Trade-off of a Two-Way Relay Network

Transcription

1 On the Energy-Delay Trade-off of a To-Way Relay Netork Xiang He Aylin Yener Wireless Communications and Netorking Laboratory Electrical Engineering Department The Pennsylvania State University University Park, PA 68 xxh9@psu.edu yener@ee.psu.edu Abstract We consider a three node netork in hich a pair of nodes ith stochastic arrivals communicate ith each other ith the help of an intermediate relay. The bi-directional nature of the traffic, in this setting, poses a ne energy delay trade-off. Namely, the relay node may choose to cache packets from one direction and send it only after packets from the other direction arrive, using an XOR netork coding scheme. Doing so ould save energy, but ould also incur some delay for the packet. In this ork, e analyze this trade-off hen the relay node queues packets from each direction and uses a first-come-firstserve policy. We sho that under an even traffic load here one ould hope for the most energy savings, to achieve the minimum energy expenditure promised by the XOR netork coding scheme, the average delay has to go to. Keyords: To-ay relay, queuing, energy-delay trade-off I. INTRODUCTION The to-ay relay netork considered in this ork consists of to nodes ho ish to communicate ith each other via a relay node. Significant recent effort has been dedicated to this model, in particular, to understanding its information-theoretic performance. The capacity region under to-phase/multiplephase protocols and decode-and-forard relaying is presented in [], []. Reference [] also computes these for Gaussian fading channels. Achievable rates for the to-ay relay netork are given in [3]. Practical relay coding schemes are proposed in [4], [5] and poer/rate optimization problems are presented in [6]. The model is extended to multiple relay nodes, and the capacity scaling la hen the relay sum poer is fixed and number of relays goes to infinity is considered in [7]. There are to implicit assumptions behind all these orks: i Traffic arriving at the to source nodes are modeled as deterministic flos ith constant rates. ii To obtain the information theoretic achievable rates, the code ord length is assumed to be infinity. While these assumptions are idely used and are essential to simplify the analysis, there are certain scenarios under hich they may not hold. A likely scenario is that the packets can arrive at the transmitters in a stochastic fashion. Hence, one can no longer guarantee that at a certain time instant, the relay ould receive data from both directions. In this case, a relevant issue is that of the characterization of the stability region [8] [] under assumption ii. In addition, commonly envisioned applications such as voice and video are delay sensitive. Therefore, the delay that the received data incurs at the relay node must be limited, rendering it impossible for the relay to code over long blocks. To address the first problem, a natural solution is to allo the relay to cache the traffic in one direction and transmit only after traffic from the other direction arrives, and employ netork coding [4]. Doing this is advantageous, since combining to or more transmissions into one ill allo the relay to save energy. Hoever, e immediately see that doing this ill increase the delay of the traffic since e must ait for the packets arriving from the more idle direction. Therefore, there is a trade-off beteen energy consumption at the relay and the end-to-end communication delay. Energy-delay trade-off has been considered previously for a number of different ireless communication scenarios. From the physical layer perspective, reference [] considers the average energy consumption hen channel coding may be done over a limited number of independent channel fading states for a multiple access channel. Reference [] explores this for a relay netork. From a medium access layer perspective as ell, it is idely accepted to ait for good channel states and save energy at the cost of delay. This trade-off is considered in [3] and the loer bound of energy consumption hen delay goes is given and shon to be tight. This result is extended to a multi-user scenario in [4]. We note that both results are restricted to single hop communication. The energy delay trade-off therein is a result of the concave relationship beteen the transmission poer and the service rate. By comparison, the energy delay tradeoff in to-ay relay netork considered here results from the lack of coordination of traffic demands at the to source nodes. In this ork, e investigate the energy delay trade-off in a bi-directional relay netork hen the relay node uses a first-come-first-serve policy. The details of the system model and the relevant assumptions are presented in Section II. In Section III, e analyze the stationary behavior of the queues at the relay node. The average packet delay and average energy consumption are related via the queue capacity at relay node. Our analysis shos that, in the case here e have a symmetric traffic load coming in from both directions, here one ould expect to harvest the energy savings of the XOR combining the most, to achieve the minimum energy, the eno-end delay goes to infinity. This is equivalent to saying the 865

2 Fig.. λ multiple access µ h h R h 3 h 4 R System Model L L Broadcast Queues at the relay node queues become unstable and leads to the conclusion that the netork coding protocol must be used in conjunction ith other protocols in order to ensure the stability of the queues. II. SYSTEM MODEL AND ASSUMPTIONS We consider a packet-based communication system, depicted in Figure. The channel is assumed to be static, and the channel state information is assumed to be globally knon. We assume the packets arrive at node and node from to independent Poisson processes ith parameters λ and µ respectively. The added redundancy and the energy of a packet are chosen in such a ay that each packet can be decoded reliably at the relay node, even if packets arrive at the relay node at the same time. Under this assumption, the traffic flo at the relay node from node and node are also Poisson processes ith the same parameters. At the relay, the decoded traffic flos from node and node are stored in to queues, ith buffer capacity L and L respectively. The policy employed by the relay to process the packets from the queues is described belo. When a packet arrives, if the queue holding the traffic from the opposite direction is not empty, the relay sends out the coded version of the binary sum of this packet and the packet in the queue from the opposite direction immediately. For the XOR netork coding to ork, e assume each packet contains same number of information bits. Again, e assume the redundancy and the energy of each packet are chosen such that they can be successfully decoded at both node and node. When a packet arrives, if the queue holding the traffic from the opposite direction is empty, then the relay stores the packet in the queue that contains packets for the arrival direction. If this queue is full, the relay sends out the oldest packet in the queue immediately and makes room to store this latest packet. Again, e assume the redundancy and energy of each packet are chosen such that it can be successfully decoded at its intended destination. In both cases, e assume the packet size is small enough such that the transmission delay is negligible. In other ords, the transmissions can be modeled as points on the time axis. Therefore, the average delay experienced by the packets equals the average amount of time they spend in the queue at the Fig.. λ λ λ λ λ λ L L µ µ µ µ µ µ State Transition Diagram relay. In the next section, e ill explore the relationship of the average delay ith the average transmission poer of the relay. Remark : We assume that the traffic generated from the to ends are independent. This assumption may not hold in a scenario here the traffic of node is triggered by the response from node. The assumption on the other hand is appropriate hen the to end nodes are simply routers in an ad hoc netork forarding data to each other. Remark : We assume the queues at the relay node have finite capacity. This is a seemingly different starting point as compared to previous ork that assumes infinite buffer size [8] []. The connection beteen these orks ill become apparent hen e let the capacity of the queue, i.e., the buffer size, go to in order to minimize the energy consumption. III. ENERGY DELAY TRADE-OFF A. Queuing Model and Stationary Distribution Based on the transmission policy at the relay described in the previous section, e remark that at most one queue at the relay node can be non-empty. Therefore, the state of queues at the relay node can be characterized ith a single number S [ L, L ]. Since the future state is independent from its past given the current state, S is a continuous time finite state Markov chain. Recall that the traffic flos arriving at the relay node are to independent Poisson processes ith parameters λ and µ respectively. Therefore, for a given time interval t, e have the transition probabilities: P S t + t a + S t a λ t + o t, a [ L, L ] P S t + t a S t a µ t + o t, a [ L +, L ] The resulting continuous time Markov chain is shon in Figure. We note that Figure is tantamount to a M/M//L + L + system [5]. Therefore, they have the same stationary distribution, given by 3 belo, here u λ/µ. P S i { uu i+l u L +L +, u L +L +, u 3 i L...L 866

3 3 B. Average Poer The average poer expended by the relay node is defined as: [ ] ε E [P] lim E t t P S t ip S t + t j St i lim ε ij t t i,j Here ε ij is the energy cost associated ith state transition i j. From the model described in the previous section, e kno that ε ij is decided by the coding scheme used ithin each packet and the channel conditions. Let ε a be the energy cost to successfully multi-cast the XOR-ed packet to both node and node. ε b is the energy cost to successfully send the packet from the relay to node. Similarly, ε c is the energy cost to send the packet from relay to node. Ho ε a, ε b, ε c are computed is not relevant to our discussion. In practice, they may be approximated by the folloing equations: 4 ε b N R /h 4 5 ε c N R /h 3 6 ε a max {ε b, ε c } 7 here N, N are the variance of the additive Gaussian noise seen by node and node. h 4, h 3 are the channel gains from the relay node to node and to node respectively see Figure. R i is the rate used from node i, i,. For the state diagram in Figure, since there are only finite number of states, e may manipulate 4 by interchanging the summation and the limit, as shon belo: lim t i,j i,j lim t i j P S t ip S t + t j S i ε ij 8 t P S t ip S t + t j S i ε ij 9 t P S i P S t + t j S iε ij L P S iµε a + P S L λε b i L + P S iλε a + P S L µε c i Substituting 3 into, for u, e obtain: E [P] ul+l+ u L+L+ µε a u L+L+ λε a u ul+l + u L+L+ λε u b + u L+L+ µε c Remark 3: From, e readily see that the average poer consumption by the relay node depends only on the traffic load from the source nodes and the total storage capacity at the relay, L +L. It does not depend on the service order. Remark 4: Minimal energy consumption is achieved hen L + L goes to. If L + L, e observe C. Average Delay E[P] λ µε b + µε a 3 Let us focus on the delay experienced by a packet sent from node to node. The case of the other direction can be addressed in a similar fashion. Suppose that the packet arrives at τ. Then the distribution of the delay T is given by P T t i P T t S τ ip S τ i 4 Note that there are only to possible ays for the packet to leave the queue. Enough packets from the opposite direction arrive so that it gets served ith XOR-ed ith a packet from the other queue. Too many packets from node arrive after this packet so that it needs to be transmitted by itself. Therefore each term P T t S τ i in the summation can be computed as: P T t S τ i P n t i + S τ i 5 +P n t < i +, n t L S τ i here n i t is the number of packets received from node i during time τ, t + τ]. The first term in 5 is given by: P n t i + S τ i i µt k e µt 6 k! k The second term in 5 can be computed as: P n t < i +, n t L S τ i P n t < i + P n t L i µt k L e µt λt n 7 e λt k! n! k n Note that, ith the Poisson flos, the residual time [5] the packet has to ait to see a packet from the opposite direction is an exponential random variable ith parameter /µ. With these preparations, e are no ready to compute E[T]. E [T] i t d P T < t 8 t d P T < t S ip S i 9 Each term of the summation in 9 can be expressed as: t d P T < t S i t d P T < t S i P T < t S i 867

4 4 We notice that P T < t S i is the sum of terms of the form ct a e bt. Therefore the integral can be computed using the folloing equation, here a is a non-negative integer. x a e λx dx λ a+ a! 3 Using this result, along ith 7, 6, becomes: i i L k L k λ λ + µ µt λt k e λ+µt 4! k! k µ λ + µ + 5 here k+! k!!. Substituting 5 and 3 back to 9, e obtain the average delay from node to node as 6 for u. The delay seen by traffic from node to node is similar, as shon in 7. E [T] L i L i k u L+ u L u L+L+ E [T] L i L i k u L+ u L u L+L+ D. Energy Delay Trade-off λ k µ λ + µ λ + µ + µ k λ λ + µ λ + µ + 6 u i 7 An immediate thought folloing the analysis above is the selection of L and L to minimize traffic delay under a relay poer constraint. Before doing that, e prove a monotonic property of the average delay, hich ill be useful later. We kno from that there is a bijection beteen the average poer and total storage capacity L + L. Also, if L +L increases, the average poer consumption decreases. Therefore, e only need to examine the behavior of the average delay 6 7 under fixed L +L. If L +L is fixed and L increases, from 7, e readily see that E [T] ill decrease. What is less obvious is that E [T] ill increase, as shon belo. Suppose L increases by. Then, as shon in 3, the first term in 6 ill increase by at least u fold. Since L decreases by, the remaining part in 6 ill decrease at least by u fold. Therefore, their product E [T] ill increase. By a similar reasoning, it can be shon that if L is fixed, then E [T] ill increase ith L. If L is fixed, then E [T] ill increase ith L. L + i L + > > i L + i L u i L + i L k i k k i λ k µ λ + µ λ + µ + λ k µ λ + µ λ + µ + L L i k λ k µ λ + µ λ + µ + λ k µ λ + µ λ + µ The minimum delay T under the average poer constraint is defined as 3 belo. We are interested in T as a function of P, hich ill yield the optimal energy delay trade-off curve achievable under this relay policy. T min L,L max {E [T],E [T]} s.t. E [P] P 3 It follos from the previous discussion that the poer constraint can be translated to a loer bound on L + L. From the monotonic properties of E [T] and E [T] e have just argued, e kno that the equivalent constraint on L + L must be binding. Therefore L + L is fixed. Under this condition, e have shon above that E [T] is a strictly increasing function of L, and E [T] is a strictly decreasing function of L. Also, it is obvious to see that hen L, E [T] and hen L, E [T]. Therefore, the optimal solution can only be of the folloing form and can be easily found via a bisection method: L is either L,a or L,a +. L,a is chosen such that E [T]L,a E [T]L,a,E [T]L,a + E [T]L,a +. Notice that T can be reduced by employing time sharing beteen L L,a and L L,a + hile keeping L +L fixed. The resulting energy delay trade-off curve is given in Figure 3 for ε a ε b ε c. The horizontal dotted lines depict the average relay poer consumption hen L +L for different λ, µ. We observe that hen λ and µ are close to each other, a ider trade-off beteen energy and delay can be achieved. When the traffic load becomes less even, the achievable trade-off range becomes smaller. E. The Symmetric Case λ µ In Figure 3, e observe that hen λ µ, to achieve the loer bound on average poer, the average delay ill go to. We next prove this observation formally via the folloing lemma. Lemma : P µεa OL, T OL ith L L L 868

5 5 P Fig λ. µ.9 λ. µ T λµ.5 Energy Delay Trade-off Curve hen L + L Proof: First e examine the average delay T. We notice that the solution of 3 yields L L L. Therefore, it suffices to derive an upper bound and loer bound of 6 hen L. T T L i L i k L i i k+ λ + µl + L +,k +k i+l k+ λ + µl + L + L i + L λ + µl + L L + λ + µl + L 36 O L 37 L i,k +k i k+ T λ + µl + L + 38 L i + i λ + µl + L L + L + λ + µl + L + 4 L + L + λ + µl + 4 O L 4 Where 35 and 39 follo from the binomial expansion formula. Remark 5: The loer bound 4 implies that T as L. Next, let us examine P µε a. Substituting 3 into, e obtain: L + L P µε a + µ ε b + ε c L + L + L + L + P µε a µ ε b + ε c ε a L + L + µ ε b + ε c ε a L + 43 From 43 and the upper bound for T, e observe that if T increases at a rate of OL, the average poer P ill not decrease faster than OL. Remark 6: As shon by Lemma, for the symmetric traffic load λ µ, the average delay to achieve the minimum energy. Remark 7: If each packet must be paired along ith a packet from the opposite direction for transmission, thus achieving minimal energy, then first-come-first-serve policy does achieve the minimal sum delay E [T]+E [T]. This can be shon as follos: Consider to sets of packets A, B and A, B paired as such to be transmitted. Suppose A arrived before A, but B arrived after B, in other ords, the packets are not paired according to first-come-first-serve. It is then easy to verify that pairing these packets as A, B and A, B instead ill not incur a greater sum delay for these four packets. On the other hand, due to the symmetry of the system, e have E [T] E [T]. This means achieving minimal sum delay is equivalent to min max delay T. Therefore e find that first-come-first-serve policy is indeed the optimal policy. This means to achieve minimal energy consumption, T ill go to regardless of hat service policy is in use. Remark 8: Our assumptions in section II dictates that the system operates at a rate that belongs to the achievable rate region C ith the coding scheme used. Therefore, it is rather surprising to see the queues become unstable, since there are many knon rate allocation policies [8] [] hich stabilize queues for all rate points inside C. Hoever, a closer look shos none of these policies use netork coding as the only coding scheme. Reference [8] uses the superposition coding scheme for the broadcast phase. In [9], the stability proof of the opportunistic netork coding scheduling algorithm relies on the fact that a multi-hopping scheme is used along ith netork coding scheme. The policy in [] is also a hybrid scheme; netork coding is used along ith direct transmission so that the resulting stability region has a nonempty interior. The stability of the queues is henceforth guaranteed via the CMDB policy []. An insight obtained is that a deterministic netork coding scheme must be used along ith other coding schemes. This hybrid approach as considered previously to obtain a larger achievable rate region of the underlying coding scheme [9], []. Hoever, it is easy to see for a symmetric channel, i.e., hen the relay is in the middle of the to source nodes, this approach ill not increase the maximal rate or the sum rate. Nevertheless, e sho here that a hybrid approach is still necessary. A multi-hopping scheme must be considered in a netork coding protocol in order to stabilize the queues. 869

6 6 IV. CONCLUSION In this paper, e have investigated the energy-delay tradeoff in a to-ay relay netork hen the relay node uses a first-come-first-serve policy and aims to harvest the energy savings by employing XOR combining of the data arriving from the sources. The trade-off is a result of the stochastic nature of the traffic from the source nodes. We have proved for the case ith an even traffic load from both directions, to fully achieve the energy saving promised by XOR netork coding scheme, the average delay ill go to. The energy-delay trade-off curve here is derived under a specific policy used at the relay node. In that sense it is an achievability result. When a different policy is used, a different curve may result. It is of interest to derive an loer bound for this energy-delay trade-off curve as future ork, along ith the optimal strategy that ill achieve this loer bound. REFERENCES [] T. J. Oechtering, I. Bjelakovic, C. Schnurr, and H. Boche. Broadcast Capacity Region of To-Phase Bidirectional Relaying, 7. submitted to IEEE Transactions on Information Theory. [] S. J. Kim, P. Mitran, and V. Tarokh. Performance Bounds for Bi- Directional Coded Cooperation Protocols. International Conference on Distributed Computing Systems Workshops, 7. [3] B. Rankov and A. Wittneben. Achievable Rate Regions for the To-ay Relay Channel. IEEE International Symposium on Information Theory, 6. [4] S. Zhang, S. C. Lie, and P. Lam. Physical Layer Netork Coding, 7. available online at [5] P. Popovski and Y. Hiroyuki. Physical Netork Coding in To- Way Wireless Relay Channels. IEEE International Conference on Communications, 7. [6] H. Ingmar, K. Marc, E. Celal, J. Zhao, A. Wittneben, and G. Bauch. MIMO To-Way Relaying ith Transmit CSI at the Relay. IEEE Signal Processing Advances in Wireless Communications, 7. [7] R. Vaze and R. W. Heath. Capacity Scaling for MIMO To-Way Relaying, 7. submitted to IEEE Transactions on Information Theory. [8] T.J. Oechtering and H. Boche. Stability Region of an Efficient Bi- Directional Regenerative Half-duplex Relaying Protocol. IEEE Information Theory Workshop, 6. [9] C.H. Liu and X. Feng. Netork Coding for To-Way Relaying: Rate Region, Sum Rate and Opportunistic Scheduling. IEEE International Conference on Communications, 8. [] E.N. Ciftcioglu, A. Yener, and R. A. Berry. Stability of Bi-Directional Cooperative Relay Netorks. IEEE Information Theory Workshop, 8. [] S. Hanly and D. Tse. Multi-Access Fading Channels: Part II: Delay Limited Capacities. IEEE Trasactions on Information Theory, 447:86 83, 998. [] D. Gunduz and E. Erkip. Opportunistic Cooperation by Dynamic Resource Allocation. IEEE Transactions on Wireless Communication, 64: , 7. [3] R. A. Berry and R. G. Gallager. Communication over Fading Channels ith Delay Constraints. IEEE Transactions on Information Theory, 485:35 49,. [4] M.J. Neely. Optimal Energy and Delay Tradeoffs for Multi-user Wireless Donlink. IEEE Transactions on Information Theory, 539:395 33, 7. [5] L. Kleinrock. Queueing Systems, Volume I: Theory. Wiley-Interscience, 975. [6] L. Tassiulas and A. Ephremides. Stability Properties of Constrained Queueing Systems and Scheduling Policies for Maximum Throughput in Multihop Radio Netorks. IEEE Trans. on Automatic Control, 37: ,