Distributed Optimal Relay Selection for QoS Provisioning in Wireless Multi-hop Cooperative Networks Yifei Wei†‡ , Mei Song‡ , F. Richard Yu† , Yong Zhang‡ , and Junde Song‡ Department of Systems and Computer Engineering, Carleton University, Ottawa, ON, Canada ‡ PCN&CAD Center, Beijing University of Posts and Telecommunications, Beijing, P.R. China Email:
[email protected],
[email protected], richard
[email protected]
†
Abstract—This paper proposes a distributed optimal relay selection scheme in wireless multi-hop cooperative networks where the wireless channels are modeled as first-order finitestate Markov channels (FSMCs) and adaptive modulation and coding (AMC) is applied. The FSMC model is used to approximate the time variations of the average received signalto-noise ratio (SNR). The state of a relay consists of the channel states of both source-to-relay and relay-to-destination links. In this scheme, a stochastic decision making approach is taken to select the optimal relay according to the states of all available relays with the quality of service (QoS) optimization goals of mitigating error propagation and increasing spectral efficiency. Simulation results show that the proposed scheme outperforms the existing scheme.
I. I NTRODUCTION Recently, cooperative relaying has been considered as a promising technique, and has been involved in IEEE 802.16j mobile multi-hop relay (MMR) systems [1] and expected to be integrated in 3GPP-LTE cellular networks called multihop cellular networks (MCN) [2]. Cooperative relaying approach provides better end-to-end performance due to spatial diversity and higher efficiency due to spatial multiplexing. However, relay selection among available relays is crucial in improving the performance and is still a big challenge in wireless multi-hop cooperative networks [3], [4]. The authors of [5] present a centralized optimization framework, in which the base station solves the joint relay strategies and resource allocations problem based on the feedback of receivers’ channel estimations, and then informs all users the appropriate power levels and cooperative strategies. In [6], a semi-distributed relaying algorithm is proposed to jointly optimize relay selection and power allocation of the system. These schemes assume that the network has the control over the behavior of the user nodes. Distributed methods are proposed in [7]–[9] to select the This work was supported in part by the National High-Tech Research and Development Plan of China (No. 2009AA01Z206), the National International Science and Technology Cooperation Project of China (No. 2008DFA12090), and the Natural Sciences and Engineering Research Council of Canada (NSERC).
“best” relay that has the maximum instantaneous value of a metric, which is the minimum or the harmonic mean of its source-to-relay (S2R) and relay-to-destination (R2D) channels’ gains. In these methods, the decision of which relay to use for the subsequent frame transmission is based on the current observed channel state information (CSI). These methods can be classified as memoryless methods [10], which assume that the channel fading is slow enough such that the channel conditions remain in the same state from the current frame to the next. However, this simplifying assumption is often not true given the time-varying nature of the mobile wireless environments. The authors of [11], [12] use a Markov chain to describe the transition activities between states from one observation time to the next. A finite-state Markov channel (FSMC) has been used extensively in recent literature to model fading channel caused by multipath propagation and shadowing, including Rayleigh fading channels [11], [12], Ricean fading channels [13], and Nakagami fading channels [14], [15]. In this paper, the first-order FSMC model is used to predict the upcoming channel quality for the subsequent packets transmission, and the adaptive modulation and coding (AMC) scheme is considered to achieve high spectral efficiency. The relay selection is based on the states of all available relays with the quality of service (QoS) optimization goals of mitigating error propagation and maximizing spectral efficiency. We formulate the optimization problem as a restless bandit system, which can be solved using linear programming (LP) relaxation and primal-dual index heuristic algorithm [16]. The solution of the restless bandit formulation has an indexability property that can dramatically reduce the computation and implementation complexity of the relay selection policy. In addition, the proposed scheme is fully distributed and scalable, and relays can join and leave from the set of relay candidates freely. The rest of this paper is organized as follows. In Section II, the system model of relay selection and the optimization objectives are described. Section III formulates the problem as a restless bandit system. Section IV discusses the dis-
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R1
tributed relay selection process. Some simulation results are presented in Section V. Section VI concludes this study.
Ȗ S2R
II. S YSTEM M ODEL OF R ELAY S ELECTION We consider a distributed cooperative wireless network where each node has the ability to relay data packets for each other using decode-and-forward (DF) mode. When viewed from the multi-hop routing diversity point of view, the first hop is more important than all subsequent hop(s) [17]. Therefore, in this study, we only consider two-hop relay: S2R and R2D as shown in Fig. 1. Assume that there are N available relays between a pair of source and destination, and denote the set of available relays as N = {1, 2, . . . , N }. The duration of the whole communication between the source and destination is divided into T time slots. The problem is to select the optimal relay in each time slot t ∈ T = {0, 1, . . . , T − 1}. The action of each relay n ∈ N in time slot t is represented by an (t), an (t) ∈ A = {0, 1}, in which an (t) = 0 means the relay n is passive (not selected) in time slot t, and an (t) = 1 means active (selected). Before each time slot, the source broadcasts an requestto-send (RTS) packet, then its destination estimates γS2D and decides the modulation and coding scheme (MCS) of source-to-destination (S2D) link (M CS2D ), and relays can estimate γS2R . After the destination broadcasts an clear-tosend (CTS) packet with the MCS information, the source and relays both receive it, and relays can estimate γR2D and decide the MCS of R2D link (M CR2D ). Each MCS has a corresponding spectral efficiency. We assume that there are totally K classes of MCSs with spectral efficiencies denoted as η0 , η1 , . . . , ηK−1 , and the minimum decoding SNRs for ∗ , respectively. different MCSs are γ0∗ , γ1∗ , . . . , γK−1 A. S2R Channel The quality of S2R channel determines the correctness of decoding at a relay. We use first-order FSMC to model S2R channel state. The channel state is characterized via the received signal-to-noise ratio (SNR) which is partitioned (quantized) into a finite number of intervals, and each interval is associated with a state of a Markov chain. That is, the average received SNR γS2Rn at relays is partitioned into L levels, and each level is associated with a state of a Markov chain. The channel varies over these states at each time slot according to a set of Markov transition probabilities. The finite state space can be denoted as C = {C0 , C1 , . . . , CL−1 }. The S2R channel state realization of γS2Rn is Γn (t) for relay n in time slot t. Let φgn hn (t) denote the probability that γS2Rn moves from state gn to state hn at time t. The L × L S2R channel state transition probability matrix of relay n is defined as: (1) Φn (t) = [φgn hn (t)]L×L , where φgn hn (t) = Pr (Γn (t + 1) = hn | Γn (t) = gn ), and gn , hn ∈ C.
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B. R2D Channel The quality of R2D channel determines the MCS of R2D link. Given the target bit error rate (BER), the ∗ , for different minimum decoding SNRs, γ0∗ , γ1∗ , . . . , γK−1 MCSs can be calculated. Since we are concerned about the MCS of R2D link, the channel state of R2D link can be divided into discrete K levels: Υ0 , if γ0∗ γR2D < γ1∗ ; ∗ . The Υ1 , if γ1∗ γR2D < γ2∗ ; . . . ; ΥK−1 , if γR2D γK−1 average received SNR of each packet can be modeled as a random variable γR2Dn evolving according to a K-state Markov chain, which has a finite state space denoted as D = {D0 , D1 , . . . , DK−1 }. The R2D channel state realization of γR2Dn is Υn (t) for relay n in time slot t. Let ψun vn (t) denote the probability that γR2Dn moves from state un to state vn at time t. The K × K R2D channel state transition probability matrix of relay n is defined as: Ψn (t) = [ψun vn (t)]K×K ,
(2)
where ψun vn (t) = Pr (Υn (t + 1) = vn | Υn (t) = un ), and un , vn ∈ D. C. Objectives We need to find out the optimal relay selection policy, which can set one relay n to be active at time slot t according to the relays’ states that contain their S2R channel state Γn (t) ∈ C and R2D channel state Υn (t) ∈ D. In this paper, we use the following QoS optimization objectives: • Mitigate error propagation. Better channel state Γn (t) ∈ C enables lower BER, and should be reflected on higher reward in our formulation. • Increase spectral efficiency. Better channel state Υn (t) ∈ D enables higher modulation and coding scheme with higher spectral efficiency, and should be reflected on higher reward in our formulation. III. R ELAY S ELECTION F ORMULATION We formulate the stochastic relay selection problem as follows. Consider a collection of N projects, each project n can be in one of finite states in (t) ∈ Sn in each time slot t. According to their states, M out of N projects are selected to work, or set to be active (an (t) = 1), and the other projects are set to be passive (an (t) = 0). The system
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a (t)
reward Rinn(t) is earned when action an (t) is taken, and their states change in a Markovian fashion, according to an transition probability matrix (into state jn (t + 1) ∈ Sn with probability pain jn ). Reward is discounted in time by a discount factor 0 < β < 1 to ensure that the expected reward is bounded and converge [16]. Projects are selected over time under a policy u ∈ U, where U is the set of all Markovian policies (which select the current action as a function, possibly randomized, of the current state and time). The problem is to determine the optimal u that maximizes the total expected discounted reward over the time horizon. In this section, we formulate the procedure of relay selection as a restless bandit problem of selecting only one relay M = 1 in each time slot. A. Relay States The state of an available relay n ∈ {1, 2, . . . , N } in time slot t ∈ {0, 1, . . . , T −1} is determined by the channel states Γn (t) and Υn (t). Consequently, the state of a relay is: in (t) = [Γn (t), Υn (t)] .
(3)
In practice, the change of S2R channel state and R2D channel state are independent. Therefore, the relay state will change in a Markovian fashion, and the finite-state space of relay n is represented as Sn , in (t) ∈ Sn , with the transition probability matrix Pn (t) = [φgn hn (t), ψun vn (t)]Gn ×Gn ,
(4)
where φgn hn (t) and ψun vn (t) are defined in (1) and (2), respectively, and Gn = L × K. The element of Pn (t) is pin jn (t), denoting the transition probability that the state of relay n changes from in to jn , where in , jn ∈ Sn . B. System Reward In the restless bandit problem, the system reward represents the optimization objectives. Here, we formulate the system reward to be the function of BER of S2R link and spectral efficiency of R2D link. The action of a relay determines whether the reward will be gained. Therefore, the system reward can be defined as:
maximizes the total expected discounted reward during the whole communication period, and the optimum value is: T −1 a (t) a2 (t) aN (t) 1 ∗ t (Ri1 (t) + Ri2 (t) + · · · + RiN (t) )β . Z = max Eu u∈U
t=0
(6)
C. Priority-Index Selection Policy Since the restless bandit problem is naturally formulated as a discounted MDC, the authors of [16] formulated it as the linear program: Riann xainn , (7) (LP) Z ∗ = max x∈X
n∈N in ∈Sn an ∈{0,1}
where X = {x = (xainn (u))in ∈Sn ,an ∈{0,1},n∈N | u ∈ U} is the corresponding performance region spanned by performance vector x under all admissible policies u ∈ U, and the performance measure xainn (u) represents the total expected discounted time that relay n take action an in state in under admissible policy u. Let αin denote the probability that the initial state is in , for in ∈ Sn , and the initial state probability vector α = (αin )in ∈Sn is given. The first-order relaxation is formulated as the linear program in [16]: Riann xainn (LP1 ) Z 1 = max n∈N in ∈Sn an ∈{0,1}
subject to xn ∈ Pn1 , n∈N in ∈Sn
n ∈ N, 1 . x1in = 1−β
(8)
The Pn1 is precisely the projection of restless bandit polytope P over the space of the variable xainn for project n, and the complete formulation of Pn1 is given by [16]. The authors of [16] interpreted the primal-dual heuristic as a priority-index heuristic under some mixing assumptions on active and passive transition probabilities. Please refer to [16] for details. We use this priority-index rule to select relays. The dual of linear program (LP1 ) is:
1 1 1 a (t) λ αjn λjn + Rinn(t) = an (t)R (ωp Pb (M CS2D , Γn (t)), ωη ηk (M CR2D , Υn (t))) , (D ) Z = min 1−β n∈N jn ∈Sn (5) subject to where |ωp | + |ωη | = 1, ωη is positive weight, ωp is negative λin − β p0in jn λjn Ri0n , in ∈ Sn , n ∈ N , weight, Pb is BER function determined by channel state jn ∈Sn Γn (t) when a modulation and coding scheme is given, ηk is the spectral efficiency determined by modulation scheme p1in jn λjn + λ Ri1n , in ∈ Sn , n ∈ N , λin − β that adapts to channel state Υn (t). jn ∈Sn a (t) The instantaneous reward Rinn(t) is earned for relay n λ 0. (9) in state in (t) when it takes action an (t) at each time slot t. Denote by u ∈ U the Markovian policy, and denote Let {xainn } and {λin , λ} the optimal primal and dual soby β the discount factor. The goal is to find a policy that lution pair to the first-order relaxation (LP1 ) and its dual
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(D1 ). Let { ainn } be the corresponding optimal reduced cost coefficients:
0in = λin − β p0in jn λjn − Ri0n , jn ∈Sn
1in
= λin − β
p1in jn λjn + λ − Ri1n ,
(10)
jn ∈Sn
which must be nonnegative. The optimal reduced costs 0in and 1in can be interpreted as the rate of decrease in the objective-value of linear program (LP1 ) per unit increase in the value of the variable x0in and x1in , respectively. The index of relay n with current state in is defined as δin = 1in − 0in .
(11)
The priority-index rule is to set active the relay that has the smallest index. IV. D ISTRIBUTED R ELAY S ELECTION S CHEME In this section, we will present the distributed relay selection scheme in wireless multi-hop cooperative networks. Before each time slot, the source broadcasts an RTS packet, then its destination estimates γS2D and evaluates the MCS of S2D link (M CS2D ), and neighbor nodes can estimate γS2R . After the destination broadcasts an CTS packet containing the M CS2D information, the source and neighbor nodes all receive it, and neighbor nodes can estimate γR2D and evaluate the MCS of R2D link (M CR2D ). The common neighbors of the pair of source and destination that can decode both RTS and CTS constitute the set of available relays N . After exchanging RTS/CTS packets, each available relay n gets its current state in (t), so it calculates its index δin and broadcasts a candidate index (CI) packet containing the index. When other available relays receive this CI packet, they will compare the received index with their own index and broadcast their own CI packet only if their own index is smaller than the received index, otherwise they will keep silent. After some time, the source receives all the CI packets or the sub-set of all CI packets due to the collision, and it will select the relay with the smallest index. Since the priority-indices can be computed and stored into a table, the relay selection process can be divided into two stages: Before any transmission is started, the set of indices {δin } can be computed with the input parameter pin jn , Riann , β and α. Relays can store these indices and the corresponding parameter in a table off-line. In the on-line stage, each relay n ∈ N lookups its index table to find out the index δin corresponding to its current state in , and then broadcasts its CI packet if it has not received any CI packet or its own index is smaller than the received indices. The relay selection operates in a distributed manner, so there is no need for a centralized control point. If a relay leaves from the set of available relays during data transmission, it does
not send the CI packet, thus it will not be selected. Another relay with the smallest index will be selected. Therefore, relay nodes can leave and join the relay candidate set freely. Although some communication overhead is added due to the necessary information exchange between the available relays, the cost is light compared with the user data transmission. The relays can share the parameters used for computing priority-index by adding their information to the existing protocols (e.g., broadcast from the base station). These off-line communication and computation actions are only needed once before real-time data transmission. Therefore, off-line communication and computation result in little cost. During data transmission, a relay just lookups the index table according to its current state and sends its current index by a rather small packet only if its own index is smaller than the received indices. Therefore, there will be at most (N −1) CI broadcast packets. Because the CI packet only contains the node ID and its index, the length of the packet is very short and the communication overhead does not have much effect on the system performance. V. S IMULATION R ESULTS AND D ISCUSSIONS In this section, we compare the proposed scheme with the random selection and the existing memoryless method [8]. We use phase-shift keying (i.e., BPSK, QPSK) and quadrature amplitude modulation (i.e., 16QAM) as the available MCSs. We assume that the state of S2R channel can be bad (s0) or good (s1), and the state of R2D channel can be good-for-BPSK (d0), good-for-QPSK (d1) or good-for16QAM (d2). Consequently, each relay has 6 states: s0d0, s0d1, s0d2, s1d0, s1d1, s1d2. We set the state transition probability matrices of S2R channel and R2D channel as: ⎛ ⎞ 0.6 0.3 0.1 0.7 0.3 Φ= , Ψ = ⎝0.2 0.6 0.2⎠ . 0.3 0.7 0.1 0.3 0.6 The relay state transition probability matrix (P) can be easily acquired according to (4). Assume that the BER of s0 and s1 is about Pb = 10−3 and Pb = 10−5 , repectively. We take lg (Pb /10−3 ) as the first component of reward function weighted by negative ωp . The spectral efficiencies of d0, d1 and d2 are η1 = 1, η2 = 2 and η3 = 4, repectively. Thus the system rewards are R0 = (0, 0, 0, 0, 0, 0) and R1 = (ωη , 2ωη , 4ωη , ωη −2ωp , 2ωη −2ωp , 4ωη −2ωp ). The network or user can adjust the values of ωp and ωη according to their preference. We set the discount factor β = 0.8 in the simulations. The initial states of the relays are random. A. Spectral Efficiency Improvement If high spectral efficiency is the most desirable QoS for the network, we should specify ωη = 1 and ωp = 0. We run the simulations for 2, 000 seconds, with N = 8 available relays. Fig. 2 is the spectral efficiency using different relay
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selection schemes. It can be seen that the proposed scheme can select a relay with d2 channel for the subsequent frame almost every decision time, and the spectral efficiency is near 4 bit/s/Hz. The existing memoryless selection method obtains the spectral efficiency round about 3 bit/s/Hz, since it selects a relay for the subsequent frame according to the current states which may change in the subsequent frame. Fig. 3 shows the average spectral efficiency with different transition probabilities, which is the probability for the R2D channel staying in the same state. It can be seen that the performance of existing memoryless method is getting closer to our proposed scheme with the transition probability increasing, and this method performs as good as our proposed scheme when the channel is absolutely static, which means the transition probability that the channel will be at the same state is 1. Our proposed scheme can achieve the highest spectral efficiency in any transition probabilities, and the random selection scheme has the spectral efficiency round about 2.3 bit/s/Hz in any case. Notice that the memoryless method is better than the random selection when the transition probability is 0.5. This is because the transition probability from d2 to d1 is three times from d2 to d0 and this method always select the relay with current d2 channel, which is better than random selection among d2, d1 and d0 states. B. Error Propagation Mitigation If we care most about the error propagation problem, we can assign ωp = −1 and ωη = 0. We run the simulations for 2, 000 seconds, with N = 8 available relays. Fig. 4 shows the performance with different transition probabilities for S2R channel staying in the same state. It can be seen that the performance of the existing memoryless method is getting closer to our proposed scheme with the transition probability increasing, and this method performs as good as our proposed scheme when the channel is absolutely static. Our proposed scheme can select a relay with s1 channel for
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the subsequent frame almost every decision time and achieve the lowest average BER in any transition probabilities. VI. C ONCLUSIONS AND F UTURE WORK In this paper, we have presented a distributed optimal relay selection scheme for wireless multi-hop cooperative networks. The optimization objectives are to mitigate error propagation and increase spectral efficiency. The optimal relay is selected according to its state, which is modeled as a first-order FSMC with stochastic property. A stochastic selection approach, restless bandit, was used in this paper to solve the optimization problem. The solution has an indexability property, which dramatically reduces the on-line computation and implementation complexity. The proposed relay selection process operates in a distributed manner, and relays can join or leave from the set of relay candidates freely. We presented some simulation results to illustrate the performance improvement compared with the existing method and random selection scheme. The future work includes taking more relay state information into our model, such as energy and security.
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