Optimal Relay Selection with Non-negligible Probing Time

# Optimal Relay Selection with Non-negligible Probing Time

Yang Liu, Yi Ouyang and Mingyan Liu Authors are from Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor. Contact information : {youngliu, ouyangyi, mingyan}@umich.edu indicates equal contribution.
###### Abstract

In this paper an optimal relay selection algorithm with non-negligible probing time is proposed and analyzed for cooperative wireless networks. Relay selection has been introduced to solve the degraded bandwidth efficiency problem in cooperative communication. Yet complete information of relay channels often remain unavailable for complex networks which renders the optimal selection strategies impossible for transmission source without probing the relay channels. Particularly when the number of relay candidate is large, even though probing all relay channels guarantees the finding of the best relays at any time instant, the degradation of bandwidth efficiency due to non-negligible probing times, which was often neglected in past literature, is also significant. In this work, a stopping rule based relay selection strategy is determined for the source node to decide when to stop the probing process and choose one of the probed relays to cooperate with under wireless channels’ stochastic uncertainties. This relay selection strategy is further shown to have a simple threshold structure. At the meantime, full diversity order and high bandwidth efficiency can be achieved simultaneously. Both analytical and simulation results are provided to verify the claims.

{keywords}

Optimal relay selection, stopping rule, diversity gain, probing times

## I Introduction

There is a quickly increasing demand for high data rates in wireless communication along with the skyrocketing usage of mobile devices. Increasing transmission diversities is among the most promising techniques and is attracting much attention [1, 2, 3]. Transmission diversity in communication systems provides more than one copy of the transmitted signal to the destination node, with which the destination can decode the transmitted signal even if some copies of the signal are distorted due to the time varying nature of transmission channels; therefore the system performance can be expected to improve significantly. More precisely the diversity order of a communication system can be measured by the relationship between the error probability, denoted by , and the Signal-Noise-Ratio (SNR) as follows : a system has diversity order if

 Pe=O(SNR−d). (1)

Cooperative communication techniques have been introduced to increase system’s diversity order (see [4]). In a cooperative network, when a node receives a packet not destined for it, instead of simply discarding the packet it can choose to help to relay and via such help, the source transmission can improve its diversity by sending signal through the relay channels. Although cooperative communications look promising, the gain is not immediate as cooperation incurs a cost of wireless resources such as frequency, active air time, and power resources to enable relaying. This trade-off between gaining diversity and conserving the wireless resources is mainly two-fold.

The first aspect is due to signal transmission. Since wireless resources are spent for relaying the signal to the destination, more bandwidth is needed when more relay nodes are involved in a transmission. To see this point more clearly, if nodes are relaying the signal, the bandwidth efficiency degrades to dues to the fact that there are all-together relays and the source are transmitting the same copy of signal. This problem of degraded bandwidth efficiency is solved by the introduction of several relay selection protocols as detailed in [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. With relay selection, only one relay instead of all is selected for the purpose of cooperation. Therefore, the bandwidth efficiency is leveraged to instead of which decreases as the number of relays increases. A relay selection protocol is called to achieve full diversity order if the resulting diversity order is for a network with relay nodes. When complete relay channel information is available, relay selection protocols with different selection metrics have been proved to achieve full diversity order in the literature (see [5, 6, 7, 8, 9, 10, 11, 12, 13]). For example in [12], best relay selection (choosing the relay with best channel SNR), best worse channel selection (choosing the relay with best worse channel condition) and best harmonic mean method have been proved to achieve full diversity order.

The second aspect, which was often neglected, comes from channel probing. Due to the dynamic nature of channel conditions, channel probing is needed for exploring each relay channel’s instantaneous transmission quality in order to find the best relay to cooperate with. Specifically we consider the following channel probing procedure. Before each transmission, the source sequentially probes the channels between the source and relays, and the channels between relays and the destination. The source can stop the probing process at any time, and select one of the probed relays to cooperate with. For each probing, a carrier sensing packet is sent to reveal each relay channel’s instantaneous quality (see [18, 19]). Despite the efforts towards reducing the size of a sensing packet, the probing time remains non-negligible. In this regards, even though probing all channels guarantees the discovery of the set of best relays, the degradation of bandwidth efficiency due to probing times is significant, especially when the number of relays is large. Therefore opportunistic channel probing (as commonly defined in the Opportunistic Spectrum Access (OSA) [20]) is needed for a bandwidth efficient system.

In this work we design a bandwidth efficient relay selection strategy in a relay network with non-negligible probing time. Technically we adopt theory of optimal stopping rule (which has been previously applied to opportunistic spectrum access, for instance [21, 22] and references therein.) to solve our relay selection problem. Our main contributions are as follows.

• We determine the optimal relay selection strategy and we show the optimal strategy is a simple threshold enabled stopping rule.

• Our relay selection strategy achieves full diversity order.

• This optimal relay selection results in bounded probing time, which further implies a relay selection scheme with high bandwidth efficiency.

The rest of our paper is organized as follows. We formulate the relay selection problem as an optimal stopping problem in Section II. In Section III, we solve the optimal stopping problem and define the corresponding relay selection strategy. We present in Section IV the analysis of diversity order and bandwidth efficiency of the proposed strategy. In Section V, we verify our results through simulation and conclude our paper in Section VI.

## Ii System Model and Problem Formulation

### Ii-a System model

We consider a relay network with one source , one destination and relays (as depicted in Fig.1) . We model the channels between any two nodes (, or ) as discrete time independent Rayleigh fading channels. Specifically the direct source to destination channel, the source to relay channels, and the relay to destination channels are modeled as follows

 ys,d=hs,d√Psxs+ηs,d, (2) ys,n=hs,n√Psxs+ηs,n, for n=1,2,...,N, (3) yn,d=hn,d√Prxn+ηn,d, for n=1,2,...,N, (4)

where and are the transmission power at the source and relays respectively; are the signals transmitted or retransmitted by the source and the th relay respectively; are the received signals at the destination from sources and relays, and the th relay , respectively. The channel noises are modeled as Gaussian random variables with zero-mean and variance .

For all the channels, are coefficients that capture the effect of path loss, fading etc. We assume these channel coefficients remain the same during the transmission of one source signal. Moreover, the channel coefficients are modeled as zero-mean, complex Gaussian random variables with unit variance for each signal transmission. For simplicity of later analysis, we use to denote the magnitude of the channel coefficients:

 ωs,d=|hs,d|2, (5) ωs,n=|hs,n|2,n=1,2,…,N, (6) ωn,d=|hn,d|2,n=1,2,…,N. (7)

For each signal transmission, besides the direct transmission to the destination, the source node selects one relay out of the candidates to cooperate using perfect Direct Forwarding (DF) protocol (for technical details please refer to [11]). That is, when relay is selected, the source will first send the signal to both the relay and the destination . Let denotes the decoded signal at , and the transmitted signal of relay is given by

 xn={xs,if xs=~xs,n,0,if xs≠~xs,n. (8)

i.e., the relays will forward the message/signal only if it has been correctly decoded.

### Ii-B The Relay Selection Problem

We formulate the relay selection problem with channel probing. For each relay we adopt the following index introduced in [11] as the criteria for selecting relay,

 ωn=2q1q2ωs,nωn,dq1ωn,d+q2ωs,n , (9)

with being constants as defined in [11]. When complete information (all s) of all channels is available, it is shown in [11] that selecting the relay with maximum gives full diversity gain, and this is also the major reason we adopt as the index for relay . Moreover if we view as an approximated channel gain of using relay , becomes the approximated signal power at the destination through .

Before each transmission, we assume the source sequentially probes the channels between the source and relays, and the channels between relays and the destination. The channel probing procedure, as shown in Fig. 2, can stop at any stage when the channels connected to relays are probed and select one relay to cooperate with. Let be the probing time to probe the channels between and (for example, could be a cycle of RTS/CTS period for IEEE 802.11 channel sensing protocol) and be the time for signal transmission of the source and the selected relay. Then for any , the time to probe relays is , and the bandwidth efficiency is given by

 cn=TtranTtran+nTs=11+nτ, (10)

where denoting the ratio between the probing time and the transmission time.

We define the signal to noise ratio (SNR) of the system to be , where is the total transmission power (including source and the selected relay). Consider a constant power scheme and for a fair comparison, the sum of transmission power at the source and the selected relay should equal the total power times the bandwidth efficiency, i.e.

 Ps+Pr=P⋅cn . (11)

The reason we take total transmission power to be in (11) is due to the fact that fraction of total power is used in channel probing. Denote as the power ratio . Then,

 Ps=rP⋅cn, (12) Pr=(1−r)P⋅cn. (13)

When the channel probing process stops at stage and relay channels are probed, the maximum relay index is , and the corresponding approximated signal power at the destination is given by

 Psmax1≤k≤nωk=rP⋅cnΩn, (14)

where

 Ωn=max1≤k≤nωk,n=1,2,…,N . (15)

The relay selection problem defined in this paper is to decide when to stop probing the relay channels and which probed relay to cooperate with to maximize the expected value of the approximated signal power at the destination.111The objective function defined in (16) is an intuitive measure. However we will show in Section IV the stopping rule maximizing this particular objective achieves full diversity order. Formally, we want to choose a stopping time with respect to the sequential channel probings and the channel realizations and we formulate the relay selection problem as an optimal stopping problem as follows.

 maxNsE[rP⋅cNsΩNs]s.t.Ns is a stopping time,Ns≤Na.s. (16)

Assuming to be a constant, then (16) is equivalent to the following optimal stopping problem.

 maxNsE[cNsΩNs]s.t.Ns is a stopping time,Ns≤Na.s. (17)

## Iii Solution to the Optimal Stopping Problem

From the theory of optimal stopping times (see [23, chap. 2]) , we define the value functions with the information state as follows.

For stage , define

 Vn(x):=maxNs is a stopping time,n≤Ns≤NE[cNsΩNs|Ωn=x] , (18)

and we can write down the backward induction for the value functions as follows,

 VN(x)=cN⋅x, Vn(x)=max{cnx,E[Vn+1(Ωn+1)|Ωn=x]},n≤N−1. (19)

From the backward induction, we obtain the structure of the optimal stopping time stated in the following theorem.

###### Theorem 1

The optimal stopping rule for the optimal stopping problem described by (17) is given by thresholds such that

 N∗s=inf{n≥1:cnΩn≥tn}, (20)

where the threshold is the unique solution of the following fixed point equations

 cntn=E[Vn+1(Ωn+1)|Ωn=tn]. (21)

Consequently, the value functions satisfy

 Vn(x)={cnx if x≥tn,E[Vn+1(Ωn+1)|Ωn=x] if x
{proof}

The proof can be found in Appendix-A.

Theorem 1 states that the optimal stopping rule to problem (17) we formulated in Section II is described by a set of thresholds , based on which we propose the relay selection strategy as follows. The decision at each stage is given by

 d∗n={Stop and choose Rk if cnΩn≥tn,ωk=Ωn,Continue  if cnΩn

and at the final stage

 d∗N={Choose Rk % if ΩN≥ωs,d,ωk=Ωn,Do not choose any relay  if ΩN<ωs,d. (24)

## Iv Performance Analysis

In this section, we analyze the performance of the relay selection strategy RS_OSR defined in Section III. We show in Section IV-A that RS_OSR achieves full diversity order. In Section IV-B we obtain an upper bound on the expected stopping time for RS_OSR.

### Iv-a Diversity Gain

To compute the diversity gain of the relay selection strategy RS_OSR, we consider the error probability and show that RS_OSR achieves full diversity order. This result is stated in the theorem below.

###### Theorem 2

The relay selection strategy RS_OSR defined by (23)-(24) achieves full diversity order. That is, when relays are available

 Diversity gain:= −limγ→∞log(Pe(γ))log(γ)=(N+1), (25)

where is the error probability under RS_OSR with being the SNR.

{proof}

In order to compute the error probability of our relay selection strategy, we first consider the error probability when the relay selection stops at stage and is selected. Let be the error probability conditional on and the relay selection stops at stage . From Appendix B there exists constants such that

 Pe,n(γ,x)≤AMexp(−BM1−r2q1xcnγ) +AMexp(−BMr2q2xcnγ):=Qn(γ,x) . (26)

Now using (26), we try to get an upper bound for the error probability when the relay selection stop at stage . Under RS_OSR, (23) implies that the relay selection stop at stage if and is selected when . Consequently, for we have

 P(error, relay selection stops at time n) = P(error,Ωn≥tn/cn) = ∫∞tn/cnPe,n(γ,x)dFΩn(x) ≤ ∫∞tn/cnQn(γ,x)dFΩn(x) ≤ Qn(γ,tn/cn). (27)

By (24), RS_OSR selects relay if and selects no relay if at stage . From [11, chap. 8], we know that the error probability if the selection stops at stage is upper bounded by

 P(error, relay selection stops at time N) ≤ (CGcNγ)−(N+1)P(relay selection stops at% time N) ≤ (CGcNγ)−(N+1), (28)

where is a constant defined in [11, chap. 8].

The combination of (27) and (28) gives the complete error probability as follows.

 Pe(γ)= N∑n=1P(error, relay selection stops at % time n) ≤ N−1∑n=1Qn(γ,tn/cn)+(CGcNγ)−(N+1) = (1+o(γ))(CGcNγ)−(N+1) , (29)

where the last equality in (29) is true because is the sum of two exponential functions that are dominated by when is large. From the upper bound (29) the diversity gain can be computed by

 Diversity gain= −limγ→∞log(Pe(γ))log(γ) ≥ −limγ→∞log((1+o(γ))(CGcNγ)−(N+1))log(γ) = (N+1). (30)

Theorem 2 states that the relay selection strategy RS_OSR achieves the full diversity order of with relays.

### Iv-B Bandwidth Efficiency

Let be the optimal stopping time in RS_OSR (defined by (20)). The expectation of can be computed as follows

 E[N∗s]= 1+N∑n=1P(N∗s>n)=1+N∑n=1P(∩nk=1{Ωk

Let and , then from (31) we further obtain

 E[N∗s]≤ 1+N∑n=1(P(ω1

If for some , the expected stopping time of RS_OSR is bounded for any number of relays. We show in the theorem below that it is indeed the case.

###### Theorem 3

There exists some positive constant such that for any . Consequently, the expected number of probing for the relay selection strategy RS_OSR described by (23)-(24) is bounded above by

 E[N∗s]≤ 1−ϵϵ . (33)
{proof}

Since , from (21) we know that satisfies

 cN−1tN−1= E[VN(ΩN)|ΩN−1=tN−1] = E[cNmax(tN−1,ωN)]. (34)

Let , we get

 h(tN−1)tN−1=cN−1cN=1+Nτ1+(N−1)τ>1. (35)

Note that the function is strictly decreasing as shown below. We first compute the derivative of as follows.

 h′(x)= (∫x0xdFω1(ω)+∫∞xωdFω1(ω))′ = xF′ω1(x)+Fω1(x)−xF′ω1(x) = Fω1(x), (36)

where is the CDF of . Moreover, . Then

 (h(x)x)′= 1x2(xh′(x)−h(x)) = 1x2(xFω1(x)−h(x)) ≤ 1x2(xFω1(x)−x)<0. (37)

Define as the inverse function of , then is also strictly decreasing.
Let be the solution to , i.e. . Then, it follows form (35) that

 tN−1=g(cN−1cN)

Furthermore, we want to show that

 tn≤tN−1 for all n≤N−1. (39)

The proof is done by induction.
(39) is true for . Assume the (39) holds for .
For stage , if , we get by the induction hypothesis. If , for any . From (22) we obtain

 E[Vn+1(Ωn+1)|Ωn=tn]= E[Vn+1(max(tn,ωn+1))] = E[cn+1max(tn,ωn+1)]. (40)

Then, from (21) for we get

 cntn= E[Vn+1(Ωn+1)|Ωn=tn] = E[cn+1max(tn,ωn+1)] = cn+1h(tn). (41)

Therefore,

 tn=g(cncn+1). (42)

Note that

 cncn+1=1+(n+1)τ1+nτ≥1+Nτ1+(N−1)τ=cN−1cN. (43)

Since is decreasing, we have

 tn=g(cncn+1)≤g(cN−1cN)=tN−1. (44)

As a result of the above analysis, we have

 tn≤tN−1

for any . Moreover,

 pN=P(ω1

We can now define , then . From (32) we obtain

 E[N∗s]≤ pN+1N1−pN≤1−ϵϵ (47)

for any total number of relays.

From Theorem 3, we know that the expected stopping time of the relay selection strategy RS_OSR is bounded. Therefore, the expected time for each transmission is bounded by

 E[Ttran(1+N∗sτ)]≤Ttran(1+τ1−ϵϵ). (48)

## V Simulation results

We simulate a relay networks and present simulation results. To implement RS_OSR, we solved the fixed point equation (21) in an offline fashion. In particular we use an iterative heuristic to solve the equation with the help of Monte Carlo sampling (to remedy the computation complexity from calculating the conditional expectation in (21)).

In the simulation we compare our proposed relay selection strategy RS_OSR with the optimal relay selection (w.r.t. indices defined by (9)) that probes all relays, which is denoted by RS_ALL. Note that the optimality (can achieve full diversity order) of RS_ALL is proved in [11].

We start with comparing the error probability . Fig.3 and Fig.4 show the comparison for error probability with and respectively. From Fig.3 and Fig.4 we observe that RS_OSR achieves full diversity order as comparable with RS_ALL. Moreover our algorithm outperforms RS_ALL at finer degree consistently. This dues to the gain of efficient bandwidth and power saving. We also observe that the advantages of RS_OSR is more obvious when is higher. This is intuitively true : our optimal stopping selection strategy helps save sensing time and power and therefore the advantage becomes more and more clear when the sensing complexity becomes higher.

Bandwidth efficiency results (rate is measured by the average over all sample path) are shown in Fig.5 and Fig.6, with and respectively. From Fig.5 and Fig.6 we conclude that under both cases, the probing time of RS_OSR is bounded, while the probing time of RS_ALL increases (thus the bandwidth efficiency decreases) as the number of relays increases.

## Vi Conclusion

In the presence of non-negligible probing time for getting relay channels’ instantaneous quality, obtaining full information for the purpose of relay selection leads to an inefficient use of bandwidth in cooperative communications. We design and implement a stopping rule based relay selection strategy RS_OSR and proved its optimality regarding achieving full diversity order. Moreover, the probing time for sensing relay channels under RS_OSR is shown to remain bounded regardless of the number of relay candidates. We establish and demonstrate the above two properties by both analytical and simulation results.

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## Appendix A Proof of Theorem 1

{proof}

First, we want to prove that the value function and are convex and increasing. The proof is done by induction.

At the final stage , the value function is obviously convex (linear) and increasing. Assume that is convex and increasing. Then, at stage we have

 E[Vn+1(Ωn+1)|Ωn=x]= E[Vn+1(max(x,ωn+1))|Ωn=x] = E[Vn+1(max(x,ωn+1))], (49)

where the last equality holds because is independent of (due to the assumption of independence among channels). Since is convex and increasing and the function is convex and increasing in , by the property of convex functions (see [24]) we know that their composition is convex and increasing in for every . Therefore, is also convex and increasing, as it is the the expectation of over . Since by (19), is also convex and increasing.

With the convexity established above, we know that for any , and can have at most two intersections because is convex and is linear. We show below that they have exactly one intersection. For that matter, we want to prove by induction

 Vn(x)≤cnE[ΩN|Ωn=x]. (50)

Inequality (50) is easily true at stage . Suppose (50) is true for . At stage , since we obtain

 Vn(x)= max{cnx,E[Vn+1(Ωn+1)|Ωn=x]} ≤ max{cnx,E[cn+1E[ΩN|Ω(n+1)]|Ωn=x]} = max{cnx,cn+1E[ΩN|Ωn=x]} ≤ max{cnx,cnE[ΩN|Ωn=x]} = cnE[ΩN|Ωn=x]. (51)

Then inequality (50) is true at any stage. Consequently, we get, as ,

 Vn(x)cnx≤ cnE[ΩN|Ωn=x]cnx = E[max(1,maxk>nωkx)] ≤ E[1+maxk>nωkx]→1, (52)

where the convergence in (52) is true because of the following. Note that converges point-wise to , and for . Since is in , by dominated convergence theorem converges to in expectation.

When we have

 cnx=0≤E[Vn+1(ωn+1)]=E[Vn+1(Ωn+1)|Ωn=0]. (53)

From (52) and (53), the linear function is above for large , and is below when . Therefore, the two functions and have exactly one intersection.

Let denote the intersection of and , we get

 Vn(x)={cnx if x≥tn,E[Vn+1(Ωn+1)|Ωn=x] if x

As a result of (54), the optimal stopping time is described by (20).

## Appendix B Proof of (26) in Theorem 2

Let be the symbol error rate function with SNR equals for each node. Since we apply perfect DF protocol, there are two cases to consider with. In the first case the relay perfectly reconstructs the signal and the destination receives both signals from the source and the relay. In the second case the relay fails to reconstruct the signal, so only the source transmitted signal is received at the destination. The probability for the first case is and the one for the second case is where is the SNR at . The error rate at the destination is given by , where is the SNR at . is can be computed as

 γd= {(rωs,d+(1−r)ωk,d)cnγ in the first case,rωs,dcnγ in the second case. (55)

Let be the error probability conditional on the channel parameters and the relay selection stops at stage and is selected. The error probability can be calculated as follows.

 Pe,n(γ,ωs,k,ωs,d,ωk,d) = (1−Φ(rωs,kcnγ))Φ((rωs,d+(1−r)ωk,d)cnγ) +Φ(rωs,kcnγ)Φ(rωs,dcnγ) . (56)

The error probability calculated in (56) is complex. However, a proper upper bound can allow us to analyze the diversity gain of our relay selection strategy. We proceed to obtain an upper bound on the error probability.

Note that from (9), the definition of , we have

 ωs,d≥ωk2q2, ωk,d≥ωk2q1 . (57)

Following which we strike an upper bound for (56) as follows.

 Pe,n(γ,ωs,k,ωs,d,ωk,d) = (1−Φ(rωs,kcnγ))Φ((rωs,d+(1−r)ωk,d)cnγ) +Φ(rωs,kcnγ)Φ(rωs,dcnγ) ≤ Φ((1−r)ωk,dcnγ)+Φ(rωs,kcnγ) ≤ Φ(1−r2q1ωkcnγ)+Φ(r2q2ωkcnγ) ≤ AMexp(−BM1−r2q1ωkcnγ)+AMexp(−BMr2q2ωkcnγ) . (58)

where are constants depending on the modulation scheme as commonly adopted (e.g., [25]) and the last inequality in (58) follows from the properties of the error probability function .

From (58) we have

 Pe,n(γ,x)≤AMexp(−BM1−r2q1xcnγ) +AMexp(−BMr2q2xcnγ):=Qn(γ,x) (59)

and (26) is established.

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