Statistical Delay Tradeoffs in Buffer-Aided Two-Hop Wireless Communication Systems
This paper analyzes the impact of statistical delay constraints on the achievable rate of a two-hop wireless communication link, in which the communication between a source and a destination is accomplished via a buffer-aided relay node. It is assumed that there is no direct link between the source and the destination, and the buffer-aided relay forwards the information to the destination by employing the decode-and-forward scheme. Given statistical delay constraints specified via maximum delay and delay violation probability, the tradeoff between the statistical delay constraints imposed on any two concatenated queues is identified. With this characterization, the maximum constant arrival rates that can be supported by this two-hop link are obtained by determining the effective capacity of such links as a function of the statistical delay constraints, signal-to-noise ratios (SNR) at the source and relay, and the fading distributions of the links. It is shown that asymmetric statistical delay constraints at the buffers of the source and relay node can improve the achievable rate. Overall, the impact of the statistical delay tradeoff on the achievable throughput is provided.
With the widespread use of smart-phones and tablets, the volume of global mobile traffic has increased explosively in recent years. The portion of multimedia data, such as mobile video and voice over IP (VoIP), has surged significantly within this wireless traffic . In such multimedia traffic, delay is an important consideration. Meanwhile, providing deterministic quality of service (QoS) guarantees is challenging in wireless systems, since the instantaneous rate of the channel varies randomly depending on numerous factors, such as mobility, changing environment and multipath fading . Therefore, providing statistical QoS guarantees is more suitable in such randomly-varying wireless environment.
Effective bandwidth theory has been developed to analyze high-speed communication systems operating under statistical queueing constraints , . The queueing constraints are imposed on buffer violation probabilities and are specified by the QoS exponent , which dictates the exponential decay rate of the queue length in the stable state. Also, Chang and Zajic have characterized the effective bandwidths of time-varying departure processes in , which can be utilized to analyze the volatile wireless systems. Moreover, Wu and Negi in  defined the dual concept of effective capacity, which provides the maximum constant arrival rate that can be supported by a given departure process while satisfying statistical delay constraints. The analysis and application of effective capacity in various settings have attracted much interest recently (see e.g. - and references therein).
In this paper, we study the achievable rate of two-hop systems operating under statistical delay constraints. In particular, we assume that there are buffers at both the source and the relay nodes, and consider the queueing delay introduced by the buffers. Note that - have also recently investigated the effective capacity of the relay channels. For instance, Tang and Zhang in  analyzed the power allocation policies of relay networks, where the relay node is assumed to have no queue, i.e., the packets arriving to the relay node are forwarded immediately. In , Liu et al. considered the cooperation of two users for data transmission, where the interchanged data goes through only the queue of the other user. Parag and Chamberland in  provided a queueing analysis of a butterfly network with constant rate for each link, while assuming that there is no congestion at the intermediate nodes. The effective capacity of the two-hop link in the presence of the statistical queueing constraints at the source and relay node is given in , and the performance for multi-relay links is analyzed in .
In this work, as a significant departure from previous works, we consider statistical end-to-end delay constraints, imposed as the limitations on the maximum delay and delay violation probability. Note that statistical end-to-end delay analysis can also be found in -. In , Wu and Negi considered statistical end-to-end delay constraints for half-duplex relays, and gave an effective capacity formulation with time allocation to the different hops. In -, the authors considered the statistical end-to-end delay constraints of multi-hop links, while assuming that the statistical delay violation probability of the queues are equal. However, it is possible that the relay can tolerate more stringent delay constraints while not affecting the system performance . Therefore, we seek to determine the optimal statistical QoS exponents of the buffers under given end-to-end delay constraints. Additionally, we note that the analysis of buffer-aided systems have attracted much interest recently (see e.g., - and reference therein). In such analysis, the authors considered the case that only the relay node has buffer, and the average queueing delay is investigated . The contributions can be summarized as follows:
We characterize the tradeoff between the statistical delay constraints at the source and relay nodes, providing a framework for dynamically adjusting the delay constraints of any two interacting queues.
With the identified interplay, we then derive the effective capacity of the two-hop links under a target statistical end-to-end delay constraint by optimizing over the statistical queueing constraints at the queues of the source and relay nodes.
We also describe a method for obtaining the effective capacity in such settings. Additionally, we show that symmetric delay constraints at the two buffers do not always lead to the optimal performance. Instead, asymmetric delay constraints, e.g., when the delay constraint at one queue is more relaxed, can lead to larger achievable rates for the two-hop system, which we verify via numerical results. Moreover, it is demonstrated that the improvement is affected by the statistical delay constraints, the signal-to-noise ratio (SNR) levels and the channel conditions of the links.
The rest of this paper is organized as follows. In Section II, the system model and necessary preliminaries are described. In Section III, we present the tradeoff between the statistical delay constraints of any two concatenated queues. We describe our main results for block-fading channels in Section IV, with numerical results provided in Section V. Finally, in Section VI, we conclude the paper.
Ii-a System Model
The two-hop communication link is depicted in Figure 1. In this model, source is sending information to the destination with the help of the intermediate relay node . We assume that there is no direct link between and (which, for instance, holds, if these nodes are sufficiently far apart in distance). Both the source and the intermediate relay nodes are equipped with buffers. Hence, for the information flow of such links, the queueing delay experienced is given by , where and denote the stationary delay experienced in the queue at the source and relay node, respectively.
We consider a full-duplex relay, and hence assume that reception and transmission can be performed simultaneously at the relay node. Note that full-duplex relaying can be achieved through some form of analog self-interference cancellation followed by digital self-interference cancellation in the baseband domain , . In the th symbol duration, the signal received at the relay from the source and the signal received at the destination from the relay can be expressed as
where for denote the inputs for the links and , respectively. More specifically, is the signal sent from the source and is sent from the relay. The inputs are subject to individual average energy constraints where is the bandwidth. Assuming that the symbol rate is complex symbols per second, we can easily see that the symbol energy constraint of implies that the channel input has a power constraint of . We assume that the fading coefficients are jointly stationary and ergodic discrete-time processes, and we denote the magnitude-square of the fading coefficients by . Above, in the channel input-output relationships, the noise component is a zero-mean, circularly symmetric, complex Gaussian random variable with variance for . The additive Gaussian noise samples are assumed to form an independent and identically distributed (i.i.d.) sequence. We denote the signal-to-noise ratios as .
Ii-B Statistical Delay Constraints
Suppose that the queue is stable and there exists a unique such that
where and are the logarithmic moment generating functions (LMGFs) of the arrival and service processes, respectively. Then, 
where is the stationary queue length. Throughout the text, logarithm expressed without a base, i.e., , refers to the natural logarithm .
We need to guarantee that the statistical delay performance of the two-hop link is not worse than the statistical delay performance specified by , where is the limitation on the statistical delay violation probability, and is the maximum tolerable delay. Note that the end-to-end delay consists of the queueing and transmission delays. As indicated in [26, Section IV], the flow of data bits are treated as the flow of a fluid in the theory of effective bandwidth, in which case the transmission delay can be negligible if . The end-to-end delay can be approximated by the queueing end-to-end delay , . Assume that the first-in first-out (FIFO) queues are saturated, and hence they always attempt to transmit . Then, the queueing delay violation probability can be written equivalently as , 
where we define when , and
is the statistical delay exponent associated with the queue, with denoting the LMGF of the service rate, and is decided by the arrival and departure processes jointly. Note that the larger , the smaller the delay violation probability is, implying more stringent delay constraints. Now, we can express the probability density function of the random variable as
where and are the LMGFs of the service rates of queues at the source and relay nodes, respectively. In the two-hop system, we can express the end-to-end delay violation probability as
Note that we should satisfy
Ii-C Effective Capacity
We can dynamically control the delay constraints at the queues of the source and relay nodes specified by and as long as the statistical end-to-end delay performance (11) can be guaranteed. At the same time, for each realization of , assume that the constant arrival rate at the source is , and the channels operate at their capacities. To satisfy the queueing constraint at the source, we must have
where is the solution to
and is the LMGF of the instantaneous capacity of the link.
In order to satisfy the queueing constraint of the intermediate relay node , we must have
where is the solution to
Above, is the LMGF of the arrival process to the queue at the relay, and is the LMGF of the instantaneous capacity of the link.
Note that we can obtain the effective capacity with following the method provided in [15, Theorem 2] (Appendix -A).111We include the theorem in Appendix -A for the reader’s convenience. Denote as the set of pairs such that (11) can be satisfied. After these characterizations, effective capacity of the two-hop communication model under statistical delay constraints can be formulated as follows.
The effective capacity of the two-hop communication link with statistical delay constraints specified by is given by
where is the set of all feasible satisfying (11). Hence, effective capacity is now the maximum constant arrival rate that can be supported by the two-hop channels under the end-to-end statistical delay constraints.
Iii Statistical Delay Tradeoffs
For the following analysis, we first characterize the relation between and the associated minimum satisfying the statistical delay constraint (11). We have the following results.
Consider the following function
where is defined as the statistical delay exponent associated with . Denoting as a function of , we have the following properties:
is continuous. Moreover, for , we have
with denoting the Lambert W function, which is the inverse function of in the range .
is strictly decreasing in .
is convex in .
, and .
Proof: See Appendix -B.
The above properties can be understood intuitively. Larger enforces more stringent delay constraints on queue 1 (i.e., the source queue), and we can loosen the delay constraints for the queue 2 (i.e., the relay queue), and vice versa. When either queue is subject to a deterministic constraint, i.e., , the delay violation occurs only at the other queue. In Fig. 2, we plot as a function of for the case with and sec for illustration. Note that only in the dark region are feasible to achieve the statistical delay performance. As can be seen from the figure, the curve given by the lower boundary matches the properties in the lemma.
Iv Effective Capacity in Block-Fading Channels
In this section, we seek to identify the constant arrival rates that can be supported by the two-hop system according to the statistical delay tradeoff characterized earlier. We consider a block fading scenario in which the fading stays constant for a block of seconds and changes independently from one block to another.
We assume that the channel state information (CSI) of the link is available at and , and the CSI of the link is available at and . The instantaneous capacities of the and links in each block are given, respectively, by
in the units of bits per block or equivalently bits per seconds. These can be regarded as the service processes at the source and relay.
Iv-a Buffer Stability and Log-Moment Generating Function of Block Fading Channels
To ensure the stability of the queues, we need to enforce the following condition 
i.e., the average arrival rate for the queue at the relay should be less than the average service rate.
Under the block fading assumption, the LMGFs for the service processes of queues at the source and the relay as functions of are given by
The LMGF for the arrival process of the queue at the relay is 
Iv-B Effective Capacity under Statistical Delay Constraints
In the following, we first assume that there exist and such that (11) is satisfied. We can identify the effective capacity associated with the given and values from Theorem 2. Reminding the statistical delay tradeoff indicated in Lemma 1, we can obtain the maximum effective capacity by looping over all possible , i.e., and , which is the effective capacity under the statistical delay constraint in Definition 1.
Consider the function
where . This function has the following properties.
is increasing in , and , i.e., the first derivative of with respect to at is given by the average service rate.
is a concave function of .
, i.e., the negative of the logarithm of the probability of the event that the service rate is 0.
Proof: See Appendix -C.
From the properties above, we can see that is equal to 0 at , and then it increases sublinearly, and approaches an upperbound, if it exists, as . Therefore, is a bijective function of , and for each value of , we can find the associated . Note that the effective capacity expressed as is decreasing in .
In the remainder of the paper, we use the following definitions
Throughout this paper, we consider the fading distributions that satisfy the following conditions: 1) ; 2) .
Under the above assumption, we can see that and approaches as increases. Note that for the continuous distributions of the fading states, such as Rayleigh and Rician fading, the above assumption is justified immediately. If the above assumption does not hold, we can see that the upper bounds for and are finite-valued, and the following analysis still holds while only considering a sliced part of of the curve characterized in Lemma 1.
According to Lemma 2 and the conditions specified in (12) and (14), we can see that the effective capacity obtained always satisfies the statistical delay constraints as long as and satisfy (11). Therefore, with the definitions of and in (24), we can find the associated and on the lower boundary curve indicated by Lemma 1. Iterating over this set of and , we can derive the maximum effective capacity under end-to-end statistical delay constraints. For other values of and , either (11) cannot be satisfied, or one of the queues is subject to a more stringent constraint than necessary, decreasing the achievable throughput.
For the following analysis, we define
We can characterize the effective capacity of the two-hop system given the statistical queueing constraints and in Theorem 2. Now, we are seeking to identify the effective capacity of the two-hop system under statistical delay constraints specified by , in which case and are unknown. Combining the behavior of given and the tradeoff between and in Lemma 1, we have the following result. Note that and denote the minimum and maximum value of , respectively.
The effective capacity of the two-hop wireless communication system subject to end-to-end statistical delay constraints specified by is given by the following:
Case I: If ,
where (,) is the unique solution pair to , and .
Case II: If ,
where is the solution to , and is the smallest value of with satisfying
Moreover, if , where is the value of with satisfying
the solution to (30) with is unique.
Case III: If ,
where is the solution to , and (,) is the unique solution to
Proof: See Appendix -D.
The above theorem covers all the possibilities in which symmetric or asymmetric delay constraints on the queues at the source and relay nodes can be optimal in the sense of achieving the maximum effective capacity of the two-hop relay system. Case I refers to the case that the maximum throughput can be achieved with symmetric delay constraints at the queues of the source and relay. Case II represents the case when the statistical delay constraints at the relay can be more stringent, while Case III shows the scenario with stricter delay constraints at the source. Recalling Theorem 2, we know that as , and , and hence
V Numerical Results
We consider the relay model depicted in Fig. 3. The source, relay, and destination nodes are located on a straight line. The distance between the source and the destination is normalized to 1. Let the distance between the source and the relay node be . Then, the distance between the relay and the destination is . We assume the fading distributions for and links follow independent Rayleigh fading with means and , respectively, where we assume that the path loss . We assume that sec, and dB in the following numerical results. The curve “Buffer-aided optimal (Asymmetric)” stands for the results in Theorem 1. We also plot the achievable rate when there is no buffer at the relay node “No-buffer” , i.e., the service rate of the queue at the source is given by , and the effective capacity with symmetric delay constraints for the two queues “Buffer-aided symmetric”, i.e., , .
In Fig. 4(a), we plot the effective capacity as a function of SNR of the relay node. We fix , in which case the and links experience the same channel conditions on average. We assume that the maximum delay violation probability is . From the figure, we can see that the effective capacity of the two-hop system increases with . Note that at small values of , the buffer at the relay introduces certain loss in the achievable rate. As increases, the buffer at the relay can be beneficial to the two-hop system under statistical delay constraints such that the achievable throughput can be larger. And, in all cases, the achievable rate of asymmetric delay constraints is greater than the one achieved with symmetric delay constraints at the two buffers. In Fig. 4(b), we plot the associated as a function of . As can be seen from the figure, increases as increases, i.e., we can impose more stringent constraints to the queue at the relay, and hence the delay constraint at the source can be relaxed. In this way, the effective capacity of the two-hop system can be improved.
We are also interested in the impact of the delay violation probability on the achievable performance. In Fig. 5(a), we plot the effective capacity as varies for dB. It is not surprising that when , the effective capacities for different are the same, since in this case. Also, when , the achievable rate with buffer at the relay is larger than the achievable rate without buffer at the relay, in accordance with the finding in  that the throughput can be improved by buffer-aided relay. Moreover, it is interesting that when is relatively large but not one, i.e., the statistical delay constraints are less stringent, the achievable throughput with buffer at the relay is larger. Therefore, buffer-aided relay can be helpful even in the presence of end-to-end delay constraints for certain cases. Also, we can find that for larger , the buffer at the relay can help improve the achievable rate at a smaller , i.e., in the presence of more stringent delay constraints. To get more insights, we also plot the associated values of and as decreases in Fig. 5(b). We can see that the increase in becomes larger in comparison with . Considering the convexity of in in Lemma 1, loosening the queueing constraint at one queue will require the other queue to operate in a much more conservative way, which provides little gain under more stringent delay constraints, i.e., for smaller .
In Fig. 6, we plot the effective capacity as varies. We assume dB, . We can see from the figure that as increases, i.e., the channel condition at the link is worse, the effective capacity decreases, and the increase of SNR at the relay node helps little. It is interesting that even for small values of , as increases, the buffer at the relay can help improve the achievable throughput. Albeit, the benefits provided by the buffer at the relay vanish as approaches 1 since the link becomes the bottleneck of the system. Finally, we plot the effective capacity as and vary in Fig. 7(a), with the associated delay tradeoff and for the proposed asymmetric delay constraints in Fig. 7(b). We assume dB. As can be seen from the figure, for all cases, effective capacity decreases as increases or decreases. The improvement in effective capacity is achieved through strong bias towards the queue at the source, in which case we have much larger in comparison with .
In this paper, we have investigated the maximum constant arrival rates that can be supported by a two-hop communication link with a buffer-aided relay under end-to-end statistical delay constraints. We have provided a unified framework for achieving the statistical delay tradeoffs imposed to the source and relay nodes while satisfying the statistical delay constraints. We have determined the effective capacity in the block-fading scenario as a function of the statistical delay constraints, the signal-to-noise ratio levels and , and the fading distributions. We have shown that asymmetric delay constraints at the two buffers can help increase the effective capacity of the two-hop system compared with symmetric delay constraints. We have found that buffer-aided relay can improve the achievable rate of the system under delay constraints when the SNR at the relay is high, the end-to-end delay constraints is loose, or when the channel conditions between the relay and destination node are more favorable.
-a Preliminary Results
() The constant arrival rates, which can be supported by the two-hop link in the presence of queueing constraints and at the source and relay, respectively, are upperbounded by
() The effective capacity of the two-hop system given and is given by the following:
Case I: If ,
Case II: If and ,
where is the unique value of for which we have the following equality satisfied:
Case III: Assume and .
where is the smallest solution to
-B Proof of Lemma 1
When , the continuity is obvious since there is no pole to (17). Consider . We can see that
(44) (45) (46) (47)
Similarly, we can show that
which gives us (19) immediately by solving the above equation with equality.
Taking the partial derivative of in and noting that the right-hand-side (RHS) of (17) is constant, we have
which, after combining the coefficients of and rearrangements, gives us
In the following, we will show that . Denote , and define
Then, we can rewrite as
Note that is positive. Taking the first derivative of , we obtain
We can show that . Suppose . Considering the numerator of the above equation, we have
(55) (56) (57) (58)
where is incorporated since it is an increasing function of , and its value at is . Therefore, for , i.e., is increasing for . In a similar way, we can show that for . Additionally, we can show by considering the Taylor expansions of and at and noting that the numerator goes to 0 in the order while the denominator goes to 0 in the order of . Therefore, is increasing in . Meanwhile,
Hence, , which in turn, tells us that in (53). Therefore, is strictly decreasing in .
We will show the convexity of by considering the branches for and