OPTIMAL DESIGN AND ANALYSIS OF A TWO-HOP RELAY NETWORK UNDER RAYLEIGH FADING FOR PACKET DELAY MINIMIZATION

In this paper, we consider a wireless network consisting of a source node, a destination node and multiple relay nodes under Rayleigh fading. Cooperative diversity in the network is achieved by selecting an opportunistic relay node with the best channel condition to the destination node. We focus on the packet level performance in this paper and analyze the average packet delay of the network for various modulation and coding schemes. We assume that the arrival process of packets follows a Markov modulated Poisson process (MMPP). To derive the average packet delay, we first derive the distribution of the number of packets that are successfully transmitted from the source node to the destination node via a relay node. Using the distribution obtained above, a queueing process of the M/G/1 type is developed to model the queue at the source node. The average packet delay is then obtained from the stationary distribution of the queueing process of the source node by applying Little’s Lemma. Based on our results on the average packet delay, the optimal modulation and coding scheme for given network parameters is determined to minimize the average packet delay. We validate our analytic model through simulation. The detailed relations between the average packet delay and network parameters such as average signal to noise ratios (SNRs) between nodes, the number of relay nodes and packet arrival rate, are investigated through numerical studies based on our analytic model as well as simulation studies. From our numerical results, we conclude that the optimal modulation and coding scheme that minimizes the average packet delay depends not only on SNRs of channels between nodes but also on the arrival rate of packets at the data link layer.

1. Introduction.In a wireless network, the channel between nodes experiences fading which is the random fluctuation in signal strength.To mitigate this fading effect in the channel, various schemes have been proposed.The cooperative diversity scheme with the help of multiple relay nodes between the source node and the destination node is one of the solutions.The cooperative diversity scheme has been extensively considered in the literature.Earlier works on the cooperative diversity scheme [1,2,3] mainly focus on the physical layer of the network.However, it is helpful to study the performance of the data link layer as well as that of the physical layer as pointed out in [4].There are a few works that concern the performance of 2 HONG IL CHO AND GANG UK HWANG the data link layer.[5] considers the relay selection problem based on packet delay and throughput for a relay network with a single relay node and a single source node.The performance analysis of a two-hop relay network with a single relay node using an adaptive modulation and coding scheme is dealt with in [7].The performance analysis of the data link layer in a multi-hop wireless relay network can be found in [6].However, the packet delay analysis in [6] is performed on simple assumptions, e.g., Bernoulli packet arrival process and binary erasure channel.
In this paper, we analyze the average packet delay of a two-hop wireless network with multiple relay nodes having a modulation and coding scheme.The rationale behind the two-hop relaying approach is that it can reduce the system complexity in practice while improving the network performance, as pointed out by [8].A practical example of a two-hop relay network is also demonstrated in [8].Besides, [16] reports that some Medium Access Control (MAC) protocols like the IEEE 802.11MAC protocol, which is widely used for WLANs, do not perform well in a multi-hop relay network.Hence, we focus on a two-hop relay network and analyze the packet delay of the network.
We assume that the packet arrival process at the source node follows the Markov modulated Poisson process (MMPP) to model the bursty characteristic of the multimedia traffic [15].The channels between nodes are assumed to follow the flat Rayleigh fading model.The probability of a successful packet transmission is determined by a packet error rate formula [9] which is the function of the signal to noise ratio (SNR) of the fading channel between two communicating nodes.
To derive the average packet delay, we first derive the distribution of the number of packets that are successfully transmitted from the source node to the destination node via a relay node.We then construct an infinite state Markov chain with states that represent the number of packets in the queue at the source node and the state of the underlying Markov chain of the arriving process at the source node.The average packet delay is derived from the stationary distribution of the Markov chain by applying Little's Lemma.Based on our results on the average packet delay, the optimal modulation and coding scheme for given network parameters, e.g., the arrival rate of packets at the source node, the average signal to noise ratio (SNR) between the source node and the relay nodes, is determined to minimize the average packet delay.From our numerical results, we conclude that the optimal modulation and coding scheme that minimizes the average packet delay depends not only on SNRs of channels between nodes but also on the arrival rate of packets at the data link layer.
The rest of this paper is organized as follows.The system model is introduced in Section 2. Our analytic model to derive the average packet delay is presented in Section 3. In particular, we obtain the distribution of the number of packets that are successfully transmitted from the source node to the destination node in Section 3.1 and 3.2.The queueing analysis for the queue at the source node is performed in Section 3.3.A validation of our analytic model through simulation and numerical results based on our analytic model are presented in Section 4. We give our concluding remarks in Section 5.
2. System Model.We consider a wireless network consisting of a source node, a destination node and N relay nodes as depicted in Fig. 1.A discrete-time system is considered in this paper.Hence, the time axis is divided into equally sized slots.In our discrete-time model, time instants at slot boundaries are indexed by In the first phase of a slot, the source node broadcasts packets to relay nodes.The transmission (forwarding) of the packets from a relay node to the destination node occurs in the second phase of the slot.We assume that the relaying uses the decode and forward scheme and that there are no direct transmissions between the source node and the destination node.The detailed description of the packet transmission from the source node to the destination node via relay nodes will be given in section 2.2.

Fading channel model.
There are N wireless channels between the source node and relay nodes, and there are the same number of wireless channels between relay nodes and the destination node.All wireless channels are assumed to be flat Rayleigh fading channels.Let X i (t) denote the SNR of the channel between the source node and the i-th relay node and let Y i (t) denote that of the i-th relay node and the destination node at the beginning of the t-th slot for 1 ≤ i ≤ N .Assumptions on X i (t) and Y i (t) are listed as follows: Assumptions 1 and 2 are reasonable when the duration of a slot is close to the coherence time of the channel [10].Since all nodes are spatially apart, assumption 3 is also justified.Let P SR be the average SNR of the channels between the source node and relay nodes and let P RD be that of the channels between relay nodes and the destination node.Since all wireless channels are flat Rayleigh fading channels by our assumption, X i (t) and Y i (t) are exponential random variables with probability distribution functions f SR (γ) and f RD (γ) given by For later use, let E be the transition probability matrix of E(t), i.e., ] .
Arriving packets are stored in the queue at the source node.The queue size of the source node is assumed to be infinite.The service discipline is assumed to be first-come-first-serviced (FCFS).The packets in the queue at the source node are modulated and coded into symbols at the beginning of a slot.At the physical layer, packets in the queue are then mapped into a frame which contains a fixed number of symbols.This frame is transmitted in the first phase of the slot from the source node.In this paper, we consider various transmission modes of modulation and coding which are adopted from [9] and summarized in Table 1.Let n denote the index of the transmission mode used by the source node.Let d n denote the number of packets that are mapped into a frame when transmission mode n is used.If the number of packets in the queue are less than d n , all packets in the queue at the source node are mapped into a frame and the remaining blocks of the frame are filled with dummy symbols.
In the first phase of a slot, the source node broadcasts a frame to relay nodes.Each relay node decodes packets from the received frame.The packet error rate w n (γ) for transmission mode n and the received SNR γ can be approximated as given by [9] where a n , g n and γ pn are transmission mode dependent parameters which are summarized in Table 1.Since we adopt the decode and forward scheme, each relay node re-maps successfully decoded packets from the received frame to a new frame.In this process, the same transmission mode n that the source node uses is applied.
Let S be the set of indexes of relay nodes that can successfully decode at least one packet from the received frame.In the second phase of the slot, a relay node with its index in the set S is selected to transmit.Let ĩ be the index of the selected relay node.To exploit cooperative diversity, the selected relay node ĩ satisfies the best channel condition, i.e., Y ĩ(t) ≥ Y i (t) for i ∈ S.An example of the relay selection is shown in Fig. 1.The selection of the relay node with the best channel condition can be processed in a distributed manner as in [11,12].The detailed analysis of this process is beyond the scope of this paper.In the second phase of the slot, the selected relay node transmits its new frame to the destination node.The destination node then decodes packets from the received frame.The packet error rate occurred in the relay-destination transmission is the same as given in (1), since the transmission mode of the selected relay node is equal to that of the source node.
After decoding the packets, the destination node transmits to the selected relay node an ACK frame which contains indexes of successfully transmitted packets.The selected relay node then forwards the received ACK to the source node.Then the source node discards the packets in the queue which are correctly received by the destination node.We assume that the transmission of ACKs is error-free.Note that the resequencing of packets at the destination node is needed to upload them to the upper layer.However, we omit the details of the resequencing procedure since it is beyond the scope of our work.
3. Performance Analysis.Throughout this section, we assume that transmission mode n is used at the source node and that there are k packets in the queue of the source node at the beginning of an arbitrary slot.Let O and O (k) be random variables that denote the numbers of packets that are transmitted successfully from the source node to the destination node via the opportunistic relay node when k ≥ d n and k < d n , respectively.Note that 0 ≤ O ≤ d n and 0 ≤ O (k) ≤ k.

The distribution of O.
In this subsection, we assume that k ≥ d n .Let R i denote the number of packets that are transmitted successfully in the first phase of the slot from the source node to the i-th relay node for 1 ≤ i ≤ N .Note that R i are i.i.d. for all i, since X i (t) is i.i.d. for all i.Since X i (t) remains invariant during a slot, the probability f l (γ) of the event {R i = l} when the SNR of the channel between the source node and the i-th relay node is equal to γ, is obtained for 0 ≤ l ≤ d n as follows: where I{•} denotes the indicator function.By averaging f l (γ) over the distribution f SR (γ) of X i (t), the distribution f l of R i is derived for 0 ≤ l ≤ d n as follows: Let |S| be the cardinality of the set S. Then the distribution b s of |S| is obtained by using (3) as follows: In the second phase of the slot, a relay node is selected to transmit to the destination node as described in Section 2.2.We denote ĩ by the index of the selected relay node.Then Y ĩ(t) ≥ Y i (t) for i ∈ S. Note that the distribution of Y ĩ(t) depends on |S|.By using the order statistics [13], the conditional probability density function f m (γ) that the destination node decodes m packets successfully from the frame sent by the selected relay node given that R ĩ = l, |S| = s and Y ĩ(t) = γ, is obtained as follows: for 1 ≤ s ≤ N .Let c (l,s) be the joint probability that the number of relay nodes that can successfully decode at least one packet and the number of successfully decoded packets at the selected relay node, i.e., c (l,s) Pr{R ĩ = l, |S| = s}.
The closed-form expression of c (l,s) is given in Theorem 3.1 below.
The distribution p m of O is obtained by using the law of total probability as follows: Theorem 3.1.The joint probability c (l,s) is given as follows: Proof.If all relay nodes fail to decode at least one packet from the received frame, i.e., s = 0, then the selected relay node (the ĩ-th relay node) also has no packet to transmit to the destination node.So we have c (l,0) = f 0 Assume that 1 ≤ s ≤ N .Since X i (t) between the source node to relay nodes are i.i.d. and X i (t) and Y i (t) are independent, the distribution of R ĩ is the same as that of R i for all i ∈ S.So we have By using (7), we have the expression of c (l,s) as follows: 3.2.The distribution of O (k) .In this subsection, we assume that k < d n .Note that all k packets are contained in a frame transmitted by the source node when k < d n .So probabilities f l (γ) and f l defined in (2) and (3) depend on k, respectively.Now we define the probabilities f for 0 ≤ m ≤ k.

3.3.
Markov model for the queue at the source node.Let Q(t) be the number of packets in the queue at the source node at the beginning of the t-th slot.For this system, we have the following evolution equation where A(t) denotes the number of packets that arrive at the source node in the t-th slot.Let α i be the probability of i packets arriving at the source node when E(t) = 1.Since the packet arrival process follows the Poisson process with rate λ (per slot) when E(t) = 1, we have The two dimensional discrete time stochastic process {Q(t), E(t)} forms a Markov chain.From (8), the transition probabilities of {Q(t), E(t)} are derived as follows: where E i,j denotes the (i, j)-th element of the matrix E.
Let P be the state transition probability matrix of {Q(t), E(t)}.It is straightforward to show that {Q(t), E(t)} is a queueing process of M/G/1 type and its state transition probability matrix P is given by , where Let π i,j denote the steady state probability that the Markov chain {Q(t), E(t)} is in state (i, j) for i ≥ 0 and 1 ≤ j ≤ 2. The steady state probability vector that there are i packets in the queue at the source node is denoted as π i (π i,1 , π i,2 ) for i ≥ 0. Then the stationary distribution of {Q(t), E(t)} is given by π (π 0 , π 1 , π 2 , • • • ).Since π is the stationary distribution, πP = π and ∑ ∞ i=0 π i e = 1 hold, where e denotes the 2 × 1 column vector whose elements are all equal to 1.The value of the stationary distribution π is obtained by the matrix analytic method in [14].
Note that q i π i e is the probability that there are i packets in the queue at the source node in the steady state for i ≥ 0. So the average number of packets in the queue at the source node, denoted by Q, is given by The average arrival rate of packets in a slot is the product of λ and the steady state probability of the event {E(t) = 1}.So the average arrival rate of packets, denoted by λ, is given by Finally, let L denote the average packet delay, i.e., the time duration between the time instant of a packet arrival at the source node and the time instant of the successful reception of the packet at the destination node.By using Little's lemma, L is derived as follows:

Numerical results.
To validate our analytic model, we simulate the wireless relay network considered in this paper.The simulation is performed using MAT-LAB.For the simulation of the Rayleigh fading channels, i.i.d.stochastic processes of SNRs of the Rayleigh fading channels are generated by a built-in m-file script exprnd.The numerical results of the average packet delay based on our analytic model are also presented in this section.For our numerical analysis and simulation, we assume that the phase duration is equal to 2 ms and the packet length P b = 1080 bits as in [9].We adopt five different transmission modes which are described in Table 1.The number d n of packets that are contained in a frame at the source node and the packet error rate parameters a n , g n and γ pn in (1) are also given in Table 1 for each transmission mode.Note that the number of packets that are contained in a frame as well as the packet error rate increase when we use higher transmission mode.Throughout this section, the transition probabilities r ON and r OF F of E(t) are set to 0.6 and 0.4, respectively.
The results on the average packet delay versus packet arrival rate λ for the simulation and our analytic model when P SR = P RD = 20(dB) and N = 5, are plotted for each transmission mode in Fig. 2. Note that λ is the arrival rate of packets at the source node when E(t) = 1.We see from the figure that the results from our analytic model and those from the simulation are well matched.From this observation, we validate our analytic model.For each transmission mode, the average packet delay grows up as λ increases.Since the number of packets mapped into a frame increases as we use higher transmission mode, the speed of the increment in the average packet delay decreases as we use higher transmission mode.However, we observe that the average packet delay for transmission mode 5 is larger than the average packet delays of the other transmission modes for small values of λ.This is due to the fact that the packet error rate of transmission mode 5 is significantly larger than the packet error rates of the other transmission modes.
In Fig. 3, the results on the average packet delay versus P SR for the simulation and our analytic model are plotted when P RD = 20(dB), λ = 2 and N = 5 for each   transmission mode.Since the queueing process for transmission mode 1 is unstable in this case, we plot the average packet delay only for the other transmission modes.As P SR increases, the average packet delay decreases for all transmission modes.Although the number of packets contained in a frame increases as we use higher transmission mode, the packet error rate also increases for higher transmission mode.Due to this reason, we see that the average packet delay of lower transmission mode is less than that of higher transmission mode for small values of P SR in Fig. 3.
In Fig. 4, the effect of the number of relay nodes N on the average packet delay for the simulation and our analytic model is plotted for each transmission mode.As the number of relay nodes increases, the average packet delay decreases for each transmission mode.This result is in consequence of the cooperative diversity in the network.Note that the average packet delay converges slowly to its minimum as N increases for transmission modes 4 and 5. On the other hand, the average packet delay converges very quickly to its minimum as N increases for transmission modes 2 and 3.The reason for this behavior can be explained as follows.As N increases, the channel condition between the selected relay node and the destination node gets better, and hence the more packets can be successfully transmitted between the selected relay node and the destination node.When the transmission mode is low such as transmission modes 2 and 3, the packet error rate between the selected relay node and the destination node becomes negligible even for small values of N due to its low packet error rate.Hence, the average packet delay reaches its minimum for small values of N in this case.On the contrary, the packet error rate for a high transmission mode such as transmission modes 4 and 5 continues to decrease even for large values of N as N increases.This affects the distributions of O and O (k) for large values of N .Hence, the average packet delay continues to decrease in this case as N increases.
To investigate a joint effect of P SR , P RD and N on the average packet delay, we give two contour plots in Fig. 5 for transmission mode 3.In Fig. 5a, the contour plot of the average packet delay when N = 1 is shown.Since we have only one relay node in this case, there is no cooperative diversity in the network.Hence, the effect of P SR and P RD on the average packet delay is symmetric as shown in Fig. 5a.On the other hand, the effect of P SR and P RD on the average packet delay is not symmetric when there are multiple relay nodes as shown in Fig. 5b where N = 5.The effect of P RD on the average packet delay for any fixed P SR is less critical than that of P SR on average packet delay for any fixed P RD when N = 5.The result shows that the channel between the source node and the selected relay node is a bottleneck in packet transmission.Hence, we can conclude that the improvement in the quality of the channels between the source node and relay nodes is important in a two-hop relay network with multiple relay nodes.
From Figs. 2, 3 and 4, it is clear that the optimal transmission mode that minimizes the average packet delay depends on the network parameters such as the arrival rate of packets at the source node, the average SNR of channels between nodes and the number of relays.In Fig. 6, the optimal transmission mode for each P SR , P RD and λ is shown.The optimal transmission mode for each P SR and λ is shown in Fig. 6a and that for each P RD and λ is shown in Fig. 6b.
In Fig. 6a, we see that higher transmission modes become optimal as P SR increases.This is due to the fact that the packet error rate is negligible for high values of P SR .In addition, the number of packets mapped into a frame for a higher transmission mode is larger than that for a lower transmission mode.However, we also see that for any fixed value of P SR , lower transmission modes become optimal as λ decreases.For small values of λ, the number of packets mapped into a frame in a lower transmission mode is large enough to manage the arriving packets at the source node.Moreover, the packet error rates of lower transmission modes are lower than those of higher transmission modes.Hence, lower transmission modes become optimal as λ decreases for any fixed P SR .
In Fig. 6b, we see the same behavior as in Fig. 6a.That is, higher transmission modes become optimal as P RD increases and lower transmission modes become optimal as λ decreases.However, we also see in Fig. 6b that the optimal transmission mode depends only on λ for higher values of P RD .This is due to the cooperative diversity in the network.That is, when the value of P RD goes beyond a certain  level, a further improvement in the channel quality between relay nodes and the destination node is meaningless due to cooperative diversity, and accordingly the optimal transmission mode depends only on λ for higher values of P RD .
The results in Fig. 6 strongly recommend that not only the channel condition between nodes at the physical layer but also the packet arrival rate at the data link layer be considered in the selection of transmission mode to minimize the average packet delay.So we conclude that a cross-layer design is mandatory to minimize the average packet delay.

Conclusions and future works.
In this paper, we consider a wireless network consisting of a source, a destination and multiple relay nodes under Rayleigh fading.To achieve cooperative diversity, an opportunistic relay selection scheme is considered in this paper.An analytic model is developed to derive the average packet  delay for various modulation and coding schemes.First, we derive the distribution of the number of packets that are transmitted successfully from the source node to the destination node via relay nodes.Using the distribution mentioned above, we model the queue at the source node as an queueing process of M/G/1 type.The average packet delay is obtained from the stationary distribution of the queueing process by applying Little's Lemma.Based on our results on the average packet delay, the optimal modulation and coding scheme for given network parameters is determined to minimize the average packet delay.Numerical results based on our analytic model and simulation results show that the optimal modulation and coding scheme to minimize the average packet delay depends not only on the parameters of the physical layer such as the average SNR of channels between nodes but also on those of the data link layer such as the packet arrival rate at the source node.
As for a future work, it is interesting to analyze higher moments of the packet delay and investigate the effect of the optimal modulation and coding scheme on higher moments of the packet delay.To obtain higher moments of the packet delay, we first tag an arbitrary packet and consider the joint distribution of the queue length and the position of the tagged packet in the queue at the slot time when the tagged packet joins the queue.Based on the joint distribution, we expect to obtain a system of equations related with higher moments of the packet delay.
A cross-layer design and analysis of our two-hop relay network is another interesting future work.In our two-hop relay network the number of packets that are successfully transmitted from the source node to the destination node during a slot, significantly depends on the numbers of successfully decoded packets at relay nodes.Hence, to achieve a more efficient low-delay transmission of packets, it will be helpful to consider the numbers of packets decoded successfully at relay nodes as well as channel conditions between nodes in the relay selection process.This cross-layer relay selection scheme is obviously of interest and needs to be analyzed.

Figure 1 .
Figure 1.The model of the two-hop network with multiple relay nodes

1 ,
RD * (γ) of Y ĩ(t) conditioned on the event {|S| = s} is given as follows: f (s) RD * (γ) = s P RD e −γ/PRD (1 − e −γ/PRD ) s−for 1 ≤ s ≤ N and γ ≥ 0. The conditional probability p (l,s) , we can obtain the conditional probability p (l,s) m that the destination node decodes m packets successfully from the received frame given that R ĩ = l and |S| = s.Note that p (l,0) m is equal to I{m = 0}.The closed-form expression of p (l,s) m for 1 ≤ s ≤ N is obtained as follows: conditional probabilities of f l (γ) and f l conditioned on k < d n , respectively.By using similar arguments as in Section 3.1, closed-form expressions of f (k) l (γ) and f (k) l are derived by substituting k for d n in (2) and (3) for 0 ≤ l ≤ k, respectively.The probabilities b (k) s and c (l,s,k) which denote the modified probabilities of b s and c (l,s) when k < d n , are derived by substituting f (k) l for f l in (4) and (6).The detailed expressions of b (k) s and c (l,s,k) are omitted.The conditional probability that the destination node decodes m packets successfully from the received frame conditioned on the event {R ĩ = l, |S| = s}, is equal to p (l,s) m in Section 3.1 for 0 ≤ m ≤ k, 0 ≤ l ≤ k and 0 ≤ s ≤ N .The distribution p (k) m of O (k) is obtained as follows:

Figure 2 .
Figure 2. Average packet delay versus λ when P SR = P RD = 20(dB) and N = 5 for each transmission mode.

5 Figure 5 .
Figure 5. Contour plots of average packet delay for transmission mode 3 versus P SR and P RD when N = 1 and N = 5.
Optimal transmission mode versus P SR and λ when P RD = 20(dB).Average SNR (dB) Optimal transmission mode versus P RD and λ when P SR = 20(dB).

Figure 6 .
Figure 6.Optimal transmission mode versus P SR , P RD and λ when N = 5.

Table 1 .
Various Transmission Modes