Opportunistic Buffered DecodeWaitandForward (OBDWF) Protocol for Mobile Wireless Relay Networks
Abstract
In this paper, we propose an opportunistic buffered decodewaitandforward (OBDWF) protocol to exploit both relay buffering and relay mobility to enhance the system throughput and the endtoend packet delay under bursty arrivals. We consider a pointtopoint communication link assisted by mobile relays. We illustrate that the OBDWF protocol could achieve a better throughput and delay performance compared with existing baseline systems such as the conventional dynamic decodeandforward (DDF) and amplifiedandforward (AF) protocol. In addition to simulation performance, we also derived closedform asymptotic throughput and delay expressions of the OBDWF protocol. Specifically, the proposed OBDWF protocol achieves an asymptotic throughput with total transmit power in the relay network. This is a significant gain compared with the best known performance in conventional protocols ( throughput with total transmit power). With bursty arrivals, we show that both the stability region and average delay of the proposed OBDWF protocol can achieve orderwise performance gain compared with conventional DDF protocol.
relay networks, mobile relays, opportunistic buffered decodewaitandforward (OBDWF) protocol, queueing theory
I Introduction
In wireless communication networks, cooperative relaying not only extends the coverage but also contributes the spatial diversity. As a result, cooperative relay is one of the core technology components in the next generation wireless systems such as IEEE 802.16m and LTEA. There are extensive studies on the theory and algorithm design of cooperative relay, and they can be roughly classified as focusing on the microscopic and macroscopic aspects. Examples of microscopic studies include the decodeandforward (DF), amplifyandforward (AF) and compressandforward (CF) protocols for single hop [1, 2, 3] as well as multihop cooperative relay networks [4]. In [5], the authors demonstrated that AF could achieve optimal endtoend DoF for the MIMO pointtopoint system with multiple relay nodes. In [6], it is shown that a variant of the CF relaying achieves the capacity of any general multiantenna Gaussian relay network within a constant number of bit. In [7], relay selection protocol is shown to achieve higher bandwidth efficiency while guaranteeing the same diversity order as that of the conventional cooperative protocols. However, all these works have focused on the physical layer performance (such as throughput) and failed to exploit the buffer dynamics in the relay. Furthermore, they have assumed all the relays are static and have ignored the potential benefit introduced by mobility in the network. On the other hand, there are also some papers focusing on studying the macroscopic behavior of cooperative adhoc networks. For example, the scaling law of the wireless adhoc network is derived in [8, 9, 10, 11] and it is shown that each node can achieve the throughput of the order ^{1}^{1}1 means that there exists a constant such that for sufficiently large , means that , and means that and . when fixed nodes are randomly distributed over a unit area. These results imply that the throughput of each node converges to zero when the number of nodes increases. Nevertheless, it is found in [12] that the pernode throughput can arbitrarily close to constant by hierarchical cooperation. In [13, 14], it’s shown that the sourcedestination throughput can scale as when all the relays amplify and forward the received packet to the destination cooperatively with total transmission power. In [15], the authors have shown that a per link throughput of can be achieved at the expense of potentially large delay when the nodes are mobile. All these works have suggested that there are potential advantage of relay buffering and relay mobility. However, there are also various technical challenges to be addressed before we could better understand the potential benefits.

Low Complexity Relay Protocol Design Exploiting Relay Buffering and Relay Mobility: Although the idea of utilizing the mobility has been studied in [15, 16] in the study of adhoc network throughput analysis (scaling laws), there is not much work that addresses the microscopic details (such as protocol design) of the problem. For example, most of the existing relay protocols have focused entirely on the physical layer performance (information theoretical capacity or DegreesofFreedom (DoF)) and they did not fully exploit the potential of relay buffering. In fact, it is quite challenging to design low complexity relay protocol that could exploit both the relay buffering and relay mobility. Furthermore, the issue is further complicated by the bursty source arrival and randomly coupled queue dynamics in the systems.

Performance Analysis: It is very important to have closed form performance analysis to obtain insights to understand the fundamental tradeoff between throughput gain and delay penalty in cooperative systems. However, it is very challenging to analyze closed form tradeoff between the throughput, stability region and endtoend delay. For instance, most of the existing papers studying delay and throughput scaling laws in adhoc network [15, 16] are focused on the macroscopic aspects of the systems and they have ignored the microscopic details such as the random bursty arrivals and queue dynamics in the systems. When these dynamics are taken into consideration, the problem involves both information theory (to model the physical layer dynamics) and the queueing theory (to model the queue dynamics), which is highly nontrivial.
In this paper, we shall propose an opportunistic buffered decodewaitandforward (OBDWF) protocol to exploit both relay buffering and relay mobility to enhance the system throughput and the endtoend packet delay under bursty arrivals. We consider a pointtopoint communication link assisted by mobile relays. By exploiting the relay buffering and relay mobility in the phase I and phase II of the proposed OBDWF, we shall illustrate that the OBDWF could achieve a better throughput and delay performance compared with existing baseline systems such as the conventional Dynamic DecodeandForward (DDF) [3] and AF protocol [17]. In addition to simulation performance, we also derived closedform asymptotic throughput and delay expressions of the OBDWF protocol. Under random bursty arrivals and queue dynamics, the proposed OBDWF protocol has low complexity with only communication overhead and total transmit power in the relay network. It achieves a throughput , which is a significant gain compared with the best known performance in conventional protocols ( throughput with total transmit power). Furthermore, both the stability region and average delay can achieve orderwise performance gain compared with conventional DDF protocol.
Ii System Model
In this section, we shall describe the system model for the pointtopoint communication system with a halfduplexing mobile relay network. Specifically, there are mobile relay nodes between the source node and the destination node, as shown in Fig. 1. Each of the source node and the relay nodes has an infinite length buffer. To facilitate the relay scheduling, transmission is partitioned into frames. Each frame is further divided into three types of slots defined as follows:

Channel Estimation Slot is used by relays for estimating the channel gains with the source and destination nodes.

Control Slot is used by relays for distributive control signaling of the OBDWF protocol. The details is given in the protocol description.

Transmission Slot is used for data transmission, and it last seconds.
Iia Relay Mobility Model
Following [8, 14], we assume that the relays are distributed on a disk with radius as illustrated in Fig. 1. The source and the destination nodes are fixed at two ends of a diameter, and the disk is divided horizontally into equalarea regions along the sourcedestination diameter. These regions are denoted as region 1, region 2, …, and region , from the source to the destination. As illustrated in Fig. 1, the movement of each relay is modeled as a random walk (Markov chain) over these regions:

At the beginning, each relay is uniformly distributed on the disk. Movements of relays can only occur in discrete frames with time index .

Let denote the region index of the th relay in the th frame, is a Markov chain with the following transition matrix
(1) 
When one relay moves into a region, its actual location in this region is uniformly distributed.
Remark 1 (Interpretation of Parameter )
The region transition probability measures how likely one relay will move into another region in the next frame, and therefore, it is related to the average speed of the relays. ∎
IiB Physical Layer Model
Let and be the small scale fading gain and the distance between the source node and the th relay respectively, and let and be the small scale fading gain and the distance between the th relay and destination node respectively.
Assumption 1 (Assumption on the Channel Model)
We assume that are quasi static in each frame. Furthermore, are i.i.d over frames according to a general distribution and independent between each link. ∎
The relay network shares a common spectrum with bandwidth Hz, and each node transmits at a peak power . Let be the transmitted symbol from the source node, the received signal at the th relay is given by: , where is the path loss exponent, and is the i.i.d noise. The achievable data rate between the source node and the th relay is given by:
(2) 
where is a constant can be used to model both the coded and uncoded systems. Similarly, the achievable data rate between th relay and the destination node is given by
(3) 
All the packets are transmitted at data rate for some constant . The th relay could correctly decode the packets transmitted from the source node if , and the destination node could correctly decode the packets transmitted from the th relay if . For easy discussion, we shall denote a link as a connected link if its achievable data rate is larger than , and otherwise a broken link.
Iii The OBDWF Protocol
In this section, we shall first describe the proposed opportunistic buffered decodewaitandforward (OBDWF) relay protocol for mobile relays.
Protocol 1 (OBDWF Protocol for Mobile Relays)

Each relay measures the current states {connected, broken} of its links with the destination node in the channel estimation slot.

The control slot is divided into minislots. If the buffer in a relay is not empty and the link state to the destination is connected, it will submit a request in a control minislot. Using standard contention mechanism, one active relay is selected to transmit its packet from its buffer to the destination node^{2}^{2}2 The algorithm can be extended to consider spatial combining from multiple relays. However, the performance gain associated with that will be quite limited due to the path loss effects. For instance, there is very low chance of having multiple relays near the source or multiple relays (having the same common packet) near the destination.. The selected relay as well as all the other relays will dequeue the same packet from their buffers.

If none of the relays attempts to compete for access to the destination node in the control slot, the source node will broadcast a new packet to the relays and the destination node using a fixed data rate for some constant . The source node will dequeue the packet from its buffer if there is an ACK from at least one of the relays or the destination node. ∎
Note that the OBDWF protocol has only communication overhead with total transmit power in the relay network, which is the same as the conventional DDF and conventional AF protocol, elaborated in Table I. Unlike conventional DDF protocol where the phase I (source to relay) and the phase II (relay to destination) of the same packet appear as inseparable atomic actions, the proposed OBDWF protocol exploits buffers in the relays to create the flexibility to schedule phase I and phase II of the same packet based on the instantaneous channel state as illustrated in Fig. 2. Coupled with relay mobility, the proposed OBDWF protocol allows the relay to buffer the packet and wait for good opportunity (when the relays is close to the destination) to deliver the packet. As a result, relay mobility allows the system to operate at a higher throughput at the expense of larger delay. We shall quantify such tradeoff in Section IV and V.
Iv Throughput Performance with Infinite Backlog
In this section, we shall first discuss the average system throughput of the proposed OBDWF protocol with infinite backlog at the source buffer. We first define the average throughput below.
Definition 1 (Average EndtoEnd System Throughput)
Let be the number of information bits successfully received by the destination node at the th frame. The average endtoend system throughput between the source node and the destination node is defined as . ∎
Iva Throughput Performance of the OBDWF Protocol
Note that for a fixed number of relay nodes , when the source node increases the data rate , the associate radio coverage and the number of relays who can decode the source packet becomes smaller. On the other hand, for fixed , the number of relays who can decode the source packet increases as increases. We shall quantify such scaling relationship in Lemma 1.
Lemma 1 (Scaling Relationship of Connected Link)
Denote the transmit data rate and for some constant . For sufficiently large , the following statements are true:

I. If , the probability that there are relays having connected links with the source node (or the destination node) tends to .

II. If , the probability that there are relays having connected links with the source node (or the destination node) is . ∎
Please refer to Appendix A.
Using Lemma 1, we obtain the closedform asymptotic average system throughput under infinite backlog at the source buffer.
Theorem 1
(Throughput Performance of the OBDWF Protocol): For sufficiently large and infinite backlog at the source buffer, the maximal average system throughput of the proposed OBDWF protocol under the random walk mobility model in (1) is given by
(4) 
This orderwise throughput is achieved when for some constant . Furthermore, is upperbounded by but infinitely close to . ∎
Please refer to Appendix B.
Remark 2 (Interpretation of Theorem 1)
Since there are infinitely large buffers at the relay nodes and the randomwalk transition probability is positive, the average system throughput is as long as there are always relays having connected links to the source and destination node (which is presented mathematically as by Lemma 1). ∎
IvB Comparison with the Conventional DDF Protocol
Similarly, we shall summarize the closedform asymptotic average system throughput for the conventional DDF protocol ( elaborated in Table I) below.
Lemma 2
(Throughput Performance of Conventional DDF Protocol): For sufficiently large and infinite backlog at the source buffer, the maximal average system throughput of the conventional DDF protocol under the random walk mobility model in (1) is given by:
(5) 
This orderwise throughput is achieved when
(6) 
Please refer to the Appendix C.
Therefore, we have the following corollary on the performance gain of the OBDWF protocol:
Corollary 1
(Comparison of the Average System Throughput): The throughput gain of the OBDWF protocol is
(7) 
Remark 3 (Interpretation of Corollary 1)
Note that when the system mobility is low (), there is an orderwise throughput gain achieved by the OBDWF protocol. ∎
V Stability and Delay Performance with Bursty Arrivals
In this section, we shall focus on the stability region and the delay performance analysis of the proposed OBDWF protocol under bursty packet arrivals. We shall first define the busty source model, followed by the analysis on the stability region and average delay performance.
Va Bursty Source Model
Let represents the number of new packets arriving at the source node at the beginning of the th frame.
Definition 2 (Bursty Source Model)
We assume that the arrival process is i.i.d over the frame index according to a general distribution . Each new packet has a fixed number of bits . The first and second order moments of the arrival process are denoted by and respectively. ∎
Let be the number of packets in the source buffer at the th frame. The dynamics of the source buffer state is given by:
(8) 
where is the number of packets transmitted to the relay network at th frame. Furthermore, let be the number of packets in the the th relay node’s buffer at the th frame. The dynamics of the relay buffer state is given by:
(9) 
where is the number of packets received by the th relay node from the source node at the beginning of the th frame, and is the number of packets dequeued from the th relay node at the th frame.
VB Stability Performance
In this section, we shall derive the stability region of the OBDWF protocol and the conventional DDF protocols. We first define the notion of queue stability [18, 19] below.
Definition 3 (Stability of the Queueing System)
The queueing system is stable, if
(10) 
where is the queue state in the queueing system at the th frame. ∎
Using Definition 3, we have the following Theorem for the OBDWF protocol.
Theorem 2 (Stability Region of the OBDWF Protocol)
For sufficiently large , the system of queues under the proposed OBDWF protocol are stable if and only if , where is given by
(11) 
Please refer to Appendix E.
Similarly, the stability region of the conventional DDF protocol is given by:
Lemma 3 (Stability Region of the Conventional DDF Protocol)
For sufficiently large , the system of queues under conventional DDF protocol are stable if and only if , where is given by
(12) 
Using Lemma 5 in Appendix C, the results in Lemma 3 follows using similar argument as in Appendix E.
The following Corollary summarizes the performance gain of the OBDWF protocol in stability region.
Corollary 2 (Stability Region Comparison)
Let and be the maximum source arrival rate the system can support and maintain queue stability using OBDWF and conventional DDF protocol respectively. For sufficiently large , the gain on the stability region is given by:
(13) 
Remark 4 (Interpretation of the Corollary 2)
Note that when the packet size is large such that , then the OBDWF protocol can obtain an orderwise gain. For example, when and , then . ∎
VC Delay Performance
In this section, we shall compare the average endtoend packet delay performance. The average endtoend packet delay of the relay network is defined below.
Definition 4 (Average EndtoEnd Packet Delay)
Let and be the frame indices of the th packet arrival at the source node and the th packet successfully received at the destination node respectively. The average endtoend packet delay^{3}^{3}3Note that it is implicitly assumed that the system of queues are stable when we discuss average delay because otherwise, the probability measure behind the expectation is not defined. of the relay network is defined as . ∎
The following Theorem summarizes the average delay performance of the proposed OBDWF protocol.
Theorem 3
Please refer to Appendix F.
Remark 5 (Interpretation of Theorem 3)
The first term in the RHS of (14) is the average waiting time (number of frames) that the packets stays in the source buffer. It is affected by the source arrival model, i.e., and . The second term , where is the average time the packet stays in the relay network. This factor depends on both the packet size and the mobility of the network (). Fig. 3 further illustrates the asymptotic relationship between , and . Specifically, the xaxis is , and the yaxis is . Observe that is an increasing function of and . ∎
Similarly, the delay performance of the conventional DDF protocol is summarized in the following Lemma:
Lemma 4
Please refer to the Appendix G.
The following Corollary summarizes the average delay gain of the OBDWF protocol.
Corollary 3 (Average Delay Comparison)
Remark 6 (Interpretation of the Corollary 3)
There are several scenarios that the OBDWF protocol will have significant orderwise gain on the delay performance. For example, when is close to the service rate , i.e., where , we have . On the other hand, even if is not so close to , i.e., , there will still be orderwise gain as long as and . Specifically, if , , where , and , then we can obtain by (16). ∎
Vi Simulation Results and Discussion
Baseline Name  Description 

Baseline 1 (Conventional DDF)  The source node broadcasts a packet from the buffer at the beginning of the frame until at least one relay or destination node returns with an ACK. If the destination node returns with an ACK, the source node start to broadcast a new packet; If the relay node returns with an ACK, the source node stops broadcasting and the relay node forward the packet to the destination node in the next frame. 
Baseline 2 (Conventional AF) 
The source node broadcasts a packet from the buffer at the beginning of a frame. All the relays listen and store the received samples from the source during the listening phase and the relay with the largest metric (, where and ) is selected to amplify and forward to the destination node in the next frame. 
Baseline 3
(AF with Spatial Combining) 
The source node broadcasts a packet from the buffer at the beginning of a frame. All the relays listen and store the received samples from the source during the listening phase and relays with the largest metric (, where and ) are selected to amplify and forward to the destination node in the next frame. 
Baseline 4 (DF with Spatial Combining)  The source node broadcasts a packet from the buffer at the beginning of the frame until at least relay nodes or destination node return with an ACK. If the destination node returns with an ACK, the source node starts to broadcast a new packet; If at least relay nodes return with an ACK, the source node stops broadcasting and all the relay nodes that have decoded the packet from the source node will forward the packet to the destination node in the next frame. 
In this section, we shall compare the performance of the proposed OBDWF protocol with various baseline schemes. Baseline 1 refers to the conventional DDF protocol[3], baseline 2 refers to the conventional AF protocol[17]. Baseline 3 and 4 are extensions of Baseline 1 and 2 respectively with spatial combining from multiple relays, which are elaborated in Table I We consider a system with the source node at and the destination node at . The relays are randomly distributed between the source node and the destination node, as shown in Fig. 1. The movement of each relay is given by the random walk mobility model in (1), where the number of relay mobility regions is . The small scale fading gain follows complex Guassian with unit variance. The pass loss exponent , and the transmit SNR dB. For bursty arrivals, we assume and . This corresponds to an arrival rate . The packet size , the frame duration ms and the bandwidth MHz. Using Lemma 2, the physical data rate at the source node is set to be , which is the optimal choice for conventional DDF.
Fig. 4 illustrates the average endtoend system throughput versus the number of relays at different mobility . Observe that the proposed OBDWF protocol has significant gain compared with the baselines. Furthermore, the performance of the OBDWF protocol is insensitive to the mobility of the network . Fig. 5 illustrates the maximal stable arrival rate versus the number of relays at different network mobility under the bursty source model. Similar significant gains over the baselines can be observed. Moreover, it can be observed in these two figures that the simulation results match with the theoretical results derived in Section IV and V.
Fig. 6 illustrates the average endtoend packet delay versus the number of relays at different mobility with finite buffer length of 25 packets for all the nodes. Note that, the delay performance is an increasing function of for all protocols and there is also a significant gain of the proposed OBDWF protocol.
Fig. 7 illustrates the average endtoend packet delay versus the average arrival rate with infinite buffer length for all the nodes. Note that the baseline 2 and 3 are not stable under the operating regime considered. The delay performance of baseline 4 quickly blows up at , which is the maximal stability input rate for this baseline. On the other hand, the delay performance of the proposed OBDWF protocol significantly outperforms all the baseline over the entire range of traffic loading. Fig. 8 illustrates the average endtoend system throughput versus the number of relays under the Random Waypoint Model, which is also widely used in [20, 21, 22]. Similar performance gains can be observed.
Vii Conclusions
In this paper, we propose an opportunistic buffered decodewaitandforward (OBDWF) protocol for a pointtopoint communication system with mobile relays. Unlike conventional DDF protocol, the proposed OBDWF protocol exploits both the relay buffering and relay mobility in the systems. We derive closedform expressions on the asymptotic system throughput under infinite backlog as well as the average endtoend delay under a general bursty arrival model, Based on the analysis, we found that there exists a throughput delay tradeoff in the buffered relay network. The system can achieve a higher throughput using the proposed OBDWF protocol at the expense of extra delay. The system mobility affects the tradeoff as below:

Effect on the Throughput/Stability Region Performance: According to Theorem 1, the maximal average system throughput of the proposed OBDWF protocol is not influenced by the relay mobility.

Effect on the Delay Performance: If the movement of the mobility is fast (large region transition probability ), the chance one relay with source’s packet gets close to the destination is high, leading to small delay in the relay network, and vice versa. This can be observed from Theorem 3.
Finally, theoretical and numerical results demonstrate the significant performance gains of the proposed OBDWF protocol against various baseline references.
Appendix A: The Proof of Lemma 1
Without loss of generality, we only study the number of connected links with the source. The connection with the destination can follow the same approach.
For a given small scale fading realization , the coverage radius of the source within which the relays have connected links to the source is given by . Therefore, the area of the source’s coverage can be approximated as , and the probability one relay falls into the source’s coverage is . Taking consideration of the random realization of , the probability that one relay falls into the source’s coverage is given by
(17) 
where is the pdf of the small scale fading gain .
Suppose there are relays having connected links with the source, therefore, . can be treated as one binomial random variable generated from Bernoulli trail where it has value 1 with probability . In other words, .
When , i.e., , by law of large number, we have
(18) 
where . Therefore, the tends to 1 when . This finishes the proof of statement I.
When , since the probability that one relay has the connected link with source node is , the probability that at least one relay can receive the transmitted packet is given by . This finishes the proof of statement II.
Appendix B: The Proof of Theorem 1
When , it is obvious that , and hence is not the optimal choice. When and , the coverage radius of the source and destination nodes can be made sufficiently small given a fixed transmit power . Only the relays close to the source node (in region 1) can decode the packet transmitted from the source node and only the relays close to the destination (in region ) can forward the packet to the destination node. Furthermore, there are (with probability ) some relays in region 1 can decode the source’s packets by Lemma 1. Similarly, there are (with probability 1) some relays in region can forward packets to the destination, and hence the average system throughput is given by: , where denotes the equality with high probability (with probability as in [9]).
When increasing the such that , The source should continue to transmit slots before relays can decode the packet by Lemma 1, and hence the average system throughput is . Since the log function is in the numerator, increasing the order of will reduce the order of . As a result, the optimal is . should be infinitely close to from below to make that there are always relays in the connected region of the source/destination node (). The corresponding maximal average system throughput is infinitely close to , when .
Appendix C: The Proof of Lemma 2
Obviously, under the conventional DDF protocol, the average system throughput for a given data rate is given by
(19) 
where is the average service time (number of frames) for a packet, i.e., the average time spent for the source node to transmit a packet to the destination node. can be divided into two part, i.e., , where is the average time spent for the source node to transmit a packet to the relay network, and is the average time spent for the relay network to forward the said packet to the destination. Specifically, we have following Lemma for the average service time .
Lemma 5
(Average Service Time of Conventional DDF Protocol): The average service time for a packet under conventional DDF protocol is given by: , where , and
(20) 
please refer to Appendix D.
Note that is an increase function of by Lemma 5. Since and due to the function in the numerator, we have , where is the optimal value that maximizes , i.e., . According to the delay expression in (20), when , we have . As a result, if , the optimal value is , leading to . Otherwise, the optimal value is , leading to .
Appendix D: The proof of Lemma 5
We provide the proof in two scenarios, and , respectively.
A. Scenario
When the source broadcasts a packet, there are relays can decode this packet with probability 1 by Lemma 1, and hence . The movement of relays with the said packet can be divided in to two stage: (1) unbalanced, which means the order of relays with the said packet in each region is not the same; (2) balanced, which means the order of relays with the said packet in each region is the same. Obviously, after frames, the system is balanced, i.e., there are relays with the said packet in each of the regions.
When the connected link mainly happens in the unbalanced stage, after frames, the number of relays with the said packet in the region is . The chance that these relays have connected link with the destination is . It can be obtained following the same approach as Lemma 1. Therefore, after frames, there is at least one relay with the said packet having connected link to the destination node. Increasing the order of , the number of relays with the said packet in the region increases, but decrease. The actual delay should satisfy: , which leads to
(21) 
As a result, the requirement of is satisfied when .
When the connected link mainly happens in the balanced stage, thus , i.e., . This requirement is satisfied when . In this case, the average delay is mainly due to the waiting time after the relays’ movement is balanced, given by .
B. Scenario
When , the source should continue to transmit slots before relay can decode the packet by Lemma 1, i.e., . Therefore, there are relays with the packet in the relay network rather than as the scenario. Following the similar approach as in the above subsection by replacing with , it can be shown that the average delay is given by Lemma 5.
Appendix E: The Proof of the Theorem 2
In this proof, we shall first study the stability region of the source buffer , and then prove that under the same stability region, the relay buffers are stable too.
From [18, 19], the queueing system is stable, if and only if the average arrival rate is smaller than the service rate , i.e., , where is the average service time to server a queueing packet out of the queueing buffer. For the source buffer , the average service time is the average number of frames for the source node to transmit a packet to the relay network, denoted as . From Lemma 1, we can discuss in the following two scenarios.
If , when the source broadcasts a packet, there are relays can decode this packet with probability 1. Therefore, .
If , the source should continue to transmit slots before relays can decode the packet. Therefore, .
Note that the queues are statistically identical, and they are either all stable or all unstable. Consider a fictitious queueing system with with the average arrival rate and service rate . Obviously, and hence, all the relay queues are dominated by the fictitious queue . The average arrival rate of the fictitious queue is , and the average number of frames for the fictitious system to deliver a packet to the destination . Therefore, if , then . In other words, the queues of the relay nodes are also stable if the queue is stable.
Appendix F: The Proof of the Theorem 3
Note that , where is the average queueing delay in the source buffer before transmitted to the relay network, and is the average waiting time in the relay network. Following the same approach as the proof of Lemma 5, it is easy to verify that given in (20). The remaining task is to find , which is discussed below.
Let be the number of frames (namely service time) to transmit a packet into the relay network, and denote and respectively. Furthermore, let denotes the number of packets in the source buffer immediately after transmitting the th packet to the relay network. is the number of packets arriving during the service time of the th packet, is the number of packets arriving in one frame given that , i.e., , and is the number of packets arriving during the service time of the th packet minus one frame (The number of arrivals during the last frames if the service time of the th packet is frames). Then will form a Markov chain with the following transitions.
(22) 
Specifically, the probability generating function (p.g.f) of is given by
(23) 
where is the probability that no packets arrives. is the p.g.f of the bursty arrival . The p.g.f of the number arriving within a service time is
(24) 
Similarly, the p.g.f of packet arrival is given by
(25) 
where is the p.g.f of the service time. From (22)(25), the p.g.f of the number in the system immediately after the service completion instants is[23]
(26) 
It is shown in [24] that the p.g.f for the packets in the queueing system immediately after an arbitrary frame is given by
(27) 
Therefore, by Little’s law[23], the average time a packet spends in the buffer will be
(28) 
Note that, to make the system stable, the arrival rate should be smaller than the service rate , i.e., . It agrees with the stability condition given in [18, 19].
If , when the source broadcasts a packet, there are relays can decode this packet with probability 1 by Lemma 1. Therefore, and .
If , the first and second order moments of the service time for a packet to enter the relay network is and respectively from Lemma 1.
Let , and note that . The average endtoend packet delay is given by
(29) 
Appendix G: The Proof of The Lemma 4
From Appendix C, the average service time to server a packet to the destination node from the source node is . In the followings, we shall prove that the seconder order moment of service time is given by . Specifically,
(30) 
where (in terms of ) is contributed by the possibilities that the service time , and is contributed by the possibilities that the service time . Clearly, is neglectable as , i.e, .
We first consider the scenario where and . In this case, the average service time is given by , and after frames, there are