Analyzing DISH for Multi-Channel MAC Protocols
in Wireless Networks
For long, node cooperation has been exploited as a data relaying mechanism. However, the wireless channel allows for much richer interaction between nodes. One such scenario is in a multi-channel environment, where transmitter-receiver pairs may make incorrect decisions (e.g., in selecting channels) but idle neighbors could help by sharing information to prevent undesirable consequences (e.g., data collisions). This represents a Distributed Information SHaring (DISH) mechanism for cooperation and suggests new ways of designing cooperative protocols. However, what is lacking is a theoretical understanding of this new notion of cooperation. In this paper, we view cooperation as a network resource and evaluate the availability of cooperation via a metric, , the probability of obtaining cooperation. First, we analytically evaluate in the context of multi-channel multi-hop wireless networks. Second, we verify our analysis via simulations and the results show that our analysis accurately characterizes the behavior of as a function of underlying network parameters. This step also yields important insights into DISH with respect to network dynamics. Third, we investigate the correlation between and network performance in terms of collision rate, packet delay, and throughput. The results indicate a near-linear relationship, which may significantly simplify performance analysis for cooperative networks and suggests that be used as an appropriate performance indicator itself. Throughout this work, we utilize, as appropriate, three different DISH contexts — model-based DISH, ideal DISH, and real DISH — to explore .
MobiHoc’08, May 26–30, 2008, Hong Kong SAR, China. \CopyrightYear2008 \crdata978-1-60558-083-9/08/05
C.4Performance Of Systems[Performance attributes] \categoryC.2.1Computer-Communication NetworksNetwork Architecture and Design[Wireless communication]
Cooperative diversity is not a new concept in wireless communications. Key ideas and results in cooperative communications can be traced back to the 1970s to van der Meulen  and Cover & El Gamal , whose works have spurred numerous studies on this topic from an information-theoretic perspective (e.g., [3, 4, 5, 6]) or a protocol-design perspective (e.g., [7, 8, 9, 10]). To date, cooperation has been intensively studied in various contexts. However, to the best of our knowledge, it has always been used as a data relaying mechanism where intermediate nodes help relay packets from a transmitter to a receiver. In fact, the wireless channel allows for much richer interaction among nodes. Consider a scenario where orthogonal frequency channels are available. A node wishes to select a conflict-free channel to transmit data, but may often fail to achieve this due to lack of sufficient information about channel usage. In this case, other nodes in the neighborhood may possess the information in need and thus could help by sharing this information.
This shows that cooperation can be used as a Distributed Information SHaring (DISH) mechanism, in addition to mere data relaying. In [11, 12], we proposed a multi-channel MAC protocol based on this idea, where performance enhancement was demonstrated via simulations. In this paper, we develop a theoretical treatment of this new notion of cooperation, in particular, the availability of cooperation. The benefit of DISH is that it can remove the need of using multiple transceivers [13, 14, 15, 16, 17] and time synchronization [18, so04, 19, 20, 21, 22] in designing multi-channel MAC protocols. This motivates us to understand DISH from a theoretical perspective.
In this paper, we define a metric which characterizes the availability of cooperation as the probability of obtaining cooperation (see Def. 3 for a more precise definition). We analytically evaluate this metric in multi-channel multi-hop wireless networks with randomly distributed nodes, and verify the analysis via simulations. We also carry out a detailed investigation of with three different contexts of DISH: model-based DISH, ideal DISH, and real DISH, in order to obtain meaningful findings.
1.1 Summary of Contributions
Our aim in this paper is to understand DISH and the availability of cooperation () from an analytical perspective. More specifically, we provide an analysis which accurately characterizes the availability of cooperation as a function of the underlying network parameters. This analysis reveals what underlying factors and how these factors affect cooperation, and can provide guidelines to provisioning the network to increase performance.
Throughput analysis for multi-hop networks is difficult (and still an open problem in general), and it gets even more complicated in a multi-channel context with DISH. Our approach in this paper is to first look at and then correlate it with network performance. The results indicate that there is a simple relationship between and several performance metrics.
The specific findings of this study are:
The availability of cooperation is high () in typical cases, which suggests that DISH is feasible to use in multi-channel MAC protocols.
The performance degradation due to an increase in node density can be alleviated due to the simultaneously increased availability of cooperation.
The metric will increase for larger packet sizes for a given bit arrival rate, but will decrease for larger packet sizes for a given packet arrival rate.
Node density and traffic load have opposite effects on but node density is the dominating factor. This implies an improved scalability for DISH networks as increases with node density.
is strongly correlated to network performance and has a near-linear dependence with metrics such as throughput and delay. This may significantly simplify performance analysis for cooperative networks, and suggests that be used as an appropriate performance indicator itself.
2 Related Work
There are three other studies most related to this work. One is CAM-MAC  which uses cooperation in the new way that we call DISH in this paper. It is a cooperative multi-channel MAC protocol requiring only a single transceiver and no synchronization. In this protocol, there is a control channel for transmitter-receiver pairs to perform handshakes in order to reserve data channels, while nodes in the neighborhood may send cooperative messages to invalidate the handshake if the selected channel or receiver is busy. As it is the only work we are aware of that explicitly uses DISH, our system model will use a protocol framework by abstracting the work in .
The second work, CoopMAC , is also a cooperative MAC protocol which exploits data relaying as many other protocols do, such as [7, 8, 9]. A protocol analysis is provided in the paper and it requires computing the probability that a relay node is available. This probability is different from in that it is determined by the static locations of nodes, i.e., whether a node exists in a specific region. The probability is computed via geometric analysis (nodes are assumed to be uniformly distributed). On the other hand, is determined not only by static node locations, but also by dynamic node behavior, e.g, a node must have acquired the specific information at a specific moment. The second main difference between the protocol analysis of CoopMAC and our work is that the problem context of CoopMAC is a wireless LAN with a single channel, whereas this paper assumes a multi-hop network with multiple channels.
The last work is by Han et al.  who considered a multi-channel MAC protocol adopting ALOHA on the control channel to reserve data channels. A queueing-theoretic approach was taken to calculate throughput for the protocol. However, there are some noteworthy limitations. First, only a single-hop scenario was considered. Second, each node was assumed to be able to communicate on the control channel and a data channel simultaneously. This essentially requires two transceivers per node, and consequently leads to collision-free data channels, which oversimplifies the problem. Third, a unique virtual queue was assumed to store the packets arriving at all nodes for the ease of centralized transmission scheduling, and the precise status of the queue was assumed to be known to the entire network. This assumption is impractical and eventually results in a throughput upper bound. Fourth, the access to the control channel adopts the ALOHA algorithm, rather than the more practical and sophisticated mechanism of CSMA/CA.
3 System Model
We consider a static and connected ad hoc network in which each node is equipped with a single half-duplex transceiver that can dynamically switch between a set of orthogonal frequency channels but can only use one at a time. One channel is designated as a control channel and the others are designated as data channels. Nodes are placed in a plane area according to a two-dimensional Poisson point process.
We consider a class of multi-channel MAC protocols with their common framework described below. A transmitter-receiver pair uses a McRTS/McCTS handshake on the control channel to set up communication (like 802.11 RTS/CTS) for their subsequent DATA/ACK handshake on a data channel. To elaborate, a transmitter sends a McRTS on the control channel using CSMA/CA, i.e., it sends McRTS after sensing the control channel to be idle for a random period (addressed below) of time. The intended receiver, after successfully receiving McRTS, will send a McCTS and then switch to a data channel (the McRTS informs the receiver of the data channel). After successfully receiving the McCTS, the transmitter will also switch to its selected data channel, and otherwise it will backoff on the control channel for a random period (addressed below) of time. Hence it is possible that only the receiver switches to the data channel. After switching to a data channel, the transmitter will send a DATA and the receiver will respond with a ACK upon successful reception. Then both of them switch back to the control channel.
In the above we have mentioned two random periods of time. Our model does not specify them but assumes that these periods are designed such that idle intervals on the control channel are well randomized. Specifically, when a node is on the control channel, it sends control messages (an aggregated stream of McRTS and McCTS) according to a Poisson process.
Note that we use McRTS, McCTS, DATA and ACK to refer to different packets (frames) without assuming specific frame formats. Since, logically, they must make a protocol functional, we assume that McRTS carries channel usage information (e.g., “who will use which channel for how long”) and, for simplicity, McCTS is the same as McRTS.
We assume that, after switching to a data channel, a node will stay on that channel for a period of , where is the duration of a successful data channel handshake. We ignore channel switching delay as it will not fundamentally change our results if it is negligible compared to (the delay is  while is more than for a 1.5KB data packet on a 2Mb/s channel). We also ignore SIFS and propagation delay for the same reason, provided that they are smaller than the transmission time of a control message.
We assume a uniform traffic pattern — all nodes have the equal data packet arrival rate, and for each data packet to send, a node chooses a receiver equally likely among its neighbors. We also assume a stable network — all data packets can be delivered to destinations within finite delay. In addition, packet reception fails if and only if packets collide with each other (i.e., no capture effect), transmission and interference ranges are equal, and the probability that neighboring nodes simultaneously start sending control messages is zero (no time synchronization).
We do not assume a specific channel selection strategy; how a node selects data channels will affect how often conflicting channels are selected, but will not affect . This is because, intuitively, we only care about the availability of cooperation () when a multi-channel coordination problem (a precise definition is given in Def. 1), which includes channel conflicting problem, has been created.
We do not assume a concrete DISH mechanism, i.e., nodes do not physically react upon a multi-channel coordination problem, because analyzing the availability of cooperation does not require the use of this resource. In fact, assuming one of the (numerous possible) DISH mechanisms will lose generality. Nevertheless, we will show in Section 5 that, when an ideal or a real DISH mechanism is used, the results do not fundamentally change. This could be an overall effect from contradicting factors which will be explained therein.
The following lists all parameters that are assumed known:
: node density. In a multi-hop network, it is the average number of nodes per where is the transmission range. In a single-hop network, it is the total number of nodes.
: the average data packet arrival rate at each node, including retransmissions.
: the duration of a data channel handshake.
: the transmission time of a control message. .
4.1 Problem Formulation and Analysis Outline
We first formally define , which depends on two concepts called the MCC problem and the cooperative node.
Definition 1 (MCC Problem)
A multi-channel coordination (MCC) problem is either a channel conflict problem or a deaf terminal problem. A channel conflict problem is created when a node, say , selects a channel to use (transmit or receive packets) but the channel is already in use by a neighboring node, say . A deaf terminal problem is created when a node, say , initiates communication to another node, say , that is however on a different channel. In either case, we say that an MCC problem is created by and .
In a protocol that transmits DATA without requiring ACK, a channel conflict problem does not necessarily indicate an impending data collision. We do not consider such a protocol.
Definition 2 (Cooperative Node)
A node that identifies an MCC problem created by two other nodes, say and , is called a cooperative node with respect to and .
See Fig. 1 for a visualization based on our system model.
Definition 3 ()
is the probability for two arbitrary nodes that create an MCC problem to obtain cooperation, i.e., there is at least one cooperative node with respect to these two nodes.
Note that, if there are multiple cooperative nodes and a DISH mechanism allows those nodes to send cooperative messages concurrently, then a collision results. However, this collision still indicates an MCC problem and thus cooperation is still deemed obtained. CAM-MAC also implements this.
We distinguish the receiving of control messages. A transmitter receiving McCTS from its intended receiver is referred to as intentional receiving, and the other cases of receiving are referred to as overhearing, i.e., any node receiving McRTS (hence an intended receiver may also be a cooperative node) or any node other than the intended transmitter receiving McCTS.
Our notation is listed in Table 1. Overall, we will determine by following the order of .
|the probability that at least one cooperative node with respect to and exists|
|the probability that node is a cooperative node with respect to and|
|the probability that a node is on the control channel at an arbitrary point in time|
|the probability that a control channel handshake (initiated by a McRTS) is successful|
|the probability that an arbitrary node successfully overhears a control message|
|node is on the control channel at time|
|node successfully overhears node ’s control message, given that sends the message|
|node is silent (not transmitting) on the control channel during interval|
|node does not introduce interference to the control channel during interval , i.e., it is on a data channel or is silent on the control channel.|
|node , which is on a data channel at , switches to the control channel in|
|is the set of node ’s neighbors, , (’s but not ’s neighbors)|
|the time when node starts to send a control message|
|the average rates of a node sending control messages, McRTS, and McCTS, respectively, when it is on the control channel. Clearly, .|
Consider first. Fig. 1 illustrates that node is cooperative if and only if it successfully overhears ’s and ’s control messages successively. Hence ,
Consider . For to successfully overhear ’s control message which is being sent during interval , must be silent on the control channel and not be interfered, i.e.,
4.2 Solving Equation (4.1)
If node is on a data channel at , then the probability that does not introduce interference to the control channel during , where , is given by
By the total probability theorem,
Let be the time when node switches to the control channel (see Fig. 2). It is uniformly distributed in because the time when started its data channel handshake is unknown, and hence
Since control channel traffic is Poisson with rate ,
where is uniformly distributed in by the same argument leading to (3). Hence
and then by substitution the proposition is proven. ∎
If node is overhearing a control message from node during , then the probability that a node does not interfere with is given by
If node (transmitter) is intentionally receiving McCTS from node (receiver) during , then the probability that a node does not interfere with is given by
4.3 Solving and
For , consider a sufficiently long period . On the one hand, the number of arrival data packets at each node is . On the other hand, each node spends total time of on data channels, a factor of which is used for sending arrival data packets. Since the network is stable (incoming traffic is equal to outgoing traffic), we establish a balanced equation:
To determine , noticing that a node switches to data channels either as a transmitter (with an average rate of ) or as a receiver (with an average rate of ), we have . Substituting this into the above yields
For (together with ), we need two lemmas.
For a Poisson random variable with mean value , and ,
For three random distributed nodes , and ,
Let be the intersection area of two circles with a distance of between their centers, and , where is the circles’ radius. It can be derived from  that
Let be the complementary area of , i.e., , and let and be the areas where and are located, respectively.
(a) See Fig. 3(a). Letting where , and be its probability density function (pdf), we have , which gives . Thus
and hence .
(b) Let where , and be its pdf. To solve , we consider instead (see Fig. 3(b)):
To determine , let where , and be its pdf. It is determined by
Substituting this and (by case (a)) into (4.3) solves , which we denote by . Then we have
(c) Proven by noticing is complementary to the area corresponding to case (a). ∎
These lemmas enable us to prove (see appendix) that
Now we solve (together with ). From the perspective of a transmitter, the average number of successful control channel handshakes that it initiates per second is . Since each successful control channel handshake leads to transmitting one data packet, we have .
From the perspective of a receiver, it sends a McCTS when it successfully receives (overhears) a McRTS addressed to it, and hence . Then combining these with yields
where and are given in (4.3).
4.4 Solving Equation (4.1) and Target Metric
Based on the proof of (4), it can be derived that
Note that , because is not an arbitrary time for due to the effect form . The reason is that implies , and thus for to happen, must stay continuously on the control channel during (otherwise, a switching will lead to staying on the data channel for , but since ’s data communication is still ongoing at , and hence can never happen).
It can be proven (see appendix for the proof) that
Let be the average of over all , i.e., is the probability that an arbitrary node in is cooperative with respect to and , Using Lemma 2-(b),
By the definition of in Table 1,
where the events corresponding to , i.e., nodes not being cooperative with respect to and , are regarded as independent of each other, as an approximation.
Thus is determined by averaging over all pairs that are possible to create MCC problems. It can be proven that these pairs are neighboring pairs satisfying ( denoting the degree of a node )
, , but not , or
, but and are not on the same three-cycle (triangle).
This condition is satisfied by all neighboring pairs in a connected random network, because the connectivity requires a sufficiently high node degree ( where is the total number of nodes) which is much larger than 2. Therefore, taking expectation of (13) over all neighboring pairs using Lemma 1 and Lemma 2-(c),
This completes the analysis.
4.5 Special Case: Single-Hop Networks
where is the number of all possible cooperative nodes with respect to and , leading to . So, as the average of ,
where is given below, by solving the equations in Section 4.3,
and is given below, by reducing (10) with ,
5 Investigating with DISH
We verify the analysis in both single-hop and multi-hop networks and identify key findings therein. We also investigate the correlation between and network performance.
5.1 Protocol Design and Simulation Setup
5.1.1 Model-Based DISH
This is a multi-channel MAC protocol based on the protocol framework described in Section 3. Key part of its pseudo-code is listed below, where is the control channel status (FREE/BUSY) detected by the node running the protocol, is the node’s state (IDLE/TX/RX, etc.), is the node’s current queue length, and they are initialized as FREE, IDLE and 0, respectively. The frame format of McRTS and McCTS is shown in Fig. 5, where we can see that they carry channel usage information. A node that overhears McRTS or McCTS will cache the information in a channel usage table shown in Fig. 5, where Until is converted from Duration by adding the node’s own clock.
As is based on the system model, this protocol does not use a concrete DISH mechanism, i.e., cooperation is treated as a resource while not actually utilized.
5.1.2 Ideal DISH
This protocol is by adding an ideal cooperating mechanism to the model-based DISH. Each time when an MCC problem is created by nodes and and if at least one cooperative node is available, the node that is on the control channel, i.e., node , will be informed without any message physically sent, and then back off to avoid the MCC problem.
5.1.3 Real DISH
In this protocol, cooperative nodes will physically send cooperative messages to inform a transmitter or receiver of the MCC problem so that it will backoff. We design this real DISH by adapting CAM-MAC. The only change that we made is that, since in CAM-MAC a transmitter will send a PRA and a CFA, and a receiver will send a PRB and a CFB, during the control channel handshake, we change these control packet sizes such that , where gives the size of a packet.
5.1.4 Simulation Setup
There are six channels of data rate 1Mb/s each. Data packets arrive at each node as a Poisson process. The uniform traffic pattern as in the model is used. Traffic load (pkt/s), node density (1/), and packet size (byte) will vary in simulations. In multi-hop networks, the network area is 1500m1500m and the transmission range is 250m. Each simulation is terminated when a total of 100,000 data packets are sent over the network, and each set of results is averaged over 15 randomly generated networks.
5.2 Investigation with Model-Based DISH
The obtained via analysis and simulations are compared in Fig. 6. We see a close match between them, with a deviation of less than 5% in almost all single-hop scenarios, and less than 10% in almost all multi-hop scenarios. Particularly, the availability of cooperation is observed to be at a high level ( in most cases), which suggests that a large percentage of MCC problems would be avoided by exploiting DISH, and DISH is feasible to use in multi-channel MAC protocols. (Finding 1)
Specifically, Fig. 6(a) and Fig. 6(c) consistently show that, in both single-hop and multi-hop networks, monotonically decreases as increases. The reasons are two folds. First, as traffic grows, each node spends more time on data channels for data transmission and reception, which reduces and hence the chance of overhearing control messages (), resulting in lower . Second, as the control channel is the rendezvous to set up all communications, larger traffic intensifies the contention and introduces more interference to the control channel, which is hostile to messages overhearing and thus also reduces .
Fig. 6(b) and Fig. 6(d) show that monotonically increases as increases, and is concave. The increase of is because MCC problems are more likely to have cooperative nodes under a larger node population, while the deceleration of the increase is because more nodes also generate more interference to the control channel.
An important message conveyed by this observation is that, although a larger node density creates more MCC problems (e.g., more channel conflicts as data channels are more likely to be busy), it also boosts the availability of cooperation which avoids more MCC problems. This implies that the performance degradation can be mitigated. (Finding 2)
In both single-hop and multi-hop networks, a larger packet size corresponds to a lower . However, note that this is observed under the same packet arrival rate (pkt/s), which means actually a larger bit arrival rate for a larger , and can be explained by the previous scenarios of versus . Now if we consider the same bit arrival rate, by examining the two analysis curves in Fig. 6(c) where we compare with respect to the same product, e.g., () versus (), and () versus (), then we will see that a larger corresponds to a higher , which is contrary to the observation under the same packet arrival rate. The explanation is that, for a given bit arrival rate, increasing reduces the number of packets and hence fewer control channel handshakes are required, thereby alleviating control channel interference. (Finding 3)
The above results indicate that node density and traffic load affect the availability of cooperation in opposite ways. This section aims to find which one dominates over the other. In Fig. 7, we plot the relationship of versus and , given and based on the analytical result for single-hop networks. We multiplicatively increase and with the same factor (two), and find that, when increasing () from (5,5) to (10,10), keeps increasing from 0.865 to 0.999, and when increasing (,n) from (10,5) to (20,10), keeps increasing from 0.724 to 0.943.
This investigation shows that is the dominating factor over that determines the variation of . This implies that DISH networks should have better scalability than non-DISH networks, since increases when both traffic load and node density scale up. (Finding 4)
5.3 Investigation with Ideal DISH
The results of comparison are shown in Fig. 8, where with ideal DISH well matches of analysis. This confirms Findings 1-3, and we speculate the reasons to be as follows. With ideal DISH, a transmitter may be informed of a deaf terminal problem and thus will backoff for a fairly long time, which leads to fewer McRTS being sent. On the other hand, a node may also be informed of a channel conflict problem and thus will re-select channel and retry shortly, which leads to more control messages being sent. Empirically, the latter case has more significant effect, which means that, overall, there will be an increase of control messages being sent. This boosts interference and thus would reduce . However, nodes will also stay longer on the control channel due to less use of conflicting data channels, which would elevate . Consequently, does not change noticeably.
We investigate how correlates to network performance — specifically, data channel collision rate , packet delay , and aggregate throughput . We consider both stable networks and saturated networks under multi-hop scenarios.
In stable networks, we measure () and when without and with cooperation (ideal DISH), respectively. Then we compute and to compare to with ideal DISH. The first set of results, by varying traffic load , is shown in Fig. 8(a). We observe that the two ascending and convex curves of and approximately reflect the descending and concave curve of , which hints at a linear or near-linear relationship between and these two performance ratios. That is, , where and are two constants. The second set of results, by varying node density , is shown in Fig. 8(b). On the one hand, and decreases as increases, which is contrary to Fig. 8(a). This confirms our earlier observations: is amicable whereas is hostile to (the smaller and , the better performance cooperation offers). On the other hand, the correlation between and the performance ratios is found again: as increases on a concave curve, it is reflected by and which decrease on two convex curves.
In saturated networks, we vary node density and measure aggregate throughput without and with cooperation (ideal DISH), as and , respectively. Then we compute (note that this definition is inverse to and , such that ) to compare to with ideal DISH. The results are summarized in Fig. 9. We see that (i) grows with , which conforms to Finding 2, and particularly, (ii) the declining and convex curve of reflects the rising and concave curve of , which is consistent with the observation in stable networks. In addition, here is lower than the in stable networks. This is explained by our earlier result that higher traffic load suppresses .
In summary, the experiments in stable networks and saturated networks both demonstrate a strong correlation (linear or near-linear mapping) between and network performance ratio in terms of typical performance metrics. This may significantly simplify performance analysis for cooperative networks via bridging the nonlinear gap between network parameters and , and also suggests that be used as an appropriate performance indicator itself. (Finding 5)
The explanation to this linear or near-linear relationship should involve intricate network dynamics. We speculate that the rationale might be that (i) MCC problems are an essential performance bottleneck to multi-channel MAC performance, and (ii) is equivalent to the ratio of MCC problems that can be avoided by DISH. In any case, we reckon that this observation may spur further studies and lead to more thought-provoking results.
5.4 Investigation with Real DISH
Due to space constraints, we discuss the results without presenting figures because the results were found to be similar to those with ideal DISH.
The deviation between simulations and analysis, which maximally reached 15%, was found to be slightly larger than that with model-based or ideal DISH. However, the overall trends still match, and Findings 1-4 are confirmed. For explanation, we speculate that, in real DISH, there are much more control messages (including cooperative messages) being sent, which result in more control channel interference and thus would diminish . But on the other hand, nodes stay longer on the control channel due to the same reason as for the case of ideal DISH (i.e., less use of conflicting data channels), which would increase . As a result, with real DISH does not deviate significantly from the analysis.
The near-linear relationship between and () was observed. This confirms Finding 5. As a possible explanation, the remark following Finding 5 in Section 5.3 applies. Although the absolute values of these quantities were found to differ from those with ideal DISH (ranging between 3-18%), this is not the primary concern.
Distributed Information SHaring (DISH) represents a mechanism different from data relaying to exploit cooperative diversity. This paper gives the first theoretical treatment of this new notion of cooperation, by addressing the availability of cooperation via a metric . Instead of directly analyzing throughput which is an open problem in general and rendered much more complicated when considered together with multiple channels and DISH, our approach is to analyze first and then correlate with performance metrics including throughput. We conduct analysis in a multi-hop multi-channel wireless network and, to verify its validity and study its implications, investigate with three different contexts of DISH: model-based DISH, ideal DISH, and real DISH. The investigation validates that our analysis accurately captures the interaction among network parameters, which allows us to draw important findings of with respect to network dynamics. It also reveals a near-linear relationship between and network performance, which may greatly aid in performance analysis for cooperative networks and also suggests to be a proper performance indicator itself.
This work is the first that explicitly presents DISH, together with a detailed study offering meaningful insights into understanding cooperation. Based on our findings, we conclude that is a useful metric capable of characterizing the performance of DISH networks. We contend that DISH is useful and practical enough to be a part of future cooperative communication networks.
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Appendix A Proofs and Derivations
a.1 Proof of Prop. 2
In the case of , no matter is on the control channel at , or is on a data channel at but switches to the control channel before , it will sense a busy control channel (due to CSMA) and thus keep silent.
a.2 Proof of Prop. 3
The case of follows the same line as the proof for Prop. 2. In the case of , the only difference from Prop. 2 is that now we are implicitly given the fact that was transmitting McRTS during . This excludes ’s any neighbor interfering in . Therefore ’s vulnerable period is instead of as compared to Prop. 2. So
Note that we condition on instead of , because is not an arbitrary time due to ’s McRTS transmission during , which leads to .
First, . This is because, as which is easy to show, will successfully overhear ’s McRTS, and hence will keep silent in the next period of to avoid interfering with receiving McCTS.
Next consider where is on a data channel at . If switches to the control channel (i) before , it will be suppressed by ’s McRTS transmission until , and thus the vulnerable period of receiving McCTS is , (ii) within , this has been solved by Prop. 1, or (iii) after , the probability to solve is obviously 1. Therefore,