Achievable Performance Improvements. Provided by Route Diversity in Multihop. Wireless Networks

Transcription

1 Achievable Performance Improvements 1 Provided by Route Diversity in Multihop Wireless Networks Stephan Bohacek University of Delaware Department of Electrical and Computer Engineering Newark, DE bohacek@udel.edu Abstract When considering many paths in a multihop wireless setting, the variability of channels results in some paths providing better performance than other paths, i.e., path diversity. While it is well known that some paths are better than others, a significant number of routing protocols do not focus on selecting the optimal path. However, cooperative diversity, an area of recent interest, provides techniques to exploit path and channel diversity. Here we examine the potential performance improvement when optimal paths are used. Three settings are examined, where the path loss can be neglected, where path loss is considered, but channel correlation is not accounted for, and where path loss and channel correlation are accounted for. It is shown that path diversity can lead to dramatic improvements in performance. For This work was prepared through collaborative participation in the Collaborative Technology Alliance for Communications and Networks sponsored by the U.S. Army Research Laboratory under Cooperative Agreement DAAD The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation thereon.

2 2 example, it is shown that in a 5-hop networks, improvements by a factor of are not uncommon and can reach a maximum of a factor of 2000 as the node density grows to infinity. An interesting feature of path diversity is that when fully exploited, paths with more hops provide better performance than those with fewer hops. It is shown that such behavior occurs when a particular map has a non-zero fixed point. I. INTRODUCTION One of the most important features of wireless networks is the variability of channels. In traditional wireless networking, great pains are taken to mitigate the impacts of the variability of channels. While all layers must cope with the effects of time-varying channels, there has been extensive effort at the network layer. For example, thereareanumberof techniques that seek to find precomputed backup paths (e.g., [1]-[9]). Thus, when the primary path fails, a new route search is not required. However, time-varying channels does not only imply that links may break, it also implies that some links are better than others. Indeed, in the context of communication theory, channel diversity means that there may be some channels between the same transmitter and receiver (but with different antennas) that have better performance than other channels. This diversity is closely related to the stochastic nature of channels. For example, popular models for the channel gain 1 include the lognormal distribution, exponential distribution, and Nakagami. If multiple transmit and/or receive antennas results in a set of channels that can be modeled as independent random variables, then the larger the set of channels, the higher the probability that a good channel can be found. While communication theory provides a clear picture of how through the use of diversity, increasing the number of channels improves performance (e.g., Chapter 11 of [10]), diversity in the setting of multihop wireless networks is less well understood. 1 The received signal power is proportional to transmitted power multiplied by the channel gain.

3 3 This paper examines multihop paths in the presence of variable propagation. Specifically, this paper determines the performance improvement that results when path diversity in multihop networks is exploited as oppose to using an arbitrary or geographically optimal route. Several insights are developed. For example, it is shown that in some settings when diversity is fully exploited, if the link behavior is fixed, then longer paths perform better than shorter paths. This behavior contrasts intuition which dictates that if the behavior of the links is fixed, then shorter paths are better than longer paths. It will be shown that the settings when longer paths are better than shorter paths depends on the existence of a fixed point of a particular onedimensional map. This paper also focuses on developing techniques to compute or estimate the performance improvement offered by diversity. In the example considered, it is shown that under the lognormal propagation model, as the density of nodes increases, exploiting diversity and improve the performance by as much as 33 db, or approximately a factor of At first glance, an improvement of a factor of several thousand seems unreasonably large. However, one must consider that the dynamic range of a physical layer such as exceeds 50 db. That is, a good channel may be 50 db better than a bad channel, i.e., a good channel is times better than a bad channel. The critical challenge for communication theorists is to achieve communication even when the channel is bad. However, at the network layer, there is the option of seeking to use other channels. If these channels are sufficiently better, then one might well expect an improvement of several orders of magnitude. In [11] a diversity seeking protocol is described that is able to attain improvements of several orders of magnitude. One important difference between [11] and this paper, is that this paper develops a number of techniques to compute the performance analytically, whereas [11] relies on simulation. As a result, the techniques developed here can be easily applied to other settings and the opportunities of path diversity are more fully explored.

4 4 This paper has three major sections, IV, V, and VII, that correspond to investigations of diversity in three settings. In the first of these, a simple and highly tractable situation is examined. Specifically, in general the channel gain has a stochastic part and a deterministic part. The deterministic part is called path loss and is a function of the distance between nodes. In this first investigation, this path loss part is neglected, only the stochastic part of the channel is considered. This setting provides insight into the performance in more general settings. For example, in this setting, the behavior of longer paths as compared to shorter paths is explored. While this behavior can be carefully examined in this simple setting, similar behavior will be noticeable in the more realistic settings that are investigated in Section V and VII. Section IV also introduces some techniques that are expanded upon in Sections V and VII. In these sections a particular topology is studied. However, the techniques can be applied other topologies. Section V focuses on the case when the nodes are reasonably far apart. Specifically, they are far enough apart that the correlation between channel gains of nearby channels can be safely neglected. The main focus of this section is examining the performance and demonstrating that if the network is large enough, then a particular dependence can be neglected. The result is that the performance of networks canbeeasilycomputedevenifthenetworkisquite large. Furthermore, Section V provides insight into when the path loss can be neglected and hence the analysis of Section IV, which neglects path loss, accurately predicts the performance. Section VII extends Section V to the case where the nodes may be closer together and hence the performance of links are correlated. Accounting for this correlation allows the evaluation of the performance when the density of the nodes grows to infinity. In this section that it is shown that the performance improvement over a particular 5-hop network may reach 33 db. The paper proceeds as follows. In the next section a brief overview of related work is presented. In Section III, the problem definition is provided along with several basic assumptions.

5 5 Sections IV, V, and VII then present the main results, as mentioned above. Section IX illustrates the performance offered by diversity when the propagation is based of realistic ray-tracing of downtown Dallas, TX. The behaviors illustrated throughout the paper is also observed in this realistic setting. However, a more detailed examination of diversity in various networks is left for future work. Finally, Section X provides some concluding remarks. II. RELATED WORK As mentioned above, communication theory provides a good understanding of the impact of exploiting diversity when the transmitter and receiver have several antennas. However, recently, there has been interest in examining the impact of allowing nearby nodes assist in a transmission [12]-[18]. Several strategies were investigated in [17] and [18]. However, in most cases, small one/two hop networks were investigated. Specifically, it is often assumed that the source can directly communicate with the destination (i.e., one hop). Furthermore, it is assumed that one or more intermediate nodes are available to increase reliability (i.e., two hops). In [14], several techniques for communicating over such networks were investigated. One technique, known as selection relaying, utilizes channel measurements to determine which nodes should transmit. While selection relaying does not require feedback, another method, incremental relaying, makes use of feedback to determine which and whether nodes should transmit. This strategy proved to yield good performance. While [14] focuses on the setting with one relay, [16] examines the one/two hop case, but with a large number of nodes available to act as the relay. There it was found that a technique referred to as best-select provided the best performance of the techniques investigated. In best-select, the node with the best channel is selected to transmit. This best-select approach is similar to the ideas pursued here. Specifically, in best-select, the best path is found among the two-hop paths. In this paper, the performance achieved when the best path among

6 6 multihop paths is utilized. Route diversity is a well known concept in multihop networks. However, route diversity, as it is often used in the literature, and exploiting diversity, as is studied in this paper, are related, but slightly different concepts. Typically, in the literature, route diversity implies that when different routes are collected during the route search phase of a routing protocol, the routes are distinct in the sense that the routes do not share common links and perhaps do not even share common nodes (e.g., see [1]-[9]). Here the set of routes considered will share common links and nodes. On the other hand, both route diversity and the type of diversity considered here embrace the stochastic nature of links. While there are a large number of protocols that precompute a set of back-up paths and select a path from this set when the current path fails, [2] selects a new route frequently in order to improve the end-to-end performance. Here, we will investigate the performance improvement that could be achieved by such an algorithm. While some performance analysis was undertaken in [2], a free-space or 2-ray model for propagation was used, whereas here, more realistic stochastic propagation is used. In [11] and [19], various performance issues related to diversity seeking protocols are investigated. III. PROBLEM DEFINITION This paper focuses on the topology shown in Figure 1. The figureshowsthesourceand destination as well as several sets of nodes that are made up of nodes placed in a vertical line. As indicated in the figure, these sets are called relay-sets. The relay-sets are numbered with the first relay-set being closest to the source. It is assumed that each transmission carries the packet fromanodeinthen-th relay-set to a node in the (n +1)-th relay-set, but the exact node within each relay-set that relays the packet can be adjusted. As is also indicated in Figure 1, the nodes are labeled with node (n, i) being the i-th node in the n-th relay-set. We assume that the number

7 7 node (n, 2 Range/h) source d h destination 2 Range node (n,1) 1 st relay-set 2 nd relay-set (N-1) th relay-set Fig. 1. The objective is to find the best path from source to destination. The nominal path is made up of the nodes along the center. In this paper, we explore the impact of being allowed to use any combination of nodes. However, each hop must be between two nodes in adjacent relay-sets and the route must always proceed toward the destination. The distance between the relay-sets is d, and the space between adjacent nodes is h. Each relay-set spans a vertical distance 2 Range. of nodes in each relay-set is fixed, and is sometimes denoted as M. Sometimes it is convenient to think of the source as a node in the zeroth relay-set, i.e., (0, bm/2c), therebm/2c is the largest integer that is less than or equal to M/2. Similarly, the destination is (N,bM/2c). There are several ways that a path can be selected. A simple approach is to utilize the path that is made up of the nodes along the center, i.e., the shortest geographical path. This is the nominal route. Alternatively, the path utilized could be the path that is best in terms of a particular route metric. Here the route metric of interest is the maximum channel loss 2 along the route. That is, each link of a path has a particular channel loss, and the route metric is the maximum of these link channel losses. To put it another way, the routemetricisthechannellossoftheworstlink along the route. This paper examines the performance gain that is achieved by using a route that is optimal in terms of this route metric as compared to the nominal route. Since the received signal strength is proportional to the transmitted power divided by the channel loss, the channel loss is closely related to the SNR. Hence, minimizing the maximum channel loss is the same as maximizing the minimum SNR experience by each receiver along 2 The channel loss is the recipocal of the channel gain. Hence, a good channel has a small channel loss.

8 8 the route. There are several motivations for trying to minimizing the maximum channel loss. For example, in most physical layer schemes, the probability of a transmission error across a link falls to zero quickly as the SNR increases beyond a threshold. Thus, the probability of successfully delivering a packet to a destination along a route is dominated by the probability of successfully transmitting a packet over the worst link along the route. Similarly, if the transmission power at each link is adjusted so that a target SNR is met (e.g., to meet a target link transmission error probability), then the energy to deliver a packet from the source to destination is dominated by the energy required to transmit across the worst link along the path. Also, if the bit-rate across each link is maximized under the constraint that the transmission error probability is less than aspecified threshold, then the throughput along the route is given by the bit-rate across the slowest link, which, again, is the worst link along the route in terms of channel gain. See [11], for further examination of metrics. It is common to decompose the channel loss into two parts, a deterministic part that depends on the distance between the nodes, and a stochastic part. In particular, the channel loss, in db, is Channel loss [db] = X α 10 log 10 (d) (1) where X accounts for the stochastic part, d is the distance between the transmitter and receiver, and α is the path loss exponent. While α hasbeenfoundtobeassmallas1.7andaslargeas 4, for sake on concreteness, we use α =2.7 following the findings presented in [20]. Clearly, the computations can be performed with a different value of a. While there are many possible distributions of L, in Sections V and VII it is assumed that X models the effect of shadowing and is a lognormally distributed. On the other hand, the techniques in Section IV can easily be applied for any distribution. One reason that the stochastic part of the channel loss in Sections

9 9 V and VII only accounts for shadowing and not multipath fading is that this paper focuses on wide band communication where the impact of multipath fading is mitigated (see, for example, page 330 in [20]). IV. PERFORMANCE WITHOUT PATH LOSS AND WITHOUT CHANNEL LOSS CORRELATION This section analyzes the case when the deterministic part of the channel loss is neglected. There are two reasons for considering this setting. First, with this simplification, analytic expressions for performance can be derived and the impact of diversity can be carefully examined. As will be seen in the following sections, the behavior that arises here, also occurs in other more general scenarios where analytic expressions are more difficult to attain. A second motivation for neglecting the deterministic part of the channel loss is when considering scenarios where the nodes within a relay-set are far enough apart so that the correlation between nearby channels can be neglected and the size of the relay-set is much smaller than the distance between the relay-sets, i.e., Range d and h À 1. Note that since h<range, this implies that d À 1. In this case, the distance between nodes in adjacent relay-sets is d + ρ Range with ρ [ 1, 1], thus from (1), the channel loss is X α 10 log 10 (d + ρ Range) X α 10 (log 10 (d)+ρ Range/d) X α 10 log 10 (d), where the last approximation holds since Range d and d À 1 and d is a fixed constant. Hence, the variability of the channel loss is only due to the variability of X, the stochastic part of the channel loss. The objective of the following analysis is to determine the expected performance, in terms of the metric discussed above. We are specifically interested in how the performance is impacted by the topology, namely, the size of relay-sets and the number of hops. The first step toward

10 10 this objective is to determine the cumulative distribution function (CDF) of the optimal value of the route metric, i.e., P (the channel loss of the worst link along the best path is less than r). To this end, we utilize the terminology of percolation theory. Fixing the channel loss at r, we saythatalinkisopen if the channel loss is less than r and is closed otherwise. Furthermore, we say that a node is occupied if there is a sequence of open links from the source to the node. Hence, letting F be the CDF of the channel loss, the probability of a link being open is F (r). With r fixed, we often denote the probability that a link is open with p and the probability the link is closed with q =1 p. Moreover, the source is always occupied and the destination is occupied if there is an end-to-end route with maximum channel loss less than r. Given that the probability that a link is open is p, we denote the probability that the destination is occupied as T M,N (p), wheretherearem nodes in each relay-set and N hops between the source and destination. Since P (the channel loss of the worst link along the best path is less than r) =T m,n (F (r)), (2) the CDF we desire can be easily computed once T is known. The following provides an efficient way to compute T. Proposition 1: With the definitions given above, T M,N (p) =Θ 1 p,m (1, :) Θ N 2 1 p,m V 1 p,m, (3) where Θ q,m is a (M +1) (M +1) matrix with the i, j element given by µ M 1 Θ q,m (i, j) := q i j q i(m j), j Θ 1 p,m (1, :) is the first row of Θ q,m,i.e.θ q,m (1, :) := Θ q,m (1, 0) Θ q,m (1,M),and

11 11 V q,m is the (M +1) 1 column vector given by V q,m (j) =(1 q j ). Note that the indexes i and j range from 0 to M. With (3), the expected impact of diversity can be determined. Letting ū n be the expect number of occupied nodes in the relay-set that is k hops from the source, then the above proposition canbeusedtocomputeū n.specifically, we have the following. Corollary 2: The expected number of occupied nodes in the relay-set that is n hops from the source is given by ū n : = E (number of occupied nodes n hops from the source) (4) = Φ q,m (1, :) Φ n 1 q,m W, where W is the (M +1) 1 column vector with W (j) =j where j =0, 1,,M. Proof: [Proof of Proposition 1] To determine T, the number of occupied nodes is represented as a Markov chain. Suppose that i nodes are occupied in the relay-set that is n hops from the source. Then, the probability that a particular node that is n +1 hops from the source is not occupied is the probability that none of the links from the i occupied nodes in the previous relay-set are open. This probability is q i. And hence, the number of occupied nodes in the relaysetthatis(n +1)hops from the source is a binomial random variable with parameters (1 q i ) and M. Thus, Φ q,m (i, j) is the probability that j nodes are occupied in the relay-set that is n+1 hops from the source given that i nodes are occupied in the relay-set that is n hops from the source. Since the source is always occupied, there is one node occupied in the relay-set that is zero hops from the source. Therefore, Φ q,m (1, :) is the probability distribution of the number of occupied nodes in the first relay-set. In general, setting v := Φ q,m (1, :) Φ n 1 q,m, we see that v (j) is the probability that exactly j nodes are occupied within the relay-set that is n hops from

12 12 the source. Finally, the probability that the destination is occupied given that i nodes within the last relay-set are occupied is 1 q i. Therefore, the right-hand side of (3) is the probability the destination is occupied. An important feature of multihop diversity isdemonstratedbythefactthatforsomep, M and N, wehavet M,N (p) >p. That is, the probability of findinganend-to-endpathsuchthateach link along the path has a channel loss that is less than some r is greater than the probability of finding a single link that achieves that same level of performance. Or to put it another way, the best path is better than the typical link. To explore this feature further, define p + to be the crossover probability, wherep + satisfies T M,N (p + )=0.5. For example, if the channel statistics are such that the probability of the channel loss being less than 40 db is p +, then with probability 0.5, the best end-to-end path consist of links where each link has a channel lossthatislessthan40db.figure2showsp + as a function of M, the relay-set size, and N, the number of hops from source to destination. Note that for large M, thep + is considerably smaller than 0.5, i.e., T M,N (p + ) >p +.Hence,in these situations, one can expect that the best end-to-end path will perform better than the typical single link. A remarkable feature of multihop diversity illustrated by Figure 2 is that for large enough M, we find that p + decreases as N increases. Consider the case of M =1, i.e., where diversity is not used. In this case, it is clear that if each link achieves a particular level of performance with probability p, then all links along the path will achieve this level of performance with probability p N, therefore p + =0.5 1/N, which increases with N, the length of the connection. This implies that in order to achieve a desired end-to-end performance, each link s performance must exceed the desired end-to-end performance. Furthermore, as the length of the connection grows, each link s performance is required to further improve if the end-to-end performance is

13 13 p N=2 N=3 N=4 N= M - size of relay-set Expected fraction of nodes occupied actual mean field 0.4 p=0.75 M=4 0.2 p=0.5 M=8 p=0.25 M=15 0 p=0.05 M= N hops from source Fig. 2. Left: Suppose that is it desired that with a probability of at least 0.5, there exists a path such that each link along the path has a particular property (e.g., minimum SNR for each hop is above r). By exploiting diversity, this objective can be met as long as each link has this property with probability at least p + (e.g., the probability of the link SNR exceeding r is p + ). The above shows the relationship between p +, N the number of hops in the path, and M the number of nodes in each relay-set. Right: Mean field estimate of the mean number of occupied nodes and the actual number of occupied nodes as a function of the number of hops from the source. to be achieved. However, as shown in Figure 2, for M>3, the opposite is true; as the length of the connection grows, each link s performance can be degraded and the desired end-to-end performance can still be achieved. Similarly, if each link s performance is held constant, then longer paths will perform better than shorter paths. This counter-intuitive behavior of diversity is further investigated in the next section. However, we note that a special case of this behavior is illustrated by T M,N (p) >p,which implies that a multihop path is better than a typical single link, which is a single hop path. 1) The Impact of Long Paths - A Mean Field Analysis: In this subsection the conditions under which longer paths provide better performance than shorter paths are investigated. It will be shown that these conditions are related to the existence of a non-zero attracting fixed point of a particular one-dimensional map. However, a more intuitive reasoning is as follows. Consider a single hop network. In this case, the performance is dictated by a single link. However, in a two-hop network, the end-to-end performance is given by the best of M disjoint paths. In a three-hop network, the end-to-end performance is given by the best of M 2 non-disjoint paths. Thus, we see that as the number of hops grows, the number of paths grows very quickly. Since

14 14 the performance of each path is random (they may be correlated, but are not perfectly correlated), the more paths considered, the better the performance. On the other hand, since the performance of a single path is dictated by the worst link along the path, the performance of a single path will decrease as its length grows. Hence, there are two contradicting mechanisms, one causes performance to increase as the path length increases, while the other causes the performance to decrease as path length increases. As will be detailed next, if the probability that the performance of links is above a threshold, then the benefit of more paths is more significant than the impact of longer paths, and performance tends to increase with path length. A detailed analysis is as follows 3. In the proof of Theorem 1, it is shown that if there are i nodes occupied in the n-th relay-set, then the mean number of nodes occupied in the (n +1)-th relay-set is (1 q i ) M. Employing atypeofmeanfield analysis, we assume that if there are i nodes occupied within the n-th relay-set, then there are exactly (1 q i ) M nodes occupied within the (n +1)-th relay-set. The right-hand plot in Figure 2 compares this mean field model to the exact computation. Under the mean field model, define u n to be number of nodes occupied in the n-th relay-set. It follows ³ that u 1 =(1 q) M, andu n+1 = 1 q u n M. This one-dimensional dynamical system can be normalized by defining ũ = u/m and q = q M, yielding ũ n+1 =1 qũn. (5) Thedynamicsof(5)canbeexaminedusingthestandard tools for analyzing one-dimensional maps of iteration (e.g., Chap. 2 of [21]). The key question is whether (5) has any stable equilibrium, that is, we seek the different possible values of lim n ũ n.inthecasethatthereisastableequilibrium,wecheckwhether 3 The remainder of this subsection can be skipped without loss of continuity.

15 15 convergence to the equilibrium is monotonic or not. It is important to note that the existence of an equilibrium ũ with ũ > 1/M and monotonic variation of ũ n implies that the mean number of occupied nodes increases with n, the number of hops from the source. Clearly, the more nodes that are occupied within the (N 1)th relay-set, the more likely that the destination will be occupied. Thus, if ũ > 1/M, then the destination is more likely to be occupied if the path is longer, that is, longer paths provide better performance (in terms of this metric). The following characterizes when this occurs. Proposition 3: Themap(5)hasanon-zeroattracting equilibrium, ũ,ifq M < 1/e. Ifq M 1/e, thenũ =0is the only equilibrium and is attracting. If ũ 0 < ũ (ũ 0 > ũ ), then the sequence ũ n for n =0, 1, is monotonically increasing (decreasing). Proof: Define g (ũ n )=ũ n+1 ũ n = 1 qũn ũn.theng 0 (ũ) = qũ ln ( q) 1. Notethat g (0) = 0, implies that ũ =0is a fixed point. Furthermore, g 0 (0) > 0 ifandonlyif ln ( q) > 1 or q <1/e, (6) or, equivalently, q M < 1/e. In this case, ũ n+1 ũ n = g (ũ n ) > 0 for ũ n 0 and positive. Thus, ũ =0is a non-attracting fixed point if and only if (6) holds. Now, suppose that (6) holds. Since g (1) = q <0 and g (ũ) > 0 for ũ small and positive, the Intermediate Value Theorem implies that there must be a point ũ between 0 and 1 such that g (ũ )=0. Thus, if (6) holds, then g has at least two zeros. To see that there are exactly zeros, note that g 00 (ũ) = qũ (ln (q)) 2 < 0. Next we show that ũ n is monotonic. If (6) holds, then g 0 (0) > 0. Andsinceg (0) = 0, we have g (ũ) > 0 for ũ (0, ũ ), or, equivalently, ũ n+1 > ũ n for ũ (0, ũ ). Similarly, since g (ũ )=0and g 0 (0) > 0, wehaveg 0 (ũ ) < 0 and hence g (ũ) < 0 for ũ>ũ, or equivalently,

16 16 1 u~ 0.5 actual fitted ~ q Number of hops to convergence Solid lines: actual Dash line: fitted q~ ~ u 0 =0.001 ~ u 0 =0.01 ~ u 0 =0.1 ~ u =0.2 0 ~ u =1/3 0 Fig. 3. Left: For q that satisfies (6), there is a non-zero fixed point that solves (5) such that lim k ũ k =ũ.theabove shows the value of ũ for different q. Also, an approximation of this relationship is shown. Right: The number of occupied nodes converges as the number of hops increases. The above shows the number of hops required to reach 10% of the final value. The number of hops to convergence depends on q and the ũ, which depend on M and q. Approximations to the required length of the connection also shown. d ũ n+1 < ũ n. On the other hand, dũ 1 q ũ = qũ ln ( q) 0. This implies that 1 q u < 1 q w if u<w.thus,ifũ n < ũ,thenũ n+1 =1 qũn < 1 qũ = ũ and if ũ n > ũ,then ũ n+1 > ũ. In summary, if ũ 0 < ũ,thenũ n < ũ and ũ n+1 > ũ n ; in other words, u n is monotonically increasing. Similarly, if ũ 0 > ũ,thenũ n is monotonically decreasing. And in both cases, lim n ũ n = ũ. In a similar fashion it can be shown that ũ n is monotonically decreasing when ũ 0 > 0 and (6) does not hold. By repeatedly evaluating (5), ũ can be estimated. Figure 3 shows the relationship between q and the non-zero fixed point ũ. This relationship was found to be well approximated by ũ 5.6 q 2 ( 5.6/e + e) q +1. (7) This approximation is also shown in Figure 3. It should be noted that for fixed q<1, wehave lim M q M = lim M q =0. Hence, lim M ũ =1,implyingthatanincreasingfractionof the nodes within the relay-set will be occupied as the size of the relay-set increases. The question of whether longer paths are better than shorter paths can be resolved by deter-

17 17 mining if (6) holds and if ũ givenby(7)exceeds1/m. On the other hand, as can be seen in left hand plot in Figure 2, the incremental increase in performance due to the incremental lengthening of the path tends to decrease as the path length increases. This is due to the convergence of the ũ n. Figure 3 shows the number of hops when ũ n reaches 10% of its final value. This relationship is shown for various q and initial values of ũ 0. For small values of q, the number of hops is approximated by 1+(1 ln (ũ 0 )) / ln (1 ln ( q)). Notethatas q increases, the number of hops until the mean number of occupied nodes converges increase. While it is not shown here, this value grows very quickly as q nears 1/e. However, for q 1/e the mean field analysis is less accurate. Figure 3 shows that reasonably short paths are sufficient to fully exploit diversity. Remark 4: While this analysis implies that long paths have some benefits, there are other drawbacks to long paths that are not modeled here. See [22] for a discussion of some advantages of shorter paths. A. Examples of multihop diversity gain The performance analysis above applies to any channel loss model. However, since much of the rest of this paper focuses on scenarios where the channel is impaired by lognormal shadow fading, this section only examines the performance under lognormal fading. As discussed in Section III, the performance metric of interest here is the expected value of the channel loss of the worst link along the best path. The average performance can easily be found from (2), where T M,N is found using Proposition 1 and F is the CDF of a lognormal distribution with mean 0 db and σ in db. The average value of the worst channel loss along the best path is compared to the average channel loss of the worst link along an arbitrarily selected path. Specifically, we consider the improvement, in db, provided bythebestpathascomparedtoanarbitrarily selected path. An improvement of 0 db implies no improvement at all.

18 18 Mean improvement in signal strength (db) σ = M - relay-set size σ = 8 14-hops hops M - relay-set size M - relay-set size σ = nodes/rs 15 nodes/rs 10 nodes/rs 5 nodes/rs σ Fig. 4. The three left-hand plots show the improvement in the minimum channel gain along a path when using the optimal path as compared to using an arbitrary path. The amount of improvement depends on the relay-set size and the number of hops. The number of hops varies from 2 hops (the lower curves) to 14 hops (the upper curves). It is assumed that the channel is impaired by shadow fading with parameters σ. The left three plots show the performance for different σ. The right-most plot shows the improvement over an arbitrary path as the shadow fading parameter σ varies when the network has three hops. The variation in performance is shown for relay-sets sizes ranging from 5 to 20. The three left-hand plots in Figure 4 show the improvement provided by the best path for different topologies and for values of σ. Each curve shows the variation of the improvement as the number of nodes in each relay-set varies. As indicated, the curve that indicates the least improvement corresponds to the case when there are two hops from source to destination while the curve that provides the largest improvement corresponds to the case when there are 14 hops. As expected, the improvement when there is one node in each relay-set is 0 db, this is the case where diversity cannot be exploited. Lognormal shadow fading is parameterized by σ, the standard deviation in db. Figure 4 shows the performance improvement for σ = 4, 8, and 11 ([20] presents experimental results that indicated that σ =11.) The right-most plot shows the performance improvement as a function of σ. As the standard deviation increases, the variability of the channel loss increases and the performance improvement provided by diversity increases. V. CORRELATED NODE OCCUPANCY While the analysis above provides a clear view into the performance gains that can be achieved by exploiting diversity, it neglects path loss and hence is only directly applicable to settings

19 19 where Range d in Figure 1 and h is not too small. In this section the impact of path loss is considered. While this section examines the actual value of the performance, the next subsection presents a computationally efficient approximation of the performance when M is large. On the other hand, this section does not consider correlated channels, which is covered in Section VII. Considering the network depicted in Figure 1, the probability of transmitting from node (n, i) q to node (n +1,j) depends on the relative distance between the nodes, d 2 + h 2 (i j) 2.As discussed in Section III, the channel loss is log-normally distributed. Hence, the probability of transmitting from node (n, i) to (n +1,j) is denoted by g Thresh ( i j ) and is given by g Thresh ( i j ) :=P µx log 10 µqd 2 + h 2 (i j) 2 <Thresh, where X is Gaussian with mean 0 and standard deviation σ, andthresh such that if the channel loss is less than Thesh, then the transmission can be decoded. As is done in the previous section, we represent node occupancy with a Markov chain. Here the state of the Markov chain is a vector of zeros and ones indicating which nodes are occupied. Thus, a particular element of this state can be represented as an integer between 0 and 2 M. Specifically, let a i =1if the i-th node is occupied and a i =0otherwise. Then, this state is represented by A 0, 2 M 1 where A = P M 1 i=0 a i2 i.thus,wecandefine Q Thresh to be the probability transition matrix via Q Thresh (A, B) :=P (moving from state A to state B) (8) = Y 1 Y (1 g Thresh ( i j )) Y Y (1 g Thresh ( i j )) {i:b i =1} {j:a j =1} {i:b i =0} {j:a j =1} To understand (8), note that the probability of node (n +1,i) not being occupied is the prob-

20 20 ability that each link from every occupied node in the previous relay-set being close, which is Y (1 q gthresh ( i j )) where {j : a j =1} is the set of nodes in the nthrelay-setthatare {j:a j =1} occupied. Furthermore, the set of nodes that are occupied in state B is {i : b i =1}. Thus,the probability of being in state B is the probability that each node i is occupied only if b i =1.Note that Q is a 2 M 2 M dimensional matrix and the probability distribution is a 2 M dimensional vector. From (8), it is straightforward to compute the probability distribution of the occupied nodes within the n-th relay-set. Specifically, we can think of the source as a node within the bm/2c-th node within the 0-th relay-set where bm/2c is M/2 rounded down to the nearest integer. Thus, we set the probability distribution of the occupied nodes within the 0-th relay-set to be V with V S =1if S =2 bm/2c. The probability distribution n hops from the source is VQ n Thresh. The destination can be thought of as the bm/2c-th node in the N-th relay-set. Given that the probability distribution in the N-th relay-set is W, the probability that the bm/2c is occupied is found by summing the elements of W over all the states that have the bm/2c occupied. Specifically, set B to be a 2 M row vector with B (S) =1if S bm/2c =1. Then the probability that the destination is occupied is P (maximum channel loss along the best path < Thresh)=V Q N Thresh B. (9) Finally, from (9) the probability distribution of the channel loss along the best channel from the source to destination can be found. Figure 5 shows the expected value of the maximum channel loss along the best path as a function of the number of hops from source to destination for M =1, 2, 4,...14 and distance between neighboring nodes within a relay-set is h =20m and h =60m. In this case the standard deviation of the shadow fading was 11 db and the distance between relay-sets was 100m. Note that the M =1case is the case when there is only one

21 21 path from source to destination. Thus, as above, the M =1case corresponds to the case when diversity is not exploited and hence represents the benchmark performance. The analysis in Section IV assumed that range of the relay-set is far smaller than the distance between relay-sets, i.e., Range d. Hence, for large d, the improvement in performance when the path loss is accounted for should be similar to the performance found in Section IV. The right-hand plot in Figure 5 examines the performance for several values of d and for relay-sets with 6 and 12 nodes. In both cases, it can be seen that d = 200 yields similar results as d =400; thecurvesnearlylieontopofeachother.however,forrelay-setswith6nodesineachrelayset, the performance for d =100or greater yields the same performance, while for 12 nodes in each relay-set, the performance does not converge until d =200. Thus, we can conclude that the analysis in Section IV is valid for reasonably distance relay-sets (recall may reach distances of m) and this is especially the case if there are relatively few nodes in each relay-set. Note that the performance shown in the right-hand plot of Figure 5 for d =400mis similar to what is shown in Figure 4. As was found in the case investigated in Section IV, we also find that the performance tends to improve as the number of hops increases and as the relay-set size increases. Another characteristic shown in Figure 5 is that as the size of the relay-set increases, the relative improvement in performance decreases. In Figure 5, the largest relay-set size is 14. In this case, there are 2 14 elements in the state-space and the state transition matrix has 2 28 elements 4. With current processors, this was the largest topology whose performance could be computed in a realistic amount of time. 4 It is possible to compute the probability distributions without computing the entire state transition matrix.

22 22 Average Best Min. Channel Loss (db) h=20m h=60m Hops Mean improvement in signal strength (db) d=50m d=100m d=200m d=400m Hops Fig. 5. Left: Average maximum channel loss along the best path. The solid curves are for the case when h =20m and the marked curves are for h =60m. The upper most curve is when there is only one node in each relay-set, i.e., when diversity is not exploited. The curve is the same regardless of the value of h. As the size of the relay-sets grows, the maximum channel loss decreases. This figure shows the performance for relay-set size of 1, 2, 4, 6,, 14. The best performance is indicated by the lowest curve and corresponds to the case where M =14. Right: The performance improvement (as compare to the nominal path) for different spacings between relay-sets. Two sets of curves are shown. The upper set is for relay-sets with 12 nodes and the lower set is for relay-sets with 6 nodes. Each set of curves consists of four curves corresponding to d =50mtod = 400m. The lowest curve in each set is for d =50m. Note that the d =50m for 12 nodes/relay-set is similar to the that of d = 400m for 6 nodes/relay-set. VI. AN APPROXIMATION A dramatic improvement in computation can be obtained by assuming that the occupancy of a node is independent of whether other nodes within the same relay-set are also occupied. It will be shown that this independence assumption provides an upper bound on performance. Furthermore, it will be observed that for large relay-sets, this approximation provides independence assumption does not have a significant impact on the accuracy. First we compute the probability of occupancy under the independence assumption. Let the probability that the j-th node in the n-th relay-set is occupied be P n,t hresh (j). Then probability that node i in the (n +1)-th relay-set is occupied is P n+1,t hresh (i) (10) MY = 1 ((1 P n,t hresh (j)) + (1 q Thresh ( i j )) P n,t hresh (j)) = 1 j=1 MY (1 q Thresh ( i j ) P n,t hresh (j)). j=

23 23 Error (db) Hops 2 node/rs 4 nodes/rs 6 nodes/rs 8 nodes/rs 10 nodes/rs 12 nodes RS 14 nodes/rs Fig. 6. The difference between the exact value of the optimal value of the route metric and an approximation where the approximation assumes that the occupancy of a node in a relay-set is independent of the occupancy of other nodes within the same relay-set. This figure shows this error for networks with 2-14 nodes per relay-set and for networks with 2 to 20 hops. Note that an error of 0 db means that there is no error and an error of 3 db means that the approximation suggests a performance that is 3 db better than the actual performance. The results here are for d =100m and h =20m. As above, the fact that the source is occupied is expressed by P 0,T hresh (i) =1for i = bm/2c and 0 otherwise and note that the probability of there exits a path from source to destination such that the maximum channel loss is less than Thresh is P N,Thresh (bm/2c). Fromthis,the performance can be efficiently approximated. This approximation yields performance relationships that are quite similar to those shown in Figure 5. Figure 6 shows the difference between the expected performance found using (10) and (4). Note that for large networks the error is less than 1 db and converges as the length of the path grows. For smaller networks, the error is larger, but for small relay-set sizes, an approximation is not needed since the actual performance can be easily computed. It can also be noticed that the error is always positive implying that the approximation is always larger than the actual value. This is always the case as explained next. Proposition 5: The assumption that the event that a node is occupied is independent of the event that other nodes within the same relay-set are occupied is leads to an upper bound on performance. Proof: Let U i be the event that node (n, i) is occupied and node i has and open link to

24 24 Ã! [ node (n +1,k). Then the probability that node (n +1,k) is occupied is P U i Ã! [ P U i = X P (U i ) X P (U i U j ). i i i6=j i and Under the assumption that the events U i are independent Ã! [ P U i = X i i P (U i ) X i6=j P (U i ) P (U j ), where P denotes the probability under the independence assumption. However, since the events U i are increasing, P (U i U j ) P (U i ) P (U j ). (11) Thus P [23]. Ã! Ã [ [ U i P U i!. For further discussion on increasing events see Chapter 2 of i i VII. CORRELATED CHANNELS Figure 5 shows that the performance improves when either the size of the relay-sets increases or if the spacing between nodes within the relay-set decreases. Both of these trends are reasonable; as the size of the relay-sets increases, there are more paths available and hence the probability that a good path is available increases. As the spacing between nodes within a relay-set decreases, the quality of the paths from any single node to all the nodes in the next relay-set improves. q Specifically, recall that the distance between nodes (n, i) and (n +1,j) is d 2 + h 2 (i j) 2, which decreases with h, the spacing between nodes within the same relay-set (see Figure 1). BasedonthetrendsshowninFigure5,thequestionnaturallyarisesastowhatthemaximum performance that can be achieved by increasing the number of nodes in the relay-sets and by decreasing the spacing between nodes within a relay-set. This question is answered in this section.

25 25 From Figure 5 it can be seen that as the size of the relay-sets grows, the incremental improvementinperformancedecreases.thisisdecreaseisduetothefactthatforverylarge relay-sets, a larger relay-set will only add nodes that are far from the shortest path (in terms of distance). Thus, it becomes less likely that these nodes will be part of a best path. A more complex issue is how the performance varies as the spacing between nodes within the relay-set decrease. The complication arises from the fact that links between nearby nodes suffer from correlated channel loss. To account for the correlation between channel loss, we use a model the channel loss as a spatial Gaussian process. A similar approach is followed in [24] and [25]. We begin by analyzing the case where there are several links from a single transmitter to several different destinations. Let L (a, b) denote the channel loss, in db, due to shadow fading from transmitter 5 at location a to a receiver at location b. Hence L (a, b) N (0,σ). Then, following the approach of [24], [25], the correlation between two links is modeled as E (L (a, b) L (c, d)) = σ 2 exp ( α ( a c + b d )), where a c is the distance between location a and c, and α 1/10 m 1. Hence, the correlation between the shadow fading part of the channel loss from (n, i) to (n +1,k) and the shadow fading part of the channel loss from (n, i) and (n +1,l) is σ 2 exp ( αh k l ). (12) and the correlation of the shadow fading part of the channel loss from (n, i) to (n +1,k) and the shadow fading part of the channel loss from (n, j) to (n +1,l) as σ 2 exp ( αh ( i j + k l )). (13) Now consider the channels from (n, i) to (n +1,j) and from (n, i) to (n +1,j+1).In 5 By recipocy, the channel gain does not depend on which node is transmitting and which node is receiving. We set one node to be the transmitter to make the presentation more concrete.

26 26 this case, the correlation between these channel gains is σ 2 exp ( αh). Thus, as h 0, these channels become highly correlated (e.g., the correlation coefficient goes to one). Thus, for small h, neighboring nodes have approximately the same channel gain and hence do not provide any diversity and do not improve performance. Therefore, as h 0, we expect that the performance does not continue to increase but converges to an asymptote. Here we seek to determine the performance limit as h 0. The correlation and the large number of nodes will pose significant computational problems. Therefore, in this section we use Monte Carlo simulations to estimate the performance. Thus, we seek to compute a realization of the channel loss of each link. Specifically, for each hop, a realization can be generatedbycomputingthem 2 dimensional vector, V, that is the channel losses from one relay-set to the next where V M i+j is the channel loss from node (n, i) to node (n +1,j). ThenV can be generated via V = L T W + C where W is the M 2 column vector with elements that are i.i.d. Gaussian with zero mean and variance of one, and L is the Cholesky factorization of R which is the covariance matrix with R M k+l,m i+j = σ 2 exp ( α ( k i + l j )).Furthermore,C is the vector of path losses with C M i+j = log 10 µq d 2 + h 2 (i j) 2.Thus,ifV M i+j <Thresh, then communication between nodes (n, i) and (n +1,j) is possible. By computing a large number of realizations, it is possible to estimate the probability that end destination is occupied for a specific value of Thresh. The solid lines in Figure 7 show resulting relationship between the end-to-end performance of a five-hop network, the node density, and the size of the relay-set. As expected, the larger the relay-set and higher the density, the better the performance. The average performance of a nominal five-hop path is -67 db. Thus, by exploiting