Reliability and Route Diversity in Wireless Networks

Transcription

1 2005 Conference on Information Sciences an Sstems, The Johns Hopkins Universit, March 16 18, 2005 Reliabilit an Route Diversit in Wireless Networks Ehsan Khanani, Etan Moiano, Jinane Abounai, Lizhong Zheng 1 Laborator for Information an Decision Sstems Massachusetts Institute of Technolog [khanani,moiano,jinane,zheng]@mit.eu Abstract We stu the problem of communication reliabilit an iversit in multi-hop wireless networks. Our aim is to evelop a new network moel that better takes into account the faing nature of the wireless phsical laer. To that en, we use the outage probabilit moel for a faing channel to evelop a probabilistic moel for a wireless link. This moel establishes a relationship between the link reliabilit, the istance between communicating noes an the transmission power. Appling this probabilistic moel to a multi-hop network setting, we efine an analze the en-to-en route reliabilit an evelop algorithms for fining the optimal route between a pair of noes. The relationship between the reliabilit of the optimal route an the consume power is stuie. The iea of route iversit is introuce as a wa to improve the en-to-en route reliabilit b taking avantage of the wireless broacast propert, the inepenence of fae state between ifferent pairs of noes, an the space iversit create b multiple rela noes along the route. We give analtical results for the improvements ue to route iversit in some simple network topologies. Our results suggest that route iversit can funamentall change the trae off between reliabilit an power in a multi-hop network. I. Introuction The area of a-hoc an sensor networks has receive a lot of attention in the research communit over the past several ears. In this paper, we look at the problem of routing an reliabilit in these networks. Motivate b moels for the propagation of electromagnetic signals in space, the amount of energ require to establish a link between two noes is usuall assume to be proportional to the istance between the communicating noes raise to a constant power. This fixe exponent, referre to as the path-loss exponent, is usuall assume to be between 2 to 4. In this moel, it is assume that the information is receive b the intene estination with certaint if the source transmits the information at a minimum power level ictate b its istance to the intene estination. We refer to this moel as the eterministic link moel in this paper since the set of noes that receive the transmitte information is known with certaint base on the transmission power level chosen b the transmitter. The eterministic moel for a wireless link, however, ma not be ver realistic for escribing one of the most important effects in wireless communication, the multi-path faing. The 1 This work was supporte b NSF ITR grant CCR , b DARPA/AFOSR through the Universit of Illinois grant no. F an b NASA Space Communication Project grant number NAG receive signal in a wireless link is the sum of signals reflecte b ifferent scatterers in the propagation environment. A link is sai to be in a eep fae state when the reflecte signals a estructivel at the receiver. Naturall, a higher transmission power is require to establish a link between two noes when the channel between them is in eep fae. Since the fae state of a link changes over time, the amount of energ require to transfer a unit of information between an two noes changes over time as well. The simple eterministic moel for a wireless link oes not take into account this time varing nature of the wireless propagation meium. Depening on how fast channel changes occur, the time varing nature of the wireless channel can be aresse in two ifferent was. If the channel changes relativel fast, coing can be one to average the effect of faing. This tpe of averaging effect is the motivation behin the ergoic capacit moel for faing channels see [9]. To achieve this tpe of average behavior, however, long elas might be impose on the transmission. In situations where the ergoic capacit is achievable, the eterministic moel for the wireless link ma still be applicable with some minor changes. We will not get into the etails here as our goal in this paper is to propose an alternative moel for a wireless link which is more suitable for scenarios in which this tpe of average behavior is not appropriate. The case of ela sensitive ata an slowl changing channel is one example. An alternative moel for the wireless link is base on the outage probabilit formulation, see [9], [5], an [6]. In this moel, the instantaneous capacit of a wireless link is treate as a ranom variable. A link is sai to be in outage when the instantaneous capacit supporte b the link is less than the transmission rate. The reliabilit of a link, i.e. the probabilit of correct reception at the receiver, is moele as a function of the transmission rate, the transmitte power, the istance between the communicating noes, an the channel fae state. In this paper, we assume that the fae state is not known to the transmitter. Uner this setting, the transmitter can control the probabilit of successful reception at its intene receiver b ajusting the transmission rate or power. We refer to this moel as the probabilistic link moel. There are several was to avoi losing ata when the channel is in outage, such as coing over a long perio of time, emploing ARQ protocols, or obtaining transmitter sie channel information. However, in this stu, we focus on the reliabilit of a link without using an of these techniques. This approach allows us to isolate the issue of obtaining iversit through routing, an the results evelope here can be reail applie in combination with other forms of iversit techniques. Our analsis starts b looking at the reliabilit of a pointto-point communication link. In section II, we evelop a moel that relates the channel fae state an the istance between the communicating noes to the probabilit of successful reception. This woul give us

2 the mathematical formulation for the probabilistic link moel. In section III, we exten the probabilistic link moel to a network setting. In a network setting, we first efine an analze the reliabilit for a fixe route an then evelop algorithms for fining the optimal route between a sourceestination pair of noes. The trae-off between route reliabilit an consume power is stuie. To our knowlege, this is the first attempt to introuce the concept of route reliabilit an the en-to-en reliabilit versus power trae-off in a network setting. More precisel, this is the first time that network laer routing algorithms an route properties, such as reliabilit an power, are stuie base on the outage probabilit moel at the phsical laer. This moel has the potential to open the oor for a wie-range of research on wireless network reliabilit. In section IV, we introuce the iea of route iversit as a wa to improve route reliabilit b taking avantage of the wireless broacast propert an the inepenence of fae state between ifferent pairs of noes. We give analtical results on improvements ue to route iversit in some simple network topologies an show how route iversit can funamentall change the trae-off between the route reliabilit an the consume power. The iea of route iversit is motivate b the work one in [1], [2], [3], an [4]. Most pervious results have been focuse on two-hop networks, an the analsis has been base on the information theor results for rela channels. References [1], [2], an [3] look at the effect of cooperation among noes in increasing the capacit or reucing the outage probabilit in a faing network. In [1] an [3], the authors escribe several protocols for taking avantage of this tpe of iversit in an a-hoc network. The look at the trae-off between the capacit an the outage improvement in a two-hop network. This analsis ignores the eterministic part of link attenuation ue to the istance between noes an assumes all link faing factors are inepenent an ienticall istribute Raleigh ranom variables. While [4] looks at the asmptotic benefit of rela noes in improving the capacit in an a-hoc network. Their analsis onl takes into account the eterministic part of link attenuation ue to the istance between noes. Their results mainl eal with how the capacit scales as a function of the number of noes in the network. II. Probabilistic Link Moel In this section, we evelop the analtical framework for the probabilistic link moel. This framework etermines the relationship between the probabilit of successful reception, the istance between the communicating noes, an the transmission power in a point-to-point single-user flat Raleigh faing link. We moel the receive signal as: a x + η, 1 where x is the transmitte signal, η is the aitive receive noise, a is the signal attenuation ue to propagation in the wireless point-to-point link, an is the receive signal. We assume the receive noise, η, is zero mean aitive white Gaussian noise with average power of σ 2 η. In general, attenuation, a, epens on the istance between the communicating noes an the fae state of the channel. We use to represent the istance between the communicating noes an f to represent the faing state of the channel. To emphasize this epenence, we express a explicitl as a function of these two parameters: af, x + η. 2 In a sstem with mobile noes an a constantl changing propagation environment, both f an change over time. However, we assume a sstem where f an remain constant for a long perio of time compare to a tpical transmission block length. Furthermore, we assume that the transmission blocks are long enough that coing can be one to average over the Gaussian noise. Given these assumptions, the link between two noes is a simple AWGN channel an the capacit, i.e. the amount of information that can be reliabl transmitte through this channel see[11], is given b: Cf,, x 2, ση 2 log1 + af, 2 x 2 3 To simplif this notation, we ecompose af, into two inepenent components corresponing to the small scale faing an the large scale path loss see [10]. More specificall, we assume: af, 2 f 2 k, 4 where k is the propagation power loss exponent, usuall assume to be between 2 to 4. Simplifing the capacit formula using this form for a, an simplifing the notation b using x 2 σ 2 η, we get: II.A Cf,, log1 + f 2 k 5 σ 2 η Outage Formulation Eq. 5 gives the instantaneous capacit of the point-topoint link efine b 2. An outage event is sai to have occurre see [5] when the transmission rate, R bits/channeluse, is above the instantaneous capacit of the link, i.e. {Outage} ef {Cf,, < R}. 6 One parameter of interest in communication sstems is the probabilit of error at the receiver. An error occurs if the channel is in outage or if the channel is not in outage but there is a ecoing error. In our analsis, we assume that the probabilit of ecoing error is almost zero when channel is not in outage. Uner this assumption, outage is the ominating error event. Hence: P Error POutage We focus our attention on calculating the outage probabilit. Base on the efinition given in 6, the outage probabilit is given b: P Outage P{Cf,, < R} P{log1 + f 2 < R} k P{ f 2 < 2R 1 } 7 k Similar to the approach taken in [1] an other works in this area, we normalize the transmission rate b absorbing its effect into the term. So we efine: norm1 2 R 1 8

3 Equation 7 simplifies to: P Outage P{ f 2 k < 1 norm1 } 9 For the case that faing, f, is ranom an istance,, is known to the transmitter, the outage probabilit simplifies to: f 2 P Outage P < 1 k norm1 k F f 2 norm1 where F f 2 is the CDF of f 2. In our analsis, we moel faing as a Raleigh ranom variable. For Raleigh faing with E [ f 2] µ, the CDF is given b: Hence: F f 2x 1 exp x µ P Outage 1 exp. 10 µ norm1 To simplif the notation, we can absorb the effect of µ into the value of norm1 b efining norm2 as: norm2 µ norm1 For notational convenience, we rop the subscript in the subsequent analsis. The probabilit of successful reception, or equivalentl the reliabilit, for a Raleigh faing link with fixe istance is given b the following simple expression: P Succ, exp k 11 Since is relate to the transmission power, through several levels of normalization to take into account the noise power an the faing parameter, µ, we use this quantit as a prox for the consume power in this paper. Furthermore, we assume that the noise power an the faing parameter are constant across the network, hence, in the network scenario that will be iscusse shortl, the total can be use as a prox for the total consume power. II.B Link Outage-Power Trae-Off Using 11 to fin the expression for link outage probabilit, we have: k P Outage, 1 exp k k where the approximation is vali for high enough values of such that k is small. For a point to point link with a single transmitting an receiving antenna, it is known that the outage probabilit ecas as 1 in the high- regime. See [8] for avance coverage of this topic an the effect of multiple antennas on this relationship. In the next section, we look at a similar relationship for a multi-hop route. III. Reliabilit at the Network Laer A multi-hop route is a sequence of noes through which the information is relae from a source noe, s, to a estination noe,, i.e. Route r 0, r 1,, r h 1, r h, where, r 0 s, r h, an h is the number of hops. We assume the network operates base on a time ivision protocol uner which successive transmissions along a route happen in consecutive transmission slots. Route s, r 1,, r h 1, is ientical to a sequence of h point-to-point links, where for the i th link, rela i 1 is the transmitter an rela i is the receiver, is the transmitte signal-to-noise power, an ri 1 r i is the istance between the noes. We efine the event of successful en-to-en transmission as the event that all h transmissions are successful an the En-to-En Reliabilit is efine as the probabilit of this event. We assume that the faing factors for ifferent links are inepenent an ienticall istribute Raleigh ranom variables. Base on this assumption an using results from 11, the en-to-en reliabilit can be written as: h Reliabilit r 0,r 1,,r h 1,r h exp k r i 1 r i exp k r i 1 r i. 12 The corresponing total amount of power spent for successful en-to-en transmission is: SNR r 0,r 1,,r h 1,r h Total 13 In the subsequent analsis, we use the route reliabilit efine b 12 an the route outage probabilit, ρ, given below interchangeabl. ρ r 0,r 1,,r h 1,r h 1 Reliabilit r 0,r 1,,r h 1,r h. 14 There are three ifferent questions in connection with the en-to-en reliabilit an power that we consier: 1. What is the en-to-en reliabilit if the maximum transmitte power per link is fixe? 2. What is the minimum total power require to achieve a guarantee level of en-to-en reliabilit? 3. What is the maximum en-to-en reliabilit for a fixe total power? The first problem is motivate b the fact that in some cases the transmitte power b each noe might be limite ue to harware constraints or to limit the interference level to other noes. The secon problem is a power allocation problem, where the objective is to minimize the total consume power subject to a guarantee level of en-to-en reliabilit. The last problem is also a power allocation problem, where the objective is to maximize the en-to-en reliabilit of a route subject to a total power constraint. The last two problems might be of interest in cooperative networks where each noe is equall likel to be source, estination, or rela of the traffic. In these scenarios, it is reasonable to minimize the total consume power subject to a reliabilit constraint, problem 2, or maximize the en-to-en reliabilit subject to a total consume power, i.e. problem 3.

4 1. Maximum En-to-En Reliabilit for a Fixe Maximum Transmission Power Per Link Assuming the transmitte signal-to-noise ratio at each link is limite to SNR Link Max, the corresponing route reliabilit can be reail calculate using 12. For a fixe route, r 0, r 1,, r h 1, r h, the en-to-en reliabilit is given b: Reliabilit r 0,r 1,,r h 1,r h exp k r i 1 r i SNR Link Max. 15 Accoring to this expression, the en-to-en reliabilit is a monotonicall ecreasing function of k r i 1 r i. This quantit can be treate as the cost metric for route selection. The most reliable route between two noes is the route that minimizes this cost metric. We refer to route selection algorithm base on this cost metric as the Minimum Outage Route, MOR, algorithm. Lemma 1 The most reliable route between noes s an in a fixe multi-hop wireless network where the faing parameters of ifferent links are inepenent Raleigh ranom variables an the maximum transmitte at each noe is limite to SNR Max Link is the route that minimizes s, r 1,, r h 1, r 0, r 1,, r h 1, r h k r i 1 r i, an the reliabilit of this route is given b Minimum En-to-En Power for a Guarantee En-to-En Reliabilit The problem of minimizing the en-to-en power for a fixe route, r 0, r 1,, r h 1, r h, an fixe en-to-en reliabilit, Reliabilit Min, is formulate as the following constraine optimization problem: min s.t. exp k r i 1 r i Reliabilit Min 16 Since exponential is a monotonicall increasing function, the constraint must be satisfie with equalit at the optimal solution. So, the optimization problem is equivalent to: min s.t. k r i 1 r i lnreliabilit Min. 17 The Lagrangian for this problem is given b: L r0 r 1,, rh 1 r h, λ k r + λ i 1 r i + lnreliabilit Min. The partial erivatives with respect to the transmitte at each intermeiate rela is: L 1 λ k r i 1r i 2 r i 1 r i. Setting these first orer conitions to 0 an solving for the optimal transmitte, we get: ŝnr r0 r 1 λ k r i 1 r i 18 Substituting these into the constraint an solving for the optimal λ, we get: λ k ri 1ri lnreliabilit Min 19 Substituting this back into 18, the optimal transmitte signal-to-noise ratio for each noe is given b: k ri 1ri ŝnr ri 1 r i k r lnreliabilit Min i 1 r i 20 The resulting optimal en-to-en power is given b: ŜNR Total ŝnr ri 1 r i h j1 k rj 1 r j k r lnreliabilit Min i 1 r i k ri 1r i 2 lnreliabilit Min 21 For easier future reference, we state this result in lemma 2. Lemma 2 For a fixe route r 0, r 1,, r h 1, r h, the minimum require total power to guarantee the en-toen reliabilit of Reliabilit min is h k ri 1r i 2 ŜNR Total lnreliabilit Min, which is achieve when the transmission powers are allocate such that k ri 1ri ŝnr ri 1 r i k r lnreliabilit Min i 1 r i From lemma 2, we know that for an route, r 0, r 1,, r h 1, r h, an uner optimal power allocation scheme, the total power require to achieve a esire level of en-to-en reliabilit is a monotonicall increasing function of h k ri 1 r i. Hence, the minimum power route is the route among all possible routes between two noes that minimizes this sum. We refer to this route selection scheme as the Minimum Energ Route, MER, algorithm. Theorem 1 In a multi-hop wireless network where the faing parameters for ifferent links are inepenent Raleigh ranom variables, the minimum power route

5 between noes s an subject to the guarantee en-toen reliabilit of Reliabilit Min is the route that minimizes s, r 1,, r h 1, r 0, r 1,, r h 1, r h k r i 1 r i, an the corresponing en-to-en power is given b Maximum En-to-En Reliabilit for a Fixe En-to-En Power The problem of achieving maximum en-to-en reliabilit for a fixe route, r 0, r 1,, r h 1, r h, an fixe ento-en power, SNR Total Max, can also be formulate as a constraine optimization problem: k r max exp i 1 r i s.t SNR Total Max 22 This problem can be solve using a technique ver similar to the approach use to solve 16. Skipping the etails of the optimization solution, we simpl present the solution to 22 in lemma 3. Lemma 3 For a fixe route r 0, r 1,, r h 1, r h an for a fixe en-to-en power of SNR Total Max, the maximum en-to-en reliabilit is h 2 k ri 1ri Reliabilit Optimal exp, 23 SNR Total Max an the optimal power allocation that achieves this reliabilit is k ri 1 r i ŝnr ri 1 r i SNR Total Max h k ri 1 r i From lemma 3, we know for an route, r 0, r 1,, r h 1, r h, an uner the optimal power allocation scheme the en-to-en reliabilit is a h monotonicall ecreasing function of k ri 1ri. Hence, the maximum reliabilit route is the route that minimizes this sum. We state this result in the following theorem. Theorem 2 The most reliable route between noes s an in a fixe multi-hop wireless network where the faing parameters of ifferent links are inepenent Raleigh ranom variables an the maximum en-to-en power is limite to SNR Total Max is the route that minimizes s, r 1,, r h 1, r 0, r 1,, r h 1, r h k r i 1 r i, an the corresponing en-to-en reliabilit is given b 23. III.A Optimal Reliabilit-Power Curve Two optimization problems that we looke at in the last section, formulate in 16 an 22, are in fact ual problems. Hence, it is not surprising that the cost metric in both cases turne out to be h k ri 1 r i. To clarif this point, we present a graphical illustration of the relationship between the en-to-en reliabilit an power uner the optimal power allocation scheme. For an fixe route, ifferent power allocation schemes result in ifferent en-to-en reliabilit an consume power. If we were to characterize each power allocation scheme onl b the total consume power an the resulting en-to-en reliabilit, each allocation scheme coul be represente b a point in the two imensional plot of the en-to-en reliabilit vs. the total power. Certain allocation schemes are optimal, i.e. either minimize the total power consume to achieve a guarantee en-to-en reliabilit or maximize the en-to-en reliabilit for a fixe consume power. In problem 2, we foun the optimal power allocation to minimizes the total power subject to a guarantee en-toen reliabilit. Graphicall, this optimization correspons to moving along the horizontal line in figure 1 an fining the allocation scheme that minimizes the total consume power subject to an en-to-en reliabilit of Reliabilit min. We foun that the reliabilit an power corresponing to the optimal allocation are relate b the following relationship: h 2 k ri 1ri ŜNR Total lnreliabilit Min 24 In problem 3, we foun the optimal power allocation to maximize the en-to-en reliabilit for a given en-to-en power. This correspons to moving along the vertical line in figure 1 an fining the allocation scheme that maximizes the reliabilit for SNR Total Max We foun that the resulting en-to-en reliabilit for this optimal allocation is: h 2 k ri 1ri Reliabilit Optimal exp 25 SNR Total Max Clearl, the curve specifie b 24 an 25 are ientical. The set of optimal power allocations can be represente b a single curve in the two imensional plot of the en-to-en reliabilit vs. total power as shown in figure 1. We refer to this curve as the Optimal Reliabilit-Power Trae-off curve. III.B Route Outage-Power Trae-off Similar to the case of a point-to-point link, it is insightful to look at the trae-off between route outage an the consume power. This tpe of analsis gives insight to the require power to achieve a esire en-to-en outage level or how fast the en-to-en outage acas with power. For the case that the maximum transmitte power at each link is limite to SNR Max Link, 15 gives the en-to-en reliabilit. We get more insight b looking at the route outage probabilit efine in 14. Writing 15 in terms of the outage probabilit, ρ, we have: 1 ρ exp k r i 1r i SNR Max Link k r i 1r i ln1 ρ. SNR Max Link,

6 Reliabilit Min Route Reliabilit vs. Power Minimum Power s r0 r1 s r0 r1 Reliabilit Maximum Reliabilit Figure 2: Simple Route Figure 3: Diversifie Route SNR Total Max SNR Total Figure 1: Route Reliabilit vs. Power For small values of ρ, we can use the approximation of ln1 ρ ρ to simplif this relation to: ρ k r i 1r i. 26 SNR Max Link The relationship between reliabilit an power with optimal power allocation, i.e. the optimal reliabilit-power curve iscusse in the last section, is given b 25. Writing 25 in terms of the route outage probabilit an following a similar approach, we fin: SNR 1 Max Link ρ h k ri 1r i 2 SNR Total. 27 From 26 an 27, we observe that route outage ecas as an SNR 1 Total, respectivel, in the high- regimes. It is not surprising that we observe this tpe of relation as these relationships are ver similar to what we observe in the first section for a point-to-point link. In the last section, we see how iversit at the route level can funamentall change this trae-off. IV. Route Diversit So far in this paper, we have taken an approach in which a multi-hop route is treate as a single en-to-en pipe. Uner this scenario, a successful en-to-en reception requires all point-to-point transmissions to be successful. Even in this ver limite scheme, it possible to improve the en-to-en route reliabilit b taking avantage of the wireless broacast propert an the inepenence between ifferent Raleigh faing links. This is the motivation behin the Route Diversit iea that we introuce in this section. To clarif this iea, let s look at a simple example. Assume that in the network shown in figure 2, the most reliable route is selecte as shown. Without iversit, a successful en-to-en relaing require 3 successful point-to-point transmissions. We refer to this strateg as the Non-Diversifie Routing Scheme. Due to the broacast an the faing nature of the wireless propagation environment, the information transmitte b s ma be receive correctl b, for example, r 1 while r 0 fails to receive that information. B accounting for this possibilit, as shown in figure 3, can receive the information irectl from s in the first transmission slot, from r 0 in the secon transmission slot or from r 1 in the thir transmission slot. We refer to this routing scheme as the Diversifie Routing Scheme. In the subsequent analsis, we assume that the maximum transmitte b each noe is fixe. This is the quantit we represente b SNR Max Link in the previous section. However, for simpler notation, we rop the subscript an enote this quantit as in the analsis that follows. Our aim is to fin how the en-to-en outage probabilit varies with the maximum transmitte power level uner the iversifie routing scheme an compare the result with the relation given in 26. To further simplif the analsis, we also assume that the path-loss exponent, k, is 2. Man of our results can be extene to have k as a parameter. We stu the benefit of route iversit b looking at two examples. IV.A Example 1: Two Hop-Networks This example focuses on a 2-hop network constructe b uniforml placing a rela noe insie a circle with raius of 1 centere at the mi-point between s an, figure 4. Base s 1 r 3 Figure 4: 2-Hop Disk Network on the results from Lemma 1, the Minimum Outage Route MOR in this network is s, r,. The non-iversifie outage probabilit for this route is: ρ s,r, Non Diversifie 1 P Succ 1, P Succ 2, where the approximation is vali in the high- regime. In the iversifie scheme, successful relaing requires either a successful irect transmission, {s }, or successful multihoping, {s r} followe b {r }. The reliabilit of this route is given b: Reliabilit s,r, Diversifie P Succ 3, + 1 P Succ 3, P Succ 1, P Succ 2,, exp 2 3 +

7 1 exp 2 3 exp where the approximation is vali in the high- regime. Hence, the route outage probabilit is simpl: ρ s,r, Diversifie This expression shows that the en-to-en outage probabilit ecas as 2 for the iversifie routing scheme. This is a significant improvement over the 1 eca observe in 28 in the absence of iversit. It shoul be note that both 28 an 30 are vali for an two hop network. Furthermore, this gain is achieve through route iversit an oes not require an coing, ARQ, or transmitter sie information. IV.B Example 2: Disconnect Probabilit Consier a network in which noes are istribute on a line an the istance between neighboring noes are inepenent exponential ranom variables with parameter λ. Assume that the estination is locate a large number of hops awa to the right of the source noe. Although it ma be possible to calculate the exact en-to-en outage probabilit as a function of the maximum transmitte power level, the location of the rela noes, an the number of hops, we will take a ifferent approach in analzing the benefit of route iversit in this network. We efine the isconnect event for a noe as the event that the noe is not connecte to an noe locate to its right. Without iversit, this event is equivalent to the event that the link between the noe an its immeiate right neighbor is in outage, see figure 5. With iversit, however, this event is equivalent to the event that all the links between the noe an all the noes to its right are in outage, see figure 6. Clearl the secon event has a lower probabilit as it is a subset of the first event. We are intereste in analticall calculating these probabilities an observing how these quantities epens on the. line segments are inepenent ranom variables, see [7] for etails. This approximation is prefect for the erivation that follows as we will take the limit of δ 0 to get the esire result. For small values of δ, let s efine the isconnect event for segment locate at istance mδ awa from the transmitter as the event that the information is not receive b an noe locate in the line segment mδ, m + 1δ]. This event inclues both the case that there is no noe in this line segment or there is a noe an the transmission fails ue to ba faing. The probabilit of this event can be calculate as: P Disconnect mδ, 1 P Succ mδ, λδ where P succ, is given in 11. Let P Disconnect x,, δ, be the probabilit that the information transmitte b a noe locate at location 0 is not receive b an noe between x, ] where this segment is broken own into segments of length δ. This probabilit can written in terms of P Disconnect mδ, calculate above as: P Disconnect x,, δ, δ P Disconnect iδ, i x δ Taking the natural log of both sies, we get: lnp Disconnect x,, δ, Taking the limit δ 0: δ lnp Disconnect iδ, i x δ δ ln1 P Succ mδλδ. i x δ lnp Disconnect x,, x λ P Succ l, l 32 where we use the approximation of ln1 x x for small values of x in the last step. For the case when the path-loss exponent k 2, the above integral can be easil calculate base on the complementar error function: 0 Figure 5: Disconnect, No- Diversit 0 Figure 6: Disconnect, with Diversit lnp Disconnect x,, λ λ π 2 x erfc e t 2 t. x erfc 33 For a given network realization, i.e. given the istance to the neighboring noe is r, the probabilit of the isconnect event without iversit is given b: P No Diversit Disconnect, r 1 P Succ r, exp 2 r 31 Calculating the probabilit of the isconnect event with iversit requires a ifferent approach. We start b iviing the line into segments of length δ. For small values of δ, the number of noes in a line segment of length δ is approximatel a Bernoulli ranom variable, i.e. there is a noe in a segment with probabilit λδ or there is no noe with probabilit 1 λδ. Furthermore, the number of noes in non-overlapping Let P Disconnect be the isconnect probabilit, i.e. the probabilit that a noe is not connecte to an noe to its right. Assuming an infinitel long line, this probabilit is obtaine b evaluating 33 for x 0 an. We have: Hence: lnp Disconnect lnp Disconnect 0,, π λ2 2 P Disconnect exp π λ Comparing 31 an 34, it is clear that without iversit, the isconnect probabilit ecas exponentiall with 1 while

8 with iversit, the isconnect probabilit ecas exponentiall with. It shoul be note that we arrive at this result without making an high- assumption. Furthermore, result is vali for an network realization, which as to its significance. V. Summar an Conclusions We stuie the problem of route reliabilit in a multi-hop wireless network. Our analsis starte b looking at the reliabilit of a point-to-point communication link. Base on this analsis, we propose a new probabilistic wa of looking at a wireless link. We use this probabilistic moel to look at the reliabilit in a wireless network. In that context, we first efine an analze the reliabilit for a fixe route an then evelope algorithms for fining the optimal route between a source-estination pairs of noes. We looke at three ifferent formulation for the routing problem: fining the most reliable route for a fixe maximum transmitte per link, fining the most reliable route for a fixe en-to-en power, an fining the minimum power route for a guarantee ento-en reliabilit. We showe that the last two problems are ual of each other. Base on this ualit, we foun the optimal trae-off curve between the en-to-en reliabilit an the en-to-en power consumption. It was shown that the trae-off between the en-to-en reliabilit an the consume power in a route is ver similar to the trae-off between the transmission power an reliabilit in a link. The iea of route iversit was introuce as a wa to improve the en-to-en reliabilit b taking avantage of the wireless broacast propert an the inepenence of the fae parameters between ifferent pairs of noes. We gave analtical results for improvements ue to route iversit in some simple network topologies. The moel propose in this paper can open the oor to a new area of research on communication reliabilit at the network laer. The trae-off among ifferent route properties, such as the en-to-en reliabilit, the ela, or the total consume power shoul be stuie to help raw a better picture of the actual limits of communication in a multi-hop wireless network. In this context, route iversit appears to have the potential to funamentall change these trae-offs. [8] Lizhong Zheng, D.N.C Tse, Diversit an multiplexing: a funamental traeoff in multiple-antenna channels, IEEE Transactions on Information Theor, Ma 2003 [9] D. Tse, P. Viswanath, Funamental of Wireless Communications, working raft. [10] T. S. Rappaport, Wireless Communications: Principles an Practice, Prentice Hall, [11] John G. Proakis, Digital Communiation, Forth Eition, Mc- GrawHill, 2001 [12] Amir E. Khanani, Cooperative Routing in Wireless Networks, Master s Thesis, MIT, Ma 2004 References [1] J.N. Laneman, Cooperative Diversit in Wireless Networks: Algorithms an Architectures, Ph.D. Thesis, Massachusetts Institute of Technolog, Cambrige, MA, August [2] J.N. Laneman, G.W. Wornell, Energ-efficient antenna sharing an relaing for wireless networks, Wireless Communications an Networking Conference, Sept. 2000, Vol.1, pp 7-12 [3] J.N. Laneman, G.W. Wornell, Distribute space-time-coe protocols for exploiting cooperative iversit in wireless networks, IEEE Transactions on Information Theor, Oct [4] M. Gastpar, M. Vetterli, On the capacit of wireless networks: the rela case, Proc. INFOCOM 2002, Vol. 3, pp [5] E. Biglieri, J. Proakis, S. Shamai, Faing channels: information-theoretic an communications aspects, Information Theor, IEEE Transactions on, Volume: 44, Issue: 6, Oct [6] L.H. Ozarow, S. Shamai, A.D. Wner, Information theoretic consierations for cellular mobile raio, IEEE Transactions on Vehicular Technolog, Ma 1994 [7] R. Gallagar, Discrete Stochastoc Processes, Kluwer Acaemic Publisher, 1996