Delay Limited Capacity of Ad hoc Networks: Asymptotically Optimal Transmission and Relaying Strategy

Transcription

1 Delay Limite Capacity of A hoc Networks: Asymptotically Optimal Transmission an Relaying Strategy Eugene Perevalov Lehigh University Bethlehem, PA 85 eup2@lehigh.eu Rick Blum Lehigh University Bethlehem, PA 85 rblum@eecs.lehigh.eu Abstract The elay limite capacity of an a hoc wireless network confine to a finite region is investigate. A transmission an relaying strategy making use of the noes motion to maximize the throughput is constructe. An approximate expression for the capacity as a function of the maximum allowable elay is obtaine. It is foun that there exists a critical value of the elay such that: ) for values of the elay below critical, the capacity oes not benefit appreciably from the motion, 2) for moerate values of the elay above critical, the capacity that can be achieve by taking avantage of the motion increases as 2/, ) the epenence of the critical elay on the number of noes is a very slowly increasing function n / ). Finally, asymptotic optimality of the propose strategy in a certain class is shown. I. INTRODUCTION A hoc wireless networks [],[2] represent a promising new technology in communications that is currently receiving significant attention ue to several successful research programs. In particular, one early effort was the DARPA packetraio network program [],[]. Important applications inclue range from rescue operations to collaborative computing in mobile environments to istribute control an comman systems to ubiquitous personal communication systems. Many researchers in the fiel of wireless communications believe that, ue to their unique features, a hoc networks will play an increasingly important role in the near future. Several attractive features of a hoc networks are: ) ease of eployment ue to the absence of require infrastructure; 2) potentially low cost ue to omission of large-scale harware such as base stations; ) very high egree of flexibility. An a hoc network is a collection of in general mobile) noes that can exchange information via a wireless channel characterize by the absence of any fixe infrastructure or hierarchy. Present an future esigners of such networks have to meet multiple challenges create by the networks very nature. Significant ifficulties arise at all levels of such networks: physical, MAC meium access control), an the network layer. The main source of such ifficulties is the same as the source of their potential avantages: essential lack of inherent organization in the network ue to the absence of infrastructure. The latter feature implies that all questions of control, ue to lack of any coorinating center, have to be aresse by the participating noes themselves. Mobility makes the situation especially complex since, uner such conitions, the topology of the network is constantly changing, an all the control ecisions have to reflect that. In this paper, we stuy the maximum information transport capacity of an a hoc wireless network. More precisely, we are intereste in the relationship between the en-to-en elay an the capacity. This work is motivate by the results of Gupta an Kumar [5] an Grossglauser an Tse [6] see also [8] for a ifferent approach). In [5], it was shown that, in an a hoc network of size n, the capacity per noe goes own with n thus making large networks impractical. In [6], it was emonstrate that, if the noes mobility is taken avantage of, the effect of ecreasing capacity can be overcome. The price one has to pay for such a ramatic increase in capacity is an en-toen elay no smaller than the time scale characterizing the noes motion. In this paper, we make an attempt to quantify the relationship between the maximum allowable elay an the transport capacity. First, we establish an upper boun on the elay limite capacity within the class of one relay strategies in the spirit of [6]. We fin an analytic expression for the upper boun an use it to get a simple approximate result vali for moerate values of the maximum elay. We then construct a transmission an relaying strategy that achieves the above upper boun asymptotically. Our approach is base on the combination of the iversity routing iea of Grossglauser an Tse [6] an the multipath routing methoology of Tsirigos an Haas [] that relies on the iversity coing approach from [9]. Namely, just as in [6] a source noe transmits to its current nearest neighbor at each time slot allocate for transmission. The ifference is that, in our approach, we o not sen a packet to its estination via one relay noe. Instea, after aing reunant information, we split the resulting enlarge packet into many blocks an sen the blocks to the estination via ifferent relay noes. As a result, in orer to achieve the esire level of service measure as the probability of correct reconstruction of the message by the estination noe within time from the

2 moment of the message origination), one nees to employ a certain reunancy level which in turn irectly affects the maximum capacity. We calculate the require reunancy level approximately. The main result that we obtain is that there exists a critical value of the elay such that for elays below the critical value, the gain in the capacity that can be achieve by making use of the motion is negligible. In other wors, for such elays the result of [5] applies: the capacity of the network goes own roughly as n. For the elays above the critical, the capacity benefits from the motion, an, for not very long elays, increases approximately as 2/. It is interesting to note that the value of the critical elay increases only very slowly as n / ) with the number of noes n which is a welcome feature. II. MODEL AND PREVIOUS RESULTS The moel we aopt is similar to to those use in [5] an [6]. The network consists of n noes locate on a sphere of area A. All noes are mobile, an we assume that the motion of any noe is escribe by the same stationary ergoic ranom process such that at each time there is no preferre irection. The trajectories of all noes are assume to be inepenent an ientically istribute. We assume that every noe i has an infinite amount of ata for its estination fi), an that the source-estination association oes not change with time. In the transmission moel we use same as in [6]), a noe i is capable of transmitting W bits/sec to noe j at time t if P i t)γ ij t) N + L k i P kt)γ kj t) >β, where P i is the transmit power of noe i, γ ij is the channel gain from noe i to noe j, N is the backgroun noise power, L is the processing gain of the system, an β is the SIR requirement for successful communication. The channel gain is assume to epen only on the shortest istance D ij between the respective noes as in γ ij t) = Dij α t), where α is a parameter greater than 2. At any time a scheuler ecies which noes transmit bits an the corresponing power levels. The objective is to ensure a high average throughput for every source-estination pair. Let us enote by M i t) the number of bits that the estination fi) receives in time slot t. We say that an average throughput of C is feasible if for every source-estination pair i, lim inf T T T M i t) C. t= Gupta an Kumar [5] emonstrate that, if a noe can transmit W bits per secon over a common wireless channel, there exist constants c an c such that lim Pr n { Cn) = cw n log n is feasible } =, an { } lim Pr Cn) = c W n is feasible =, n i.e. up to a factor of log n the throughput per source-estination pair goes to zero as W n. Grossglauser an Tse [6] constructe a scheuling policy accoring to which, in any time slot t, n S = θn where θ is the sener ensity parameter to be etermine) noes are esignate as seners an the remaining n R noes as potential receivers. Each sener noe then transmits packets to to its nearest neighbor among potential receivers using unit transmit power. Among the n S sener-receiver pairs the policy retains those for which the interference generate by the other seners is low enough so that a successful transmission is possible. If N t is the number of such pairs then, as was shown in [6], E[N t ] lim = φ>. n n III. UPPER BOUND ON DELAY LIMITED CAPACITY A. Maximum achievable capacity Let us enote by C the per noe capacity of the network achieve by the one relay noe approach [6] in the absence of en-to-en elay constraints. We now fin an upper boun on the capacity in the presence of a uniform en-to-en elay constraint. Theorem : In the class of relaying strategies where each packet goes through at most one relay noe, the maximum capacity C of an a hoc network uner the constraint that the en-to-en elay not excee is upper boune by C C u) C γ p), ) for sufficiently large n, where p) is the ensemble average of the probability that two noes come within range of irect transmission in a time not exceeing, an γ is the corresponing capture probability. Proof: Since the relative contribution of irect source to estination transmissions to the total capacity is negligible of the orer n ), we will ignore it in the proof. Let us concentrate on a fixe source noe i with its associate estination fi). Suppose noe k is the current potential recipient of a packet intene for noe fi). If it were known in avance that noe k woul come within transmission range of fi) in time, an the transmission to fi) woul be successful, the transmission to k woul have to take place. Otherwise, the transmission from i to k woul be useless the packet woul not reach its estination in time. So, in the ieal case of complete knowlege of all the trajectories the capacity of C γ p) woul be achieve. To en the proof, we note that state expression for the upper boun formally approaches γc as.inthe following, we will see that, for an optimally chosen transmission range form relay to estination, γ as. Moreover for large values of allowable elay, the two given noes come close on multiple occasions, thus making the effective probability of capture approach unity even faster.

3 B. Probability of close range transmission We woul like to fin an approximate expression for the ensemble average pt) of the probability of a close range transmission within time t. More precisely, we wish to fin the ensemble average of the probability that a given noe comes within istance r or less from another fixe noe in an interval of time of length t. It turns out to be easier to fin the probability of the opposite event: qt) = pt). Let us enote by q x, y t) the probability of two noes not coming within range r provie that at time the two noes in question are locate at points x an y respectively. Then the the ensemble average qt) can be efine as the expectation of q x, y t) with respect to all possible uniformly istribute on the sphere) starting points x an y which we enote as qt) E U x,u y [q x, y t)]. 2) Due to symmetry of the sphere, we can concentrate on the relative motion of the noes by assuming that the secon noe is at rest at the north pole at all times. Then the above efinition can be replace by qt) E U x [q x t)], ) where q x t) is the probability that the two noes will not come within range r in time t provie that at time the first noe is at point x an the secon is at the north pole. Now fix a point in time s such that < s < t. Then provie the noes i not come within r between an s an the first noe is at the relative to the north pole) position x, the probability of not coming within range r between an t can be written as q x t) =q x s)q x t s), ) an taking expectations of both sies in ) over all initial points an assuming the motion is inepenent over nonoverlapping increments an inepenent increment process, a common assumption) we obtain E U x [q x t)] = E U x [q x s)]e P x [q x t s)], 5) where E P x stans for the expectation with respect to the istribution of the position x of the first noe at time s provie the noes i not come within range between an s. Now let f x s) be the probability ensity function characterizing the above probability istribution. We will write it as f x s) = A + h xs)), 6) where h x s) is the ifference from the uniform istribution such that h x ) =. Substituting 6) into the expression for the expectation over all possible starting points, we arrive at where E P x [q x t s)] = qt s) + ɛs, t s), 7) ɛs, t s) = A q x t s)h x s) x accounts for the ifference of the initial istribution in 7) from the uniform. Now, substituting 7) into 5), we obtain qs) qt s) + ɛs, t s)) = qt). 8) Note that in the absence of the term ɛs, t s) the solution of 8) with the proper initial conition woul be qt) = e λt for some constant λ. Writing qs) = e λs + δs) an taking the limit t s + in 8) we can arrive at the following orinary ifferential equation for δs): δ s) = λδs)+zs)δs)+e λs zs), 9) where zs) = λh x s), with x being a point at a istance r from the north pole. The equation 9) with the initial conition δ) = has the solution δs) =e λs+ s s zs )s e s zu)u zs )s. ) Thus, for qt) we obtain qt) = e λt +e t t zs)s e s ) zu)u zs)s. ) The latter expression involves the function zs) whose exact form epens on the particular moel of ranom motion of the noes. However, the fact that zs) = λh x s) combine with the observation that h x s) < for all s allows us to justify the approximate expression for the ensemble average of the probability of no irect nearest neighbor transmission within time t vali for moerate times t such that for λt we have pt) e λt, 2) where the parameter λ characterizes the noes mobility. In orer to estimate the value of λ, we note that, from ), that λ = pt). t t= To evaluate λ from this efinition, we must calculate the number of noes that enter a circle of raius r uring a ifferential time interval assuming uniformly istribute noes over a sphere of raius R which are moving at spee v. Given this interpretation the parameter λ can be shown to be equal to λr) = 2vr A = vr 2πR 2, ) Introucing a imensionless parameter x r/r we can rewrite the above expression as λx) = v x, ) 2πR Let us also introuce notation for the average length of time uring which the ientity of a nearest neighbor of a noe remains unchange. We will enote such an average by τ. Obviously, in the orer of magnitue, τ is equal to the time it takes a noe to travel an average istance between the noes so that A cr τ n v = 2 n v, 5) where c is some constant epening on the etails of the motion moel.

4 C. Probability of capture A typical information block on its way from the source noe to the estination performs two hops: from the source to a relay noe, an from the relay to the estination. We escribe these two stages in turn below. For simplicity we assume, following [6], that in o time slots the first stage is effecte, an in even time slots the secon. ) Source to relay: The source to relay transmission is effecte to the nearest neighbor as escribe in [6]. There, it was shown that the capture probability approaches a finite number for very large number of noes n. Let us enote this number η. The fact that η is substantially less than reuces the elay limite capacity by effectively increasing τ the average time between two successful transmissions in the first stage. In this paper, we o not go into the etail of transmission policy at this stage postponing it to the future work. 2) Relay to estination: In the secon stage our transmission policy is to transmit to the estination once it is at the istance r from the relay. It is clear that choosing higher value of r increases the probability of coming into the range within limite time an, at the same time, ecreases the probability of capture. So one can hope to be able to choose the optimal value of r given the parameters of the system. The probability of coming in the range was stuie in the previous section. Here, we approximately compute the probability of capture. We fix the parameter α escribing the ecay of the signal in space to α =. First, note that if any other transmitter is at a istance no more than r = rβ / from our estination then the capture is impossible. Let us enote by A the event that none of the other transmitting relays are within istance r or less from our estination. The probability of A can be compute as ) βr 2 q P A) = R 2, 6) where q is the number of simultaneously transmitting relays which is on average equal to q = p) θηn, where θ is the sener ensity in the first stage. Using the fact that r R as a consequence of large n) we can write an approximate expression for 6): P A) =e /) βnθη p) x 2, 7) where x r/r. If A is true, i.e. none of other transmitting noes is within istance r from our estination, the capture may still be impossible ue to total interference power from all the other transmitting noes. Since the number of such noes is large proportional to n), we can approximate the istribution of the interfering power by the normal one with the mean equal to qp an stanar eviation equal to qstp ), where P an StP ) are the mean an stanar eviation of the interfering power of one other noe, respectively. If the transmitting power of each noe is equal to, we can calculate P an StP ) approximately as an P = R 2 r 2, StP )= 2 Rr. So, given that A is true, the probability that the transmission is successful, can be compute as Φ βr q R 2 r 2, 8) q 2 Rr with Φ.) enoting the stanar normal cf. Finally, combining 7) an 8), we obtain an approximate vali for large n) equation for the probability of capture γx): γx) =e /) βnθη p) x 2 Φ 2 β p) θηn x 2 ) β /. p) θηn x 9) It is convenient to introuce the imensionless parameter v 2πR w which measures the elay in natural units counting how many times a noe coul traverse the spherical region if it move in a straight line. Using this new parameter an remembering that, in the first approximation, the quantity p) is equal to λ, we can rewrite 9) as γx) =e /) βθηnwx Φ 2 βθηnw x ) β /. 2) θηnw x /2 IV. ASYMPTOTICALLY OPTIMAL RELAYING STRATEGY A. Strategy escription We now escribe our relaying strategy. The goal is to get close to the maximum capacity. We achieve it by using a ifferent approach from that in [6]. Specifically, we sprea the packet traffic between many noes. Namely, after aing reunant information, we split the resulting packet into blocks an sen the latter via ifferent routes relay noes). We employ the coing scheme use in [] in which Y extra bits are ae to the packet of X information bits as overhea thus resulting in B = X + Y bits that are treate as one new network-layer packet. The aitional bits are calculate as a function of the original X bits so that the original bits can be correctly reconstructe from any subset of the B bits of size no less than X. The quantity z = B 2) X is the overhea factor. Our strategy consists of splitting the resulting B-bit packet into m equal size blocks an sening them via ifferent consecutive) relay noes. Similarly, the blocks are communicate from the relays to the estination.

5 B. Approximate capacity calculation The key question we have to answer is, given the maximum allowable elay, how o we select the overhea ratio in orer to achieve the require probability close to ) that a correct message is receive in the require time. If we are able to fin the minimum sufficient overhea ratio z min then obviously the capacity such a strategy can achieve will be equal to C = C. 22) z min In this paper we will limit ourselves to the case when, for each B-bit packet, exactly one block is sent to the estination via each relay noe. Observe that, in establishing the upper boun, we assume the perfect knowlege of the future trajectories of the noes. Thus we coul get the information to the corresponing estination in time % of the time. Here, ue to the lack of such knowlege, all we can aim at is to get the packets to the estination in time with some fixe although arbitrary) average probability. Note though that if % on-time elivery were neee that coul be achieve by combining our relaying strategy with occasional multihop transmissions. The asymptotic result that we report below woul still hol. Let us fix the esire level of service Q, the average probability that a packet will reach the estination within time. Our task is to etermine the minimum overhea ratio such that the esire level of service Q can be achieve. Using the results of [], we can write an approximate expression for the probability of successful reconstruction of a packet at the estination within time t as P m t) = 2 + m 2 erf i= γp it i ) m/z +/2 m 2 i= γp, it i ) γp i t i )) 2) where m is the number of blocks the packet is split into, p i t i ) is the probability the i-th block reaches the estination within time t i, an z is the employe overhea ratio. We eman that the ensemble average of P m t) be equal to Q so that 2 + m ) 2 i= erf γp it i ) m/z +/2 m 2 i= γp = Q. it i ) γp i t i )) 2) The LHS of 2) seems to be har to evaluate, so we simplify it by first noting that m i= γp it i ) γp i t i )) m/ an hence, if we eman that erf m i= γp it i ) m/z +/2 ) = Q 25) then the resulting level of service will be no less than Q. Next, we assume that the probabilities p i t i ), i =,...,m,areinepenent an ientically istribute 2. Uner this assumption, t i is the amount of time it has left until the ealine t. Since the transmission from the source to the i-th relay for i> happens later than time, t i <tunless i =. 2 This assumption is a reasonable approximation provie τm. ) TABLE I µq) FOR SOME LEVELS OF SERVICE Q. Q µ the sum m i= p it i ) is approximately normally istribute with mean m i= p it i ) an stanar eviation not exceeing since the stanar eviation of pi t i ) is no more than /2 for all i). Therefore, the argument of erf ) in 25) is approximately normally istribute with the mean of m i= µ = γ p it i ) m/z +/2 an stanar eviation s / 2. This allows us to fin the value of µ such that 25) hols from π 2 + ) 2 erfx) e x µ)2 x = Q. 26) Table I shows the numerical values of µ for some fixe levels of service Q. We can now fin the value of z min by solving the following optimization problem an equating the objective value to µq): m i= max γ p it i ) m/z +/2 = µq). 27) m The above equation states that we want to reach the esire level of service Q for at least one optimal) value of the number of blocks m which we enote m. The maximization over m will lea to the smallest value of z min as illustrate in Fig.. For further convenience, let us enote m i= gm, r) γ p it i ) m/z +/2 so that 27) reas max gm, z) =µq). 28) m We can now substitute the approximate expression 2) for p i t i ) into 27). Strictly speaking, since the time perios uring which the ientity of the nearest neighbor of a noe is unchange are ranom variables, we woul have to take the averages over the corresponing istributions for calculating the quantities p i t i ) for all values of the inex i. However we can use the fact confirme by the results) that the typical number m of the blocks into which a packet is split is large for large n an, therefore, we can get a goo approximation by replacing the ranom variables by their means. In this way, we obtain gm, z) = γm e λ m i= ei )λτ ) m/z +/2 29) Obtaining a close form solution of 28) still seems to be a ifficult task. Therefore, we use yet another approximation. For a large total number of noes n, the optimal value of m

6 g.5.5 z>z min z = z min z<z min µ = m Fig.. gm, z) as given by 29)) as a function of m for ifferent values of r an Q =.99. We can see that for z<z min the require level of service µ =2.) is not achieve for any m, while for z>z min it is achieve or exceee for a whole range of values of m. is going to also be large. Thus we: ) replace the sum by an integral; 2) replace m/z in 29) by m/z; ) rop terms which are small compare to m. As a result of the above approximations, 29) becomes 2 gm, z) =γ m ) e λ ) m γz λτ emλτ ). ) Differentiating ) with respect to m, an Taylor expaning the result to first orer in the small parameter λτ, we obtain the value m maximizing gm, z) for a fixe z. m = 2 e λ ) ). ) λτ γz Now substituting ) back into ) we obtain gm,z)= γ e e λ λτ λ ) ). 2) γz Finally, solving 28) for z using 2) we arrive at the following expression for z min : z min = e λ ) 2 µ e λτ λ, γ γ ) provie the expression in the outmost brackets is positive. Otherwise, the require level of service Q cannot be achieve. Thus, the capacity uner the constraint that the en-toen elay not excee with probability no less than Q is approximately equal to C = γ e λ ) 2 µq) e λτ λ C, γ ) if the expression in the brackets is positive. Otherwise, the require level of service Q cannot be achieve, an we can consier the corresponing capacity to be equal to. C. Optimal transmission range Analyzing eq. 2), we can see that, for a given level of service Q, there exists a critical elay cr such that: For < cr, the transmission strategy iscusse above is unable to achieve the require level of service. For > cr, the capacity increases with as illustrate in Fig. 2. We can estimate the value of cr from ) by equating C to as ) 2 µq) τ ) cr = γ λ 2. 5) Expaning ) to the first orer in λ, we see that C as a function of elay behaves approximately as { C if <cr =, 6) C γλ cr ) if > cr provie λ. It is interesting to compare 6) with the corresponing approximate expression for the upper boun C u) : C u) = γλ, which can be obtaine from 6) by setting cr =. So far we have not chosen the transmission range r. Recall that the quantities λ an γ in 5) an 6) epen on r or its imensionless version x) as shown in ) an 2), respectively. So we can choose the value of x such that the capacity is maximize for any fixe value of the elay. By inspection of 6) we can see that the capacity for values of the elay greater than cr epens linearly on the combination γx)λx). At the same time, the value of cr from 5) is inversely proportional to a positive power of the same combination. Therefore, one can maximize the capacity by choosing the value of x so that γx)λx) is maximize. In orer to achieve this goal, note that, as x increases, the first factor in the expression 2) for γx) rops much faster than the secon one which stays at nearly until the argument of Φ ) becomes roughly less than 2 Φ2) =.977, Φ.5) =.9). Therefore we can try to maximize γx)λx) by setting the secon factor in γx) to an later verifying that the resulting argument of Φ ) is large enough to justify the approximation. Thus we nee to maximize the function v 2πR xe /) βθηnwx with respect to x which yiels the optimal x equal to x opt = βθηnw, 7) so that the argument of Φ ) in the expression for γx) is equal to 2 an our approximation is well justifie. Substituting 7), ) an 2) into 5), an solving the resulting equation for cr, we obtain: cr = µ ) 6 7 π 2 e βθη) 2 7 c n R v, 8)

7 or, in the natural units: µ w cr = ) 6 7 π 2 e βθη) 2 7 2π c n. 9).8 We also get: γx opt )λx opt ) = v 2 2 π 2 e βθηnr 2 ). ) Thus the behavior of the capacity as a function of for moerate values of is such as shown in Fig R n.8 Fig.. The ratio R = C /C u) as a function of number of noes n. Note that the ratio approaches rather slowly. This example uses Q =.99 an λ =.2..6 y Fig. 2. The ratio y = C /C as a function of elay. Several observations are now in orer. The epenence of the critical elay cr on the number of noes n turns out to be very slow n / ) which is a esirable feature. Beyon cr, the epenence of the capacity on is as 2/ so significant capacity is reache relatively quickly. As can be seen from 7), the optimal transmission range ecreases as n /, i.e. for large values of n it can be significantly larger than the typical internoe istance that scales as n /2. It is also interesting to note that the optimal transmission range ecreases with as /. From the above expressions, we can easily fin that, for the optimal value of the transmission range, λτ = c π 6 βθηw ) n 5/6, an the upper boun on the elay limite capacity C u) can be approximate as C u) = e λ )γc.soifwefix C u) an increase the number of noes n we see that the ratio C /C u) approahes. Thus we have prove the following theorem. Theorem 2: There exists a transmission strategy that asymptotically achieves the upper boun C u) on the elaylimite capacity of an a hoc network. We shoul note that this ratio approaches rather slowly as shown in Fig.. V. CONCLUSION In this paper we have conucte a preliminary exploration of the problem of the influence of the en-to-en elay on the maximum capacity of a wireless a hoc network confine to a certain area. Aiming at general results, we have mae a number of simplifying assumptions. Thus, we aopte a totally ranom moel of motion ignoring both the etails of the corresponing istributions an possible patterns in the noes motion. We also limite ourselves to one relay class of strategies in the spirit of ref. [6]. Confining the analysis to the above class of strategies we foun an expression for the upper boun of the elay limite capacity that involves the ensemble average of the probability of two noes coming within certain range within the maximum elay time an the corresponing capture probability. In orer to establish that upper boun we assume the perfect knowlege of the future trajectories of the noes. We then obtaine a general expression for that ensemble average which epens on the transmission range from the relay to the estination. Next, we foun an approximate expression of the average probability of success of that transmission as a function of the transmission range. We then proceee to construct a relaying strategy that woul asymptotically achieve the upper boun. We use the iversity coing approach in combination with the seconary iversity routing of [6] in orer to asymptotically achieve the upper boun for this class of strategies in the absence of any information about the noes motion. We also mae use of a number of approximations that allowe us to en up with close form expressions. Analyzing the epenence of the resulting capacity on the transmission range, we foun an approximate expression for the optimal range that maximizes the capacity for any fixe value of the elay. We foun that, for moerate elays, the epenence of the optimal capacity on the elay is characterize by the critical elay below which our relaying an transmission strategy oes not lea to any appreciable capacity. For the values of the elay higher than the critical value, the capacity grows approximately as 2/.

8 The critical elay has a very slow epenence on the number of noes n so that it practically is inepenent on n. Itis interesting to note that the existence of that minimum elay is precisely the price we have to pay for the lack of the knowlege of the noes motion. Finally, we have shown that our transmission an relaying strategy is asymptotically optimal in the sense that the ratio of the achieve capacity an the upper boun approaches albeit rather slowly) as the number of noes n grows. REFERENCES [] A. ALWAN ET AL., Aaptive mobile multimeia networks, IEEE Personal Communications Magazine, 2, no. 996). [2] Z.J. HAAS AND M. PEARLMAN, Panel report on a hoc networks, Mobile Computation an Communication Review, 2, no ). [] A. EPHREMIDES, J.E.WIESELTHIER, AND D. BAKER, A esign concept for reliable mobile raio networks with frequency hopping signaling, Proceeings of IEEE, 75, no. 987) 56. [] J. JUBIN AND J.D. TORNOW, The DARPA packet raio network protocols, Proceeings of IEEE, 75 no. 987). [5] P. GUPTA AND P.R. KUMAR, The capacity of wireless networks, IEEE Transactions on Information Theory, 6, no. 2 2) 88. [6] M. GROSSGLAUSER AND D. TSE, Mobility increases the capacity of a hoc wireless networks, InINFOCOM 2 Proceeings, 2) 6. [7] P. GUPTA AND P.R. KUMAR, Towars an information theory of large networks: an achievable rate region, In Proceeings of 2 IEEE International Symposium on Information Theory, 2) 59. [8] S. TOUMPIS AND A. GOLDSMITH, Capacity regions for wireless a hoc networks, working paper. [9] E. AYANOGLU ET AL. Diversity coing for transparent self-healing an fault-tolerant communications networks, IEEE Transactions on Communications, no. 99). [] A. TSIRIGOS AND Z. HAAS, Multipath routing in the presence of frequent topological changes, IEEE Communications Magazine, 9, no. 2) 2.