Delay characterization of multi-hop transmission in a Poisson field of interference

Transcription

1 1 Delay characterization of multi-ho transmission in a Poisson field of interference Kostas Stamatiou and Martin Haenggi, Senior Member, IEEE Abstract We evaluate the end-to-end delay of a multi-ho transmission scheme that includes a source, a number of relays and a destination, in the resence of interferers located according to a Poisson oint rocess. The medium-access control (MAC rotocol considered is a combination of TDMA and ALOHA, according to which nodes located a certain number of hos aart are allowed to transmit with a certain robability. Based on an indeendent transmissions assumtion, which decoules the queue evolutions, our analysis rovides exlicit exressions for the mean end-to-end delay and throughut, as well as scaling laws when the interferer density grows to infinity. If the source always has ackets to transmit, we find that full satial reuse, i.e., ALOHA, is asymtotically delay-otimal, but requires more hos than a TDMA-ALOHA rotocol. The results of our analysis have alications in delayminimizing joint MAC/routing algorithms for networks with randomly located nodes. We simulate a network where sources and relays form a Poisson oint rocess, and each source assembles a route to its destination by selecting the relays closest to the otimal locations. We assess both theoretically and via simulation the sensitivity of the end-to-end delay with resect to imerfect relay lacements and route crossings. Index Terms Multi-ho, end-to-end delay, throughut, Poisson oint rocess, queueing. I. INTRODUCTION The main question ertinent to wireless multi-ho networks is determining the delay at which a certain throughut can be achieved, at the end-to-end level. The question is related to the following fundamental tradeoff: On the one hand, a smaller hoing distance rovides more robustness to interference and noise, resulting in better link reliability; on the other hand, each node that is added between the source of ackets and their final destination also incurs additional delay, as a acket tyically has to wait in line before it is transmitted to the next node [1]. The treatment of the roblem deends on a number of diverse factors, among which are the emloyed routing and medium-access (MA control (MAC rotocols, the channel model and, quite imortantly, the toology of the network. This aer obtains concrete end-to-end delay and throughut results for multi-ho networks with randomly laced nodes, taking into full account the effects of fading, interference and queueing delays due to acket buffering. We otimize the delay over the number of hos between the source of ackets and their destination, and other network arameters; obtain asymtotic delay-throughut tradeoffs as the density of nodes goes to infinity; and roose a delay-otimal routing algorithm for networks with randomly laced nodes. A. Related work and motivation The delay and throughut of multi-ho networks has been a toic of intense investigation, in articular in the last decade [1] [4]. An imortant line of work, surred by [5], considers the network as a collection of m nodes randomly distributed in a unit-area disk, where source-destination airs are randomly formed, and focuses on obtaining asymtotic results as m grows large. Following [5] and a number of other aers that dealt exclusively with the issue of achievable throughut (see [2] for an overview, [3] raised the question of delayconstrained throughut. In articular, under an ideal scheme that can schedule transmissions throughout the network, they showed that, for almost all network realizations, the otimal delay-throughut tradeoff is given by D(m = Θ(mT(m, where D(m and T(m are the delay and throughut scaling, resectively. A similar result was derived in [6], albeit within a different framework where nodes were allowed to move throughout the network in an indeendent and identically distributed (iid fashion. Although useful in shedding light on fundamental erformance trends, the revious aroach falls short in roviding concrete results for given design choices, which are based on realistic routing and MAC rotocols. In [7], it was argued that a functional network caacity theory should take into account issues of delay and overhead, since these dramatically affect the erformance of ractical networks. An aroach ioneered in [8] was to consider the network as a collection of transmitters, each with a distinct receiver, which are distributed on the lane as a Poisson oint rocess (PPP. The PPP framework is well suited for networks with no articular structure and uncoordinated transmissions, i.e., a random access MAC (ALOHA. A significant amount of work has been devoted to the study of single-ho PPP networks (see [9] for a comrehensive overview and the evaluation of metrics such as the exected acket rogress [8], the transmission caacity [10] and the satial density of rogress [11]. Given the tractability of the PPP framework, some extensions have been roosed to accommodate multi-ho transmission. In [12], an oortunistic routing strategy was advocated where the relay with the most favorable channel is selected in each ho, and the end-to-end delay was evaluated via simulation. In [4], the end-to-end throughut was derived and otimized over the number of hos, assuming that in each ho the interferer locations are drawn indeendently according to a PPP. The authors coined the term randomaccess transort caacity for the otimized throughut, to emhasize that, as in [5], the metric reflects the rate at which

2 2 ackets are transorted from the source to the destination, but in the secific setting where interferer locations and their transmissions are random. In [13], the multi-ho roblem was studied from an end-to-end connectivity ersective and bounds were determined on the time required for a ath to form between the source and the destination. The common trait of these aers is their throughut-centric aroach; it is assumed that nodes always have ackets to transmit and queueing delays resulting from acket buffering are ignored. Other related work includes [14] [19], which have studied the line network consisting of a source, a number of relays and a destination. The common assumtion here is that the line network oerates in a stand-alone fashion, i.e., interference from other such lines, which are exected to be resent in a network environment, is not considered. Assuming a channel model with ath-loss, fading and noise, and no delay constraints, [15], [16] determined the end-to-end rate, i.e., the minimum achievable rate over all hos, when a TDMAaccess rotocol is emloyed. Alternatively, under a given delay constraint, [17] secified the number of hos and the rate allocation among them, such that the total ower consumtion is minimized. A similar roblem was studied in [14], under an end-to-end success robability requirement. In [19], a decomosition aroach was emloyed to decoule the line network into isolated queues and the end-to-end delay of timedivision multile access (TDMA and ALOHA rotocols was evaluated. B. Contributions In this aer, we study the end-to-end delay erformance of a multi-ho transmission system (or route, in routing terminology consisting of a source, a number of relays and a destination, in a network where interferers are located according to a PPP. In this manner, we bridge the ga between existing endto-end delay results for line networks that do not account for interference [19], and existing end-to-end throughut results for PPP multi-ho networks that do not account for queueing delays [4]. Our main dearture oint from revious work is the introduction of buffers at the nodes, which leads to the exlicit evaluation of the associated acket service and waiting times. The couling of the queue evolutions renders the evaluation of the end-to-end delay a very challenging roblem; consequently, we assume that transmissions across nodes are indeendent, which allows the use of the framework develoed in [20], in order to evaluate the steady-state distribution of the size of each node queue. Section VI-D is devoted to verifying the validity of our aroach through simulations. The MAC rotocol considered is a combination of TDMA and ALOHA. In each slot, the rotocol schedules nodes which are searated by a given number of hos, and the scheduled nodes are allowed to transmit with a certain robability. It is selected in light of the fact that, in ractice, while intraroute coordination is fairly easy, inter-route coordination is hard, hence a random-access olicy for scheduled nodes is easily imlementable. Moreover, slotted ALOHA arises as a secial case, when all nodes in the route are simultaneously scheduled. In this manner, we model and analyze in a mean sense a network consisting of an infinite number of mutually interfering routes, which emloy the TDMA-ALOHA MAC rotocol. We study in detail a scenario where the source always has ackets to transmit ( backlogged source and show how the analysis can also be adated for the case of sources with geometric arrivals. In summary, our main contributions consist of: Obtaining analytical exressions for the ho success robability and the end-to-end delay as functions of the number of hos, the source MA robability and the intraroute satial reuse factor. Deriving the delay-otimal values of these arameters, and delay-throughut scaling laws when the density of interferers grows to infinity. In articular, it is shown that, in the limit of a large interferer density, maximum intraroute reuse, i.e., slotted ALOHA, minimizes the end-toend delay. Using the theoretically obtained delay-otimal number of hos in order to erform routing in a network where both sources and relays form a PPP. The routing algorithm consists of each source selecting the relays closest to the otimal locations on the source-destination line. We assess theoretically and via simulation the sensitivity of the end-to-end delay with resect to imerfect relay lacements and the utilization of given relays by more than one source-destination airs. Moreover, we verify via a number of exeriments the validity of the assumtions that form the backbone of our analysis, in the small MA robability regime. C. Paer outline and notation In Section II, the system model is described in detail. Section III is devoted to the evaluation of the ho success robability. In Sections IV and V, the mean end-to-end delay is derived and otimized over the relevant network arameters for the cases of backlogged sources and geometric arrivals, resectively. In Section VI, we resent our simulation results and in Section VII we summarize our conclusions. Table I includes a list of the main symbols emloyed throughout the aer. Note that the following conventions are emloyed for x x o : If lim x xo f(x = lim x xo g(x, then f(x g(x; if lim x xo f(x/g(x = 1, then f(x g(x. II. SYSTEM MODEL A. Toology, source traffic and MAC rotocol A source node emloys N 1 relays, N N, to communicate with a destination at distance R. The relays are laced equidistantly on the source-destination line so that the hoing distance is R/N (if N = 1 we have single-ho transmission. A node in the source-destination ath is secified by the index n = 0,...,N, where n = 0 corresonds to the source, n = 1 to the first relay and so on. As in [4], [11], we assume that time is slotted and nodes are synchronized to a common clock. We define the intraroute satial reuse factor d = 1,...,N, which determines the airwise distance (in hos between nodes in the route

3 3 TABLE I COMMONLY USED SYMBOLS Interference from PPP Φ(t Symbol Φ(t λ ex λ o ρ a s R N d = 1,...,N γ δ(d b θ Meaning interferer PPP at time t interferer density (extrinsic interference source density (intrinsic interference source MA robability relay MA robability node transmission robability source acket arrival robability (geometric arrivals ho success robability source-destination distance number of hos intra-route satial reuse factor satial contention intra-route satial contention roagation exonent SINR threshold for successful recetion S S SOURCE RELAY 1 RELAY 2 DEST. o o R1 R2 (a Worst interfered hos t D that may simultaneously transmit in a slot. By definition, there are d such grous of nodes: P 0 = {0,d,2d,...}, P 1 = {1,d+1,2d+1,...},..., P d 1 = {d 1,2d 1,...}. The value d = 1 corresonds to maximum intra-route reuse, i.e., a slotted ALOHA rotocol, while d = N corresonds to no intra-route reuse, i.e., the case where only one node may transmit at any given time, resectively. When d < N, simultaneous transmissions create intra-route interference, which, on the average (due to the resence of fading, is larger for smaller values of d. Each node is equied with an infinite-caacity buffer 1, where received ackets are stored in a first-in first-out fashion. We consider two different cases regarding acket traffic at the source: backlogged, where the source always has ackets to transmit, and geometric arrivals, where a new acket arrives at the source buffer with robability a every d slots, i.e., traffic intensity a/d. The first case models a scenario where a large amount of information rests at the source, e.g., a large file in an FTP-tye alication. The second one models in a simle manner the bursty nature of acket traffic in other tyes of alications. The MAC rotocol is a combination of TDMA and ALOHA and is described below Set t = 0 and randomly select k {0,...,d 1}. 2. Set P(t = P k. If the source is in P(t, it is allowed to transmit with robability o. If a relay is in P(t, it is allowed to transmit with robability. 3. A acket is successfully sent over a ho if the receiver SINR is larger than a threshold θ. If it is not, the transmitting node is informed via an ideal feedback channel and the acket remains at the head of its queue. 4. For geometric arrivals only: If the source is in P(t, a new acket arrives at the end of its queue with robability a at t+1 ǫ, where 1 ǫ > Set t t+1 and k mod(k +1,d. Reeat At low-traffic, the assumtion of infinite buffer caacity has negligible imact on the derived results. 2 The roosed rotocol can be imlemented in a distributed fashion as follows. Once the route is established, send a test acket to the destination that includes a ho counter. Each relay increases the ho counter by one, thus learning its osition and corresonding time slot in the route. (b Fig. 1. (a TDMA-ALOHA MAC rotocol with N = 3, d = 2. In the first slot, the source and the second relay are scheduled (P 0 = {0,2}; in the second slot, the first relay is scheduled (P 1 = {1}, and so on. When the source (relay is scheduled, it accesses the medium with robability o (. If the source is backlogged it always has ackets to transmit; if arrivals are geometric, a new acket arrives at the end of its queue every time it is scheduled, with robability a. (b An examle for N = 8 and d = 3. The worst-interfered hos are 4 and 5, thus I = { 1,1} and the resective distances are d+1 = 4, d 1 = 2. The rotocol for both traffic scenarios is deicted in Fig. 1(a, when N = 3,d = 2. It is emhasized that a node in P(t transmits only when it is allowed to (by the ALOHA art of the MAC and there is at least one acket in its queue. (The two events are equivalent only for the backlogged source. We denote by ρ n, n = 0,...,N 1, the robability that node n transmits, given that it is scheduled by the TDMA art of the MAC. By definition, ρ 0 o and ρ n, n = 1,...,N 1. We model network or inter-route interference by assuming that, in slot t, the locations of inter-route interferers are drawn from a PPP Φ(t of density λ PPP, where {Φ(t} are iid across t. We consider two cases, one of extrinsic and one of intrinsic interference, which are defined below: Extrinsic interference: λ PPP = λ ex. Inter-route interferers are randomly located on the lane with arbitrary density λ ex. Intrinsic interference:λ PPP = λn ρ/d. The network consists of an infinite number of randomly located and mutually interfering routes, whose nodes observe the MAC rotocol described above in a slot-synchronous manner. In articular, λ is the density of sources (or routes in the network, and N ρ/d reflects the fact that, with intra-route reuse d, there are on average N ρ/d interferers er route, where ρ = N 1 N 1 n=0 ρ n. If d = N, only one node er route is scheduled at any given slot, so λ PPP = λ ρ/n; if d = 1, all nodes er route are simultaneously scheduled and λ PPP = λ ρ. Since λ PPP is roortional to ρ, we exlicitly take into account that an interroute node is an interferer only when it is allowed to transmit and it has at least one acket in its queue.

4 4 B. SINR-based acket successes The channel between two nodes at distance r includes Rayleigh fading and ath-loss according to the law r b, where b > 2 is the ath-loss exonent. The fading coefficients are satially iid, with a coherence time that takes values in [1,d] (i.e., the fading is assummed to change at least as frequently as a node is allowed to transmit. All the nodes have the same transmit ower and the transmit signal-to-noise-ratio is β. Suose that node n 1 is scheduled at time t, i.e., n 1 P(t and its queue is not emty. Without loss of generality, assume that node n is located at the origin. A acket is successfully received by n if where SINR n (t A(t(R/N b > θ (1 I n,o (t+i n,i (t+β 1 A(t is the fading coefficient between n 1 and n, exonentially distributed with unit mean. I n,o (t is the total inter-route interference ower I n,o (t = A x (t x b, (2 x Φ(t where A x (t, exonentially distributed with unit mean, is the fading coefficient between the interferer at location x Φ(t and n. I n,i (t is the total intra-route interference ower I n,i (t = e m (ta m (t x m b, (3 m P(t\{n 1} where e m (t = 1 if m is a transmitter (and zero otherwise, A m (t is the fading coefficient between m and n, and x m is the location of m. Note that the recetion model based on (1 has an embedded half-dulex constraint. If d = 1, n P(t, thus, if e n (t = 1, SINR n (t = 0. III. A GENERAL EXPRESSION FOR THE HOP SUCCESS PROBABILITY In order to simlify the analysis, we ignore the favorable fact that nodes at the edge of the route are subject to less intraroute interference and assume that the success robabilities are equal to the one of the worst-interfered ho, which we denote by s. Due to symmetry, the robabilities of transmission are equal, i.e., ρ 1 = = ρ N 1 ρ and ρ = ρ. In the following roosition, we derive an exression for s, under the assumtion that transmissions occur indeendently with robability ρ. Proosition 1 If nodes scheduled by the TDMA art of the MAC transmit indeendently with robability ρ, then where s = s,o s,i s,n (4 s,o = e λpppc( R N 2, (5 with c = Γ(1+2/bΓ(1 2/bπθ 2/b, ( 1 ρ+ with I = and { 1 2 s,i = i I N, 1,1,..., d ρ 1+ di 1 b θ N 1 d 2, (6 } N 1, d (7 s,n = e ( R N b θβ 1. (8 Proof: For the roof, we emloy the aroach in Section III.B of [9]. From (1, the success robability can be written as s = P ( A(t θ(r/n b( I n,o (t+i n,i (t+β 1. Due to the indeendence of A(t,I n,o (t,i n,i (t, and the exonential distribution of A(t, we have that [ s = E e ( N ] [ R b θi n,o(t E e ( N ] R b θi n,i(t e ( N R b θβ 1. (9 Each term in this roduct corresonds to the success robability taking into account only inter-route interference ( s,o, intra-route interference ( s,i, and noise ( s,n. Since Φ(t is a PPP with density λ PPP, s,o is given by (5 (see [9, Eq. (9]. Moreover, the index of the transmitter with the worst-interfered receiver is n = 1 2 N d, where N d is the maximum number of concurrently scheduled nodes given N and d. The otential intra-route interferers are thus located at distances (R/N id 1, where i I and I is the set defined in (7. Due to the indeendence of transmission events, from [21, Eq. (19], we obtain (6. This concludes the roof. Remarks on Proosition 1: 1. The seti defined in (7 determines the distances of the intraroute interferers for the worst-interfered ho. In Fig. 1(b, an examle is shown for N = 8, d = 3. When d = N, I =, and s,i = The assumtion of indeendent transmission events is made for the sake of analytical tractability as the exact tandem queueing system is a very involved roblem [19], [22]. When d < N, the queue states are correlated due to (a intraroute interference, and (b the common to all scheduled hos interference rocess Φ(t. Regarding (a, we maintain that the assumtion is reasonable when the nodes are not allowed to transmit often, and this is the regime considered in the rest of the aer; indicatively, max{ o,} 0.1. Regarding (b, as shown in [23], the satial correlation coefficient of the interference ower resulting from a PPP is zero (under the ath-loss and fading model of this aer. This indicates that the deendence between acket successes at a given time slot due to Φ(t is very weak. On the grounds of these observations, acket successes, determined by the SINR criterion in (1, are considered indeendent across n and t. Note that when d = N, the indeendence of acket successes (hence transmission events is exact, since only one node is scheduled at a time, {Φ(t} are indeendent across t and the coherence time of the fading is at most N slots.

5 5 Based on (6, we now derive a lower bound to s,i. Proosition 2 If d < N, then s,i can be lower-bounded as where δ = i Z\{0} The bound is tight for ρ 0. Proof: Taking the inverse of (6 1 s,i = ( 1+ i I s,i e δρ, (10 1 (1 + di 1 b. (11 θ ρ 1 ρ+ di 1 b /θ. Alying the logarithm to both sides and using the inequality log(1+x < x, x > 0, we obtain that ( s,i > ex ρ 1 1 ρ+ di 1 b. /θ i I Since ρ < and I Z\{0}, (10 follows. When ρ 0, s,i 1 and e δρ 1, which roves the tightness of the bound. As shown in Fig. 2, e δρ rovides a good aroximation to s,i for sufficiently small values of ρ. For analytical convenience, we (conservatively set s,i = e δρ when d < N. Since s,i = 1 for d = N, from (4, we have the following general exression for s, which is emloyed throughout the rest of the aer, s = ex ( λ PPP c ( 2 R δ ρ N ( b R θβ 1, (12 N where δ = δ for d < N and δ = 0 for d = N. Based on (12, we define the arameter γ s / ρ ρ=0 as the satial contention [24]. It measures how steely the success robability decreases with the transmission robability ρ. If d < N, γ = δ for λ PPP = λ ex, and γ = λcr 2 /(Nd+δ for λ PPP = λnρ/d. In the latter case, i.e., intrinsic interference, γ consists of both an inter- and an intra- route comonent. Hence, δ is termed the intra-route satial contention, which, as seen from (11, is a decreasing function of d. In order to emhasize the deendence of δ on d, we also emloy the notation δ(d. Note that, if d = N, γ = λcr 2 /N 2 for intrinsic interference. Armed with (12, in the next two sections we examine searately the cases of a backlogged source and geometric arrivals. In each case, we evaluate ρ, derive exressions for the mean end-to-end delay and throughut, and minimize the delay over the relevant network arameters. Since the noisedeendent term in (12 does not deend on ρ, in the remainder of the aer, we focus on the interference-limited regime, i.e., we let β ( s,n = 1. Closing, in Table II, we have listed and commented on the main assumtions made in this section, which rovide the backbone for (12 and the analysis of the following sections. The validity of each assumtion is checked via simulation in Section VI-D; Table II also lists the figures where the resective results can be found s,i, eq. (6 e δρ, eq. (10 d = 1 d = ρ d = 3 d = 4 Fig. 2. Success robability, taking into account only intra-route interference, as a function of ρ, for N = 10 and d = 1,2,3,4. The lower-bound calculated in Proosition 2 becomes tighter as d increases. (b = 3, θ = 6 db, = 0.1 Assumtion Inter-route interference: PPP and iid across time Ho success robabilities: equal to success robability of worst ho Transmission events: indeendent TABLE II MAIN ASSUMPTIONS OF SECTION II Comments Crucial; reasonable for small MA robabilities (see Fig. 9 Conservative; can be relaxed, but would lead to cumbersome exressions; reasonable for range of interest of ath-loss exonents (see Fig. 10 Crucial; reasonable for small MA robabilities (see Fig. 11 IV. BACKLOGGED SOURCES A. Evaluation of the mean end-to-end delay We first determine the robability of transmission ρ when the source is backlogged. Proosition 3 If the source is backlogged and o <, then ρ = o. Proof: Recall the analysis in [20]. Since acket successes are indeendent events with robability s, if o s < s, ackets arrive to the first (and all subsequent relays with robability o s. Hence the robability that a relay has a non-emty queue is o s /( s = o /, which yields ρ = o / = o (same as the source. Setting ρ = o in (12, we readily obtain s. Note that the condition o < is necessary for the stability of the relay queues, as it ensures that the acket arrival rate does not exceed the acket service rate. We now evaluate the mean end-to-end delay D, defined as the mean total time (in slots that it takes a acket to travel to the destination from the moment of its first transmission attemt at the source. Proosition 4 If the source is backlogged, the end-to-end delay is given by D = d +d(n 1 1 o s N(d 1. (13 o s s ( o

6 6 Proof: Since a dearture occurs from the source every d slots indeendently with robability o s, the mean service time measured from the first transmission attemt till the acket is successfully received by the first relay, is H s = d/( o s d+1. For o <, ackets arrive at a relay every d slots with robability o s and are serviced with robability s. The mean service time for the head-of-line (HOL acket at a relay is thereforeh r = d/( s d+1. The mean waiting time at a relay, W r, defined as the mean total time from the moment a acket arrives at the end of the queue till it becomes the HOL acket, is calculated with standard queueing theory. The robability that there are k ackets in the queue is π k = ( o/ k 1 o s ( k 1 1 s (1 o /, k 1. (14 1 o s By Little s theorem, W r is the average queue size, excluding the HOL acket, divided by the arrival rate, in this case o s /d. Using (14, we find that W r = d o s (k 1π k = d o 1 s s ( o. (15 k=2 By definition, D = H s +(N 1(H r +W r, and (13 follows. Remarks on Proosition 4: Since a acket is received by the destination every d slots with robability o s, the first term in (13 is the inverse of the end-to-end throughut T = o s /d. The second term is the mean total time from the moment a acket arrives at the end of the queue of the first relay till it arrives at the destination. It is roortional to ( s ( o 1, i.e., the inverse of the difference between the acket service and arrival rates at each relay buffer. Hence, if N > 1, a necessary condition for finite D is o <. From (13, the following uer bound can be readily obtained, which is tight for small. Corollary 1 If the source is backlogged, D D, where The bound is tight for 0. D = d d(n 1 +. (16 o s ( o s In the next section, we ursue the otimization of D over the arameters N,d, o for the cases of extrinsic and intrinsic interference. We obtain two kinds of results: (a Exact exressions or tight bounds on the delay-otimal value of each arameter, keeing the other arameters fixed, and (b asymtotic exressions for the jointly delay-otimal arameter values, as λ ex for extrinsic interference, and λ for intrinsic interference. Note that, in an interferencelimited network, (12 deends only on the roduct λ ex cr 2 for extrinsic interference and λcr 2 for intrinsic interference. Hence, all asymtotic results may equivalently be derived letting λ ex cr 2 and λcr 2, resectively. The delay-otimal arameter values and the resective delay and throughut are denoted by the suerscrit 3. For analytical 3 We do not emloy different notation for the otimal and jointly-otimal arameter values. To make the distinction clear, we state when the arameters are searately or jointly otimized. tractability, we relax the integer constraints on N and d and let N [1,+, d [1,N]. We close this section by suggesting how the framework resented in this aer can also be emloyed to comute the delay in a network where the distance R of each source-destination air is drawn in an iid fashion from a given distribution. For each R, we let N(R = R/r, where r is an inter-relay distance r that does not deend on R. Therefore, on average, the number of hos erformed in the network ise[r]/r, where the exectation is taken with resect to the distribution of R. The relevant interferer density is λ PPP = λ o E[R]/(rd, and the mean delay in the network can readily be comuted by (13, where the otimization arameters are now r,d, o, with E[R] in lace of the common distance R of the homogeneous setting. B. Extrinsic interference We consider the cases of no intra-route satial reuse (d = N and intra-route satial reuse (d < N searately. 1 No intra-route satial reuse (d = N: In the following roosition, the delay-otimal N, o are determined. Proosition 5 Let λ PPP = λ ex and d = N. If λ ex cr 2 > 1, then, for given o, {[ N λex cr, 2λ ex cr o (0,/2], [ 1, λex cr o (/2,. For given N > 1 o = (17 1+ N 1. (18 Proof: See Aendix A. Remarks on Proosition 5: 1. For given λ ex, in the light-traffic regime, i.e., o 0, D 1/T, so N 2λ ex cr, which is the value of N that maximizes the end-to-end throughut T = o s /N = o e λexc(r/n2 /N. As we move into the high-traffic regime, i.e., o > /2, the second term of (16, which increases with N 2, dominates the delay. Therefore, a smaller number of hos is more delay-efficient and N < λ ex cr. 2. The delay-otimal o decreases as Θ(1/ N. For a given N, (18 achieves the best tradeoff between throughut and total time sent in the relay queues. We now determine the jointly delay-otimal (N, o as λ ex. Proosition 6 Let λ PPP = λ ex and d = N. The jointly delayotimal (N, o for λ ex are The resective minimum delay is ( 1+ N λ ex cr (19 o (λ ex cr 2 1/4. (20 D λ excr 2 e Proof: See Aendix B. 2. (21 (λ ex cr 2 1/4

7 7 Remarks on Proosition 6: 1. From (21, it is seen that D = Θ(λ ex. The linear scaling is due to the factor N 2 in (16 and the fact that N = Θ( λ ex. Intuitively, a HOL acket has to wait at least N slots before a retransmission attemt, and there are N buffers in the route. The resective delay-otimal throughut is T = Θ( o /N = Θ(λex 3/4. 2. The throughut-otimal strategy for all λ ex is to set N = 2λ ex cr (see remark on Proosition 5 and o = (if o = the delay is infinite, though. So, asymtotically, the throughut-otimal number of hos is larger than the delayotimal number of hos by a factor 2. The resulting maximum throughut is T = Θ(1/ λ ex. Hence, a throughut enalty of Θ(λ 1/4 ex is incurred by the delay-otimal olicy due to the fact that o = Θ(λex 1/4. 2 With intra-route satial reuse (d < N: Given the inefficiency of a rotocol which allows only one node to be scheduled at a time, we now let d < N. In the following roosition, we determine the delay-otimal N, o. Proosition 7 Let λ PPP = λ ex and d [1,N. If 2λ ex cr 2 > 1, then, for given o, {[ 2λex N cr, 2(/ o λ ex cr o (0,/2], [ 1, 2λex cr o (/2,. (22 For given N,d o 2 δ2 +4(N 1 1 (δ + 1 +δ. (23 The bound is tight for N. Proof: See Aendix C. Remarks on Proosition 7: For givenλ ex, if o 0, the uer bound in (22 goes to infinity. Indeed, for o 0, N is delay-otimal, since D (see (45 in roof is dominated by 1/T, where T = e λexc(r/n2 δ o /d, and setting N maximizest. Also, note that N does not deend on d, which is easy to see from (45. In contrast, o in (23 is a decreasing function of δ (i.e., an increasing function of d, as well as N. Based on Proosition 7, we now derive the jointly delayotimal (N,d, o, as λ ex. Proosition 8 Let λ PPP = λ ex and d [1,N. The jointly delay-otimal (N, o,d for λ ex are d 1 and N 2λ ex cr (24 (25 (1+δ(1(2λex cr 2 1/4 o The resective minimum delay is ( D 2λex cer 1+δ(1 1+. (26 (2λ ex cr 2 1/4 Proof: See Aendix D. Remarks on Proosition 8: 1. In the limit λ ex, (slotted ALOHA is the delayotimal MAC rotocol. The reason that maximum reuse minimizes the delay is that, o (as well as the busy robability of the relay buffers o / goes to zero whenλ ex. Hence, s,i = e δo 1, and D in (16 is roortional to d, making d = 1 the otimal choice. From (12, it is also seen that the otimal ho success robability is s e 1/2. 2. N in (24 is larger than the resective one in (19 by a factor of 2. This is the rice aid in terms of resources, i.e., relays, for allowing intra-route satial reuse. 3. The minimum delay scales as D = Θ( λ ex, i.e., there is a delay gain ofθ( λ ex comared to the case of no reuse. Since T = o s /d, the resective delay-otimal throughut scales as T = Θ(λex 1/4, so the throughut gain is also Θ( λ ex. 4. The throughut-otimal strategy selects (N,d, o to maximize T = o e λexc(r/n2 δ o /d. It is clear that, for a given λ ex, N maximizes T, which reduces the roblem to selecting (d, o to maximize o e δ(do /d. Since the maximum throughut is a constant with resect to λ ex, the throughut enalty incurred by the delay-otimal olicy is Θ(λ 1/4 ex, as in the case of no reuse. C. Intrinsic interference We now study the case of intrinsic interference, i.e.,λ PPP = λn o /d. For lack of sace (and similarity of the relevant derivations, we only state the asymtotic results for λ. As in the case of extrinsic interference, we consider d = N and d < N searately. 1 No intra-route satial reuse (d = N: The interferer density is λ PPP = λn o /N = λ o. The jointly delayotimal (N, o when λ are determined in the following roosition. Proosition 9 Letλ PPP = λ o andd = N. The jointly delayotimal (N, o for λ are ( ζλcr N 2 1/3 (27 2 ( 2 1/3 o 4ζλcR 2, (28 where ζ is a constant in (1,2. The minimum delay is ( ζλcr D e 1/ζ 2 1/3 ( ( ζλcr 2 1/ ( /3 Proof: See Aendix E. Remarks on Proosition 9: 1. The minimum delay scales as D = Θ(λ 2/3, and the resective delay-otimal throughut as T = Θ(λ 2/3. We can interret this result by defining the delay and throughut exonents = lim λ log D(λ/logλ and τ = lim λ logt(λ/logλ and letting o = λ κ, where λ > 1. From (17, it is seen that, for a given o, the delay-otimal N must satisfy N = Θ( o (λλ = Θ(λ 1 κ 2. Substituting in (13, we have that (κ = max{(κ + 1/2,1 κ} and τ(κ = (κ+1/2. The value of κ that minimizes (κ is 1/3, which yields (1/3 = τ(1/3 = 2/3. 2. The constant ζ arises due to the fact that N is in the range ( λc o R,2 λc or (see roof.

8 8 TABLE III SCALING LAWS FOR BACKLOGGED SOURCE, FOR THE CASES OF EXTRINSIC AND INTRINSIC INTERFERENCE extrinsic: λ ex intrinsic: λ Metric d = N d < N d = N d < N D Θ(λ ex Θ( λ ex Θ(λ 2/3 Θ( λ T Θ(1/ λ ex Θ(1/ λ ex Θ(λ 2/3 Θ(1/ λ N Θ( λ ex Θ( λ ex Θ(λ 1/3 Θ( λ d o Θ(λex 1/4 Θ(λ 1/4 ex Θ(λ 1/3 Θ(1/ λ s Θ(1 Θ(1 Θ(1 Θ(1 Delay (slots Extrinsic Intrinsic No reuse Max. reuse 3. The throughut T = e λco(r/n2 /N is maximized for o = and N = 2λcR. The maximum throughut scales as T = Θ(1/ λ, hence the delay-otimal olicy incurs a throughut enalty of Θ(λ 1/6. 2 With intra-route satial reuse (d < N: The interferer density is λ PPP = λn o /d. Proosition 10 Let λ PPP = λn o /d and d [1,N. The jointly delay-otimal (N, o,d for λ are d 1 and N 2λcR (30 o 2λcR2. (31 The resective minimum delay is D 2eλcR 2 2. (32 Proof: See Aendix F. Remarks on Proosition 10: 1. As in the case of extrinsic interference, ALOHA is asymtotically delay-otimal. The minimum delay scales as D = Θ( λ, i.e., there is a delay gain of Θ(λ 1/6 comared to the case of no reuse (Proosition 9. Since T = o s /d, the resective delay-otimal throughut scales as T = Θ(1/ λ, so the throughut gain is also Θ(λ 1/6. These gains are achieved by increasing the number of hos from N = Θ(λ 1/3, when d = N, to N = Θ( λ. 2. The throughut-otimal strategy selects (N,d, o to maximize T = o e λcor2 /(Nd δ o /d. Since N maximizes T, the maximum throughut is a constant with resect to λ. Hence, the throughut enalty incurred by the delay-otimal olicy is Θ( λ. The scaling laws derived throughout Section IV are summarized in Table III. We now rovide numerical results for secific values of the network arameters. D. Numerical results Let R = 500 m, b = 3, θ = 6 db and = 0.1. In Fig. 3, D in (13 is lotted vs. λ ex for the case of extrinsic interference, and λ for the case of intrinsic interference. D is numerically otimized over N and o when d = N (no reuse and d = 1 (maximum reuse, or slotted ALOHA. Fig. 4 shows the resective delay-otimal numbers of hos. The theoretical exressions for D and N, derived in this section, are also Density (nodes/m 2 Fig. 3. D in (13 lotted in solid lines vs. λ ex for extrinsic interference, and λ for intrinsic interference. Maximum reuse (d = 1 corresonds to a slotted ALOHA MAC. For each density, D is numerically otimized over N and o. The exressions for D given, from left to right, in (21, (26, (29 and (32 are also lotted for comarison (dashed. (R = 500 m, = 0.1, b = 3, θ = 6 db Otimal N 10 1 Extrinsic No reuse Max. reuse Intrinsic Density (nodes/m 2 Fig. 4. Delay-otimal N, corresonding to Fig. 3. The solid lines (staircase curves corresond to the delay-otimal N found numerically. The dashed lines corresond, from left to right, to (19, (24, (27 and (30. For extrinsic interference, the ratio of the delay-otimal N for maximum reuse and no reuse is 2; for intrinsic interference, this ratio increases as λ 1/6 (R = 500 m, = 0.1, b = 3, θ = 6 db lotted for comarison. Note that, even though asymtotic, they rovide good aroximations of the resective minimum delay and delay-otimal number of hos for a realistic range of densities. Indicatively, for the case of intrinsic interference and λ = 10 4 m 2, Fig. 4 shows that 3 hos are required when no reuse is emloyed, while 9 hos are required with maximum reuse, when the source-destination distance is 500 m. It is also aarent that, for the selected arameter values, maximum reuse outerforms no reuse for all density values, but the required number of hos is larger. In the case of extrinsic interference, the ratio of the delay-otimal N for maximum reuse and no reuse is aroximately 2, while, for intrinsic interference, this ratio increases as λ 1/6. These observations are in agreement with the scaling laws listed in Table III.

9 9 V. GEOMETRIC ARRIVALS In the revious section we examined in detail a heavy-traffic scenario, where the source always has ackets to transmit. In this section, we briefly treat the case of geometric arrivals at the source. The analysis follows closely the one of Section IV. We focus on the case of intrinsic interference, i.e., λ PPP = λnρ/d, and o = (the cases of extrinsic interference or o can be treated very similarly. The main result is stated in the following roosition. Let W(x, x e 1, denote the rincial branch of the Lambert function [25]. Proosition 11 Assume that a new acket arrives at the source every d slots with robability a and interference is intrinsic. If a < ex( γ, where γ = λcr 2 /(Nd + δ, then s = ex(w( aγ (33 and ρ = aex( W( aγ. (34 The mean end-to-end delay D, measured from the moment a acket arrives at the end of the source queue, is D = D N(d 1, where D = Nd 1 a s a. (35 Proof: The condition a < ex( γ ensures that the queues are stable see [26, Pro. 1]. In this case, the acket arrival robability to each relay is a. Hence, the robability that a node is a transmitter is ρ = (a/ s = a/ s. From (12, this results in the following fixed-oint equation over s s = ex ( γ as. (36 Eq. (36 has two solutions if and only if a < (γe 1, which always holds if a < ex( γ. The smaller solution is increasing in a, on the basis of which it is rejected, since it reresents a network where the ho success robability increases with increasing traffic. Rewriting (36 as aγ/ s ex( aγ/ s = aγ and alying the Lambert function to both sides, we obtain (33. Since ρ = a/ s, (34 follows. The roof of (35 follows the one of Proosition 4. If a < s, the acket arrival robability to all nodes is a and the acket service robability is s. Due to symmetry, D = N(H +W, where H = d/( s d+1 and W = d a s 1 s s a, (37 and (35 follows. Remarks on Proosition 11: 1. s is a decreasing function of a. In the extreme case a = 0, the throughut is zero and s = The fixed-oint equation (36 is the result of the assumtion that acket successes are iid. Note that similar decouling assumtions emloyed in [27] and [28] also resulted in fixedoint equations for the transmission robability. 3. If the queues are stable, the end-to-end throughut is T = a/d, since a acket arrives at the destination every d slots with robability a. For a given end-to-end throughut requirement T = T o, we can show that the number of hos N that minimizes (35 satisfies the relation N = λcr 2 T o (1+W( γ dt o ( e W( γ dt o dt o, (38 where γ = λcr 2 /(N d +δ. It follows that N = Θ(λT o and D = Θ(λT o. This is a manifestation of the scaling law derived in [3], in the context of our model, which assumes erfectly laced relays and interferers located according to a PPP. A. Simulated network setting VI. APPLICATION In the revious sections, we develoed an analytical framework to evaluate and otimize the mean total delay from the source to the destination in the resence of interferers that form a PPP. In articular, the case of intrinsic interference was considered, in order to evaluate the delay in a network with mutually interfering routes. We now examine how the results of Section IV can be alied in a setting where backlogged sources have to route ackets to their destinations by emloying a common ool of relays. We consider a network where both the source and relay locations are drawn from a PPP Π t of total density λ t, and a node is a source with robability µ, or a relay with robability 1 µ. Therefore, sources and relays form two indeendent PPPs, Π s and Π r, with densities λ = µλ t and λ r = (1 µλ t, resectively. Each source has a destination at distance R and random orientation, and selects out of the available relays the ones which are closest to the delay-otimal locations. In each route formed in this manner, the nodes observe the MAC rotocol described in Section II, in a slot-synchronous manner 4. The simulated network dearts from the theoretical model as relays are not erfectly laced on the line between the source and destination, and two or more routes may utilize the same relay. We first discuss the imact of these factors on the theoretical erformance, and then describe our simulation camaign and results. B. Imerfect relay lacements For ease of exosition, we consider N > 2, and no reuse, i.e.,d = N. Assume that the second relay selected is dislaced by x from the ideal osition on the source-destination line, where x < r and r = R/N is the hoing distance. We derive the incurred delay enalty for small erturbations x r. Proosition 12 Let x r = R/N, x > 0, N > 2, be the dislacement of the second relay from the ideal osition. When x 0, the delay increase δ D = D dis D is given by δ D = 4Nλc oe λcor2 ( o 2 ( 1+2λc o r 2+ o o x 2 +O(x 4, (39 4 The slot boundaries are synchronized, but not the TDMA schedules, i.e., P(t (see Section II is generally different across routes in a given slot t.

10 10 m Pcr,N Theor. arox Simulation N = 4 N = 3 N = Source Dest 800 Selected relay Relay Fig. 5. Crossings of different source-destination airs at common relays for a network with λ = 10 4 m 2, N = 3, R = 500 m and a relay density λ r = 32λ. Utilized (non-utilized relays are shown with circles (crosses. where D is defined in (16. Proof: From (16, δ D is found to be m Ne λcor 2 δ D Ne λcor2 = e + λco(x2 +2crx e 2Neλcor. λco(x2 2crx o Taking the Taylor series exansion at x = 0, we obtain (39. Remarks on Proosition 12: Eq. (39 imlies that the delay enalty due to imerfect relay lacement is more severe if o, i.e., if the system is oerated close to caacity, and it is roortional to N, if no intra-route reuse is emloyed. Moreover, for x R/N, the enalty is aroximately roortional to x 2 (and an even function of x, due to symmetry. If we set x = (2 λ r 1, which is the exected distance of the closest relay to the desired oint, it follows that the delay enalty is also roughly inversely roortional to the density of relays in the network. C. Route crossings Each source selects the relays which are closest to the desired locations on the source-destination line. As shown in the examle of Fig. 5, this results in the utilization of articular relays by more than one source-destination air. If C is the number of times the tyical relay node is actually emloyed as a relay in a network where the desired number of hos is N, we define the crossing robability P cr,n = P(C > 1 C > 0. The exact evaluation of P cr,n aears comlicated, hence we resort to the following aroximation. Let x be the tyical relay in Π r. Denote the oint rocess of ideal relay locations as Π r,ideal, and let z be the closest oint of Π r,ideal to x, and z the second closest. We define as P cr,2, the robability that, in a two-ho system (i.e., one ideal relay location er sourcedestination air, x is the closest neighbor of Π r to z, given that it is also the closest neighbor to z. Mathematically, P cr,2 (arg = P min x z = x arg min x z = x. x Π r x Π r λ r /λ Fig. 6. Crossing robability vs.λ r/λ for N = 2,3,4. The simulation results were obtained for λ = 10 4 sources/m 2 and R = 500 m. The theoretical aroximation in (40 is lotted for comarison. For N 2, we then aroximate P cr,n by P cr,n = (N 1P cr,2, (40 since, for a sufficiently large relay density, (a a relay is likely to be utilized by its neighboring oints in Π r,ideal, and (b the robability of a crossing should increase roughly roortionally with the desired number of hos. In the following roosition, we derive an exression for P cr,2. Proosition 13 P cr,2 is given by P cr,2 = 4/π 1+ λr λ t 1 t 2 e t2 2 λr y = + 0 dt 1 + t 1 dt 2 π 0 dθ λπ(t 2 1 (π φ+t2 2 (φ+θ+yt1 sinφ (41 t 2 1 +t2 2 2t 1t 2 cosθ φ = tan 1 ( t2 sinθ t 1 t 2 cosθ. Proof: See Aendix G. P cr,n was evaluated by simulation over different relay densities and network realizations, for N = 2, 3, 4 hos, λ = 10 4 sources/m 2 and R = 500 m. The results are lotted in Fig. 6, as a function of the ratio λ r /λ. Is is seen that P cr,n in (40, rovides a good aroximation ofp cr,n. We can verify that P cr,n roughly follows the trend (N 1(1+2λ r /λ 1. D. Simulation results We let λ = 10 4 m 2, = 0.1, b = 3, θ = 6 db and erform a number of simulations for different values of N, d, R, o and λ r. The network area is square, with size such that, on average, 2000 sources are included; for λ = 10 4 m 2, this corresonds to a square side of 4.5 km. For each oerating oint, we generate one network toology and run an exeriment with duration slots (at the beginning of each exeriment the node buffers are emty. In order to resolve conflicts when a relay is selected by more than one sources, ackets with different destinations are stored in

11 11 Delay (slots o =0.01, R=500m o =0.005, R=1000m d=1 d=2 d=3 d=4 Throughut (ackets/slot d=1 d=2 d=3 d=4 o =0.01, R=500m o =0.005, R=1000m N N Fig. 7. Delay vs. number of hos for ( o = 0.01, R = 500 m and ( o = 0.005, R = 1000 m, and various reuse factors. The markers corresond to simulation results obtained for λ r = 4Nλ for each N and the solid curves corresond to (13. a common queue and the following rule is alied at any given slot: if the relay is a receiver for one route and a transmitter for another, the recetion fails; in all other cases, successful recetion follows the SINR criterion (1. In order to avoid edge effects, for each toology, samle-metrics are only collected for the routes with the 200 innermost sources. For each oerating oint, the lotted metrics are obtained by averaging over routes (where alicable and time slots. We first select a relay density of λ r = 4Nλ, where N is the desired number of hos, such that, for a given number of hos, relays are found close to the desired locations with high robability. We consider two scenarios: R = 500 m and R = 1000 m, which are 10 and 20 times the exected closestneighbor distance in the source PPP, i.e., 1/(2 λ = 50 m, resectively. According to (30 and (31, the corresonding otimal values of (N, o are (10,0.01 and (20, Since the relay MA robability is set to = 0.1, these values of o corresond to a traffic generation rate at 10% and 5% of caacity, resectively (Proosition 3. In Figs. 7-8, we have lotted the theoretically comuted delay (13 and throughut (T = o s /d, along with the simulation results, for N ranging from 3 to 10 hos, and various reuse factors. Figs 7-8 illustrate the general agreement between theory and simulation; the discreancy is largest for small numbers of hos and reuse factors. The main message is that d = 1 (maximum reuse is otimal once N is sufficiently large; for small N, it is more advantageous to sace out transmissions by imosing a d > 1, e.g., for o = 0.01 and R = 500 m, d = 2 yields a smaller delay than d = 1 for N < 4, while for o = and R = 1000 m, d = 3 yields a smaller delay than d = 1 for N < 6. In addition, note from Fig. 8 that, while there exists a delay-otimal number of hos, the throughut increases with N since the distance er ho decreases for fixed R. The agreement between theoretical and simulation results in Figs. 7-8 imlicitly demonstrates that the theoretical aroach Fig. 8. Throughut vs. number of hos corresonding to Fig. 7. is valid in the considered regime. In Figs. 9-11, we look more closely at the assumtions that underly our analysis, which are listed in Table II. In order to validate the iid comonent of the first assumtion 5, in Fig. 9 we select N = 4 and d = 1,..., 4, and lot: the squares of the cdfs of the simulated interference ower at the origin at odd and even time slots, P 1 (x = P(I n,o (t + I n,i (t x and P 2 (x = P(I n,o (t I n,i (t + 1 x; the simulated joint interference ower cdf at the origin, over odd and even time slots, i.e., P 12 (x = P(I n,o (t+i n,i (t x,i n,o (t+1+ I n,i (t+1 x; and the roduct of the individual simulated cdfs, i.e., P 1 (xp 2 (x. The match between the curves is very good for the whole range of interference values and reuse factors, which imlies that, in the considered regime, the temoral iid assumtion is reasonable. This result also agrees with a recently discovered rule of thumb that the interference may be considered aroximately temorally indeendent, if (1 2/b < 0.1 [29]. Fig. 10 shows the simulated success robabilities of the last ho and the worst-interfered ho, i.e., the ho with index 1 2 N d +1, that corresond to the set of curves ( o = 0.01, R = 500 m of Fig. 7. In almost all cases, the curve corresonding to the worst-interfered ho lies very slightly below the corresonding one for the last ho, indicating that the second assumtion is also quite reasonable in the considered regime. Our interretation of these results is that, while edge nodes suffer from less intra-route interference, the inter-route interference is the same (on average; for a ath loss exonent b = 3, it dominates the total interference, such that edge effects can be safely neglected. Fig. 11 is concerned with the last assumtion of Table II. To this extent, we have lotted the simulated joint robability of transmission of two nodes in the same route at distancedhos, as well as the roduct of the resective individual robabilities of transmission. As in Fig. 10, the lotted curves corresond to the set of curves ( o = 0.01, R = 500 m of Fig. 7. The 5 Since the source and relay locations form two indeendent PPPs, the interference ower is generated from a PPP with very good aroximation.

12 d = 3 d = 2 d = 1 d = 4 2 P (x 1 2 P (x 2 P (x 12 P (xp (x 1 2 Probability of transmission d=1 d=2 d=3 d= Normalized interference ower, x N Fig. 9. Simulated interference ower (at origin cdfs for N = 4, d = 1,...,4 and ( o = 0.01, R = 500 m. The x-axis is normalized to one. P 1 (x and P 2 (x are the interference ower cdfs at odd and even time slots, and P 12 (x is the joint interference ower cdf over two consecutive slots. The conclusion is that the iid assumtion is reasonable in the considered oeration regime. Probability of success d=1 d=2 d=3 d= N Fig. 10. Simulated success robabilities of last (solid line and worst (dashed line hos corresonding to the set of curves ( o = 0.01,R = 500 m of Fig. 7. The discreancy is small due to the dominance of inter-route over intra-route interference for ath-loss exonent b = 3. results indicate that the joint robability of transmission is smaller than the roduct by at most 50% for all considered values of N and d. Finally, in Fig. 12, we examine the sensitivity of the delay with resect to the relay density. We select two oerating oints from those in Fig. 7: (N = 5,d = 3 and (N = 10, d = 1, and let the relay density λ r vary between λ and 64λ (note that for these two oints, the results in Fig. 7 were obtained for relay densities 20λ and 40λ, resectively. We observe that the simulated delay converges to the theoretically redicted value when λ r takes values larger than Nλ. Moreover, consistent with intuition, erforming 10 hos (at maximum reuse requires a larger relay density than 5 hos Fig. 11. Simulated joint robability that two nodes at distance d hos within a route are simultaneously transmitting (dashed line and roduct of resective simulated individual robabilities of transmission (solid line, corresonding to the delay curves of Fig. 7 for ( o = 0.01, R = 500 m. The maximum difference between the two curves is about 50% for all reuse factors. Delay (slots Relay density (relays/m 2 N=5, d=3 N=10, d=1 Fig. 12. Simulated delay vs. relay density for (N = 5,d = 3 and (N = 10, d = 1. The delay converges to the theoretically redicted value when λ r > Nλ. (R = 500 m, o = 0.01 (at reuse factor 3 for convergence, as the attemted number of hos is larger. In articular, for λ r < 20λ, the bottlenecks that occur at overutilized relays by multile sources incur a significant delay enalty, and thus erforming 5 instead of 10 hos results in smaller delay. VII. CONCLUSIONS We evaluated the end-to-end delay of multi-ho transmission in the resence of interferers that form a PPP, under a TDMA-ALOHA MAC rotocol. We considered the case of an arbitrary interferer density, as well as the case where the density deends on the number of scheduled nodes er route and the transmission robability. The delay-otimal number of hos was determined, and asymtotic exressions were derived

13 13 for large values of the interferer density. Consistent with intuition, we obtained that, when the source is backlogged, a random access olicy is asymtotically delay-otimal, but requires more hos than a TDMA-ALOHA rotocol. The theoretical results were alied to a delay-minimizing routing algorithm for networks with randomly distributed nodes. We simulated a static network setting, where both sources and relays formed a PPP, and each source erformed the delay-otimal number of hos to its destination, by routing to the relays closest to the otimal locations. We confirmed that the main assumtions on which the analysis is based are reasonable for small enough MA robabilities; as a consequence, the match between the theoretical and simulated delay is also satisfactory in this regime. In addition, we assessed the sensitivity of the delay with resect to imerfect relay lacements and relay-utilization by more than one sourcedestination airs. In conclusion, this aer combined elements from queueing theory and the theory of PPPs in order to obtain exlicit endto-end delay results for multi-ho networks with randomly laced nodes, fading and interference. These results fill in the ga between existing work on multi-ho networks, that has focused exclusively on scaling laws, and existing work on PPPs that has focused on throughut, without taking into account the effects of acket buffering. ACKNOWLEDGMENTS The artial suort of the DARPA/IPTO IT-MANET rogram (grant W911NF and the U.S. National Science Foundation (grant CNS is gratefully acknowledged. A. Proof of Proosition 5 APPENDIX Setting d = N in (12 and (16 ( 1 D(N, o = N + N 1 e λexc( N R 2. (42 o o The sign of D/ N is determined by the function f(n = (/ o 2(N 2 2λ ex cr 2 +2N(N 2 λ ex cr 2. We examine the two ranges of o searately: o (0,/2]: If o (0,/2, D/ N < 0 for N λex cr and D/ N > 0 for N 2λ ex cr. If = o /2, then f(n = 0 yields N = λ ex cr. This roves the first branch of (17. o (/2,: We set N = αλ ex cr, α 1. Then f(n/(λ ex cr 2 = 2 αλ ex cr 2 (α 1+(/ o 2(α 2 > 2 αλ ex cr 2 (α 1 (α 2 > 2 α(α 1 (α 2 > (2 α 1(α 1 0, since / o > 1, λ ex cr 2 > 1 and α 1. This roves the second branch of (17. Finally, setting D/ o o= = 0 yields (18. o B. Proof of Proosition 6 Substituting o = /(1+ N 1 in (42, we obtain D = N 1( 1+ N 1 2e λ exc( R N 2. (43 Setting D/ N N=N = 0 N + N (N 1 N 1 λ excr 2 N ( 1+ 2 N 1 N = 0. (44 If λ ex, (44 is satisfied only if N. Letting N, we obtain (19. Substituting (19 in (18 and (43, we obtain (20 and (21, resectively. C. Proof of Proosition 7 From (12 and (16, ( 1 D(N,d, o = d + N 1 e λexc( N R 2 +δ o. (45 o o The sign of D/ N is determined by the function f(n = N(N 2 2λ ex cr 2 2λ ex cr 2 (/ o 2. (46 We examine the two ranges of o searately: o (/2,: D/ N > 0, for N 2λ ex cr, which roves the second branch of (22. o (0,/2]: If o (0,/2, D/ N < 0 for N 2λ ex cr. If o = /2, D/ N = 0 yields N = 2λ ex cr. This roves the lower bound in the first branch of (22. In order to rove the uer bound, we set N = 2α(/ o λ ex cr, α 1. Then f(n/(λ ex cr 2 = 2α(/o λ ex cr(2α/ o 2 2(/ o 2 > 2( 2(/o 1 (/ o 2 > 2( 2 1(/ o 1 > 0, since α 1, / o > 2 and λ ex cr 2 > 1/2. This concludes the roof of the uer bound. For the roof of the second statement, we set D = 1 o 2 + δ o + N 1 1δ o ( o 2+(N = 0. (47 o o= o This equation holds only if Solving over o, γ o < N 1 o 2 (N 1 (N 1δ. (48 ( δ ( o 2 +δ o 1 < 0, which is equivalent to (23. For N, o 0, therefore (48 becomes an equality. This roves the tightness of the bound.

14 14 D. Proof of Proosition 8 Assume that, for λ ex, N. Then, from (23, it follows that o / N (1+δ(d. Setting (46 equal to zero at N = N and substituting o, we obtain (N 3 2λ ex cr 2 N 2λ ex cr 2( N (1+δ(d 2 0, or N 2λ ex cr. Finally, substituting N and o in (45 D d 1( N (1+δ(d +N e, Hence d 1 and (26 follows. We now return to the assumtion that N for λ ex. If N = Θ(1 for λ ex, then (45 imlies that D = e Θ(λex. Therefore, N = Θ(1 is rejected, and the roof is concluded. E. Proof of Proosition 9 Setting λ = λ o and d = N in (12 and (13, we obtain that s = e λco(r/n2 and ( 1 D(N, o = N + N 1 e λco( N R 2. (49 o o The jointly otimal (N, o are found by solving the system D/ N = 0 and D/ o = 0. After some maniulations, we obtain λcr 2 ( 1 (N 2 o λcr 2 ( 1 (N 2 o + N 1 o + N 1 o = 1 ( N 1 o 2 ( o 2, = 1 ( 1 2 o o + 2N 1 o. Equating the right-hand sides 2N 1 / o 1 + 2N 2 (/ 2 = 1. (50 o 1 For λ, we either have o = Θ(1 or o 0: If o = Θ(1, (50 demands that N = Θ(1. From (49, this imlies that D = e Θ(λ. If o 0, then, from (50, it is necessary that N, which also imlies that o /(2N. Moreover, using the same stes as in the roof of Proosition 5, we can show that, since o < /2, it is necessary that N = αλ o cr, where α (1,2. Setting N = αλ o cr in o /(2N, we obtain (27 and (28. Substituting (27 and (28 in (49 results in (29. Since, in this case, D = Θ(λ 2/3, the case o = Θ(1 is rejected, which concludes the roof. F. Proof of Proosition 10 Setting λ PPP = λn o /d in (12 and (16, we obtain ( 1 D(N,d, o = d + N 1 ( λcr 2 Nd e +δ o. (51 o o Setting D/ o = 0 and D/ N = 0 at (N,d, o = (N,d, o, we have 1 ( N 1 o 2 ( o 2 δ(d K = λcr2 N d K, (52 N o ( o = λcr2 N d K, ( 1 where K + N 1 o. Equating the right-hand sides and o rearranging terms N o ( o + N 1 ( + δ(d (N 1 o 2 = 1 o ( δ(d o 2. o (53 For λ, we either have o = Θ(1 or o 0: If o 0, then (53 yields N o +(N 1( o 2( 1 +δ(d, or N / o. Substituting this condition in (52 yields o d /(2λcR 2, so N 2λcR 2 /d. From (51, we obtain D 2 2d e 1 λcr 2. Therefore,d 1 and (30-(32 follow. If o = Θ(1, it is necessary that N d = Θ(λ, otherwise, from (51 D = e Θ(λ. Therefore, D = Θ(λ. Since this scaling is worse than Θ( λ, this case is rejected, which concludes the roof. 2. d = Θ(N : From (33, it is necessary that N = Θ(λT o for a non-vanishing s. From (35, it follows that D = Θ(N d = Θ((λT o 2. Since this scaling is worse than Θ(λT o, this case is rejected, which concludes the roof. G. Proof of Proosition 13 Since Π r is a PPP, we assume, without loss of generality, that the tyical relay is located at the origin, i.e., x = (0,0. Then P cr,2 can be written as [e ] λra(b(z,r1 B(z,r 2} P cr,2 = E r1,r 2,θ E r1 [ e λ ra(b(z,r 1 ], where r 1 = z, r 2 = z, θ = (z,z, B(z,r 1 is the disc with center z and radius r 1, and A( denotes area. If θ [0, π, geometric calculations lead to A(B(z,r 1 B(z,r 2 = (π φr 2 1 +(φ+θr2 2 +yr 1sinφ, where y = z z and φ = (z,z z. For N = 2, by the dislacement theorem [9], Π r,ideal is a PPP with density λ. Therefore θ is uniformly distributed in [0, 2π, the joint df of (r 1,r 2 is f(r 1,r 2 = 4(λπ 2 r 1 r 2 e λπr2 2, r2 > r 1, and the df of r 1 is f(r 1 = 2λπre λπr2 1, r1 > 0. Performing the exectations over r 1,r 2,θ, taking into account the symmetry for θ [0,π and θ [π,2π, and making the change of variables t 1 = λπr 1, t 2 = λπr 2, we obtain (41. REFERENCES [1] M. Haenggi and D. Puccinelli, Routing in ad hoc networks: a case for long hos, IEEE Commun. Mag., , Oct [2] F. Xue and P. R. Kumar, Scaling laws for ad hoc wireless networks: an information theoretic aroach, series of Foundations and Trends in Networking, Now Publishers Inc., vol. 1, , Jul [3] A. E. Gamal, J. Mammen, B. Prabhakar, and D. Shah, Otimal throughut-delay scaling in wireless networks - art I: the fluid model, IEEE Trans. Inf. Theory, , Jun [4] J. G. Andrews, S. Weber, M. Kountouris, and M. Haenggi, Random access transort caacity, IEEE Transactions on Wireless Communications, vol. 9, no. 6, , Jun [5] P. Guta and P. R. Kumar, The caacity of wireless networks, IEEE Trans. Inf. Theory, , Mar [6] M. J. Neely and E. Modiano, Caacity and delay tradeoffs for ad hoc mobile networks, IEEE Trans. Inf. Theory, , Jun

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Haenggi, Outage robability of general ad hoc networks in the high-reliability regime, IEEE/ACM Transactions on Networking, vol. 19, , Aug [25] R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, On the Lambert function, Adv. Comutational Maths., vol. 5, , [26] K. Stamatiou and M. Haenggi, Random-access Poisson networks: stability and delay, IEEE Communication Letters, vol. 14, no. 11, , Nov [27] G. Bianchi, Performance analysis of the IEEE distributed coordination function, IEEE J. Sel. Areas Commun., vol. 18, , Mar [28] S. Weber, A. Kini, and A. Petroulu, A new aroximation for slotted buffered Aloha, in 42nd Annual Conference on Information Sciences and Systems (CISS, 2008, [29] M. Haenggi and R. Smarandache, Diversity olynomials for the analysis of temoral correlations in wireless networks, IEEE Transactions on Wireless Communications, 2013, submitted. Available at htt:// mhaenggi/ubs/twc14.df. Kostas Stamatiou received his Diloma in Electrical and Comuter Engineering from the National Technical University of Athens in 2000, and the M.Sc. and Ph.D. degrees in Electrical Engineering in 2004 and 2009, resectively, from the University of California San Diego (UCSD. From 2009 to 2010 he was a ost-doctoral scholar in the Deartment of Electrical Engineering at the University of Notre Dame, South Bend, Indiana, and from 2010 to 2012 he held a research aointment at the Deartment of Information Engineering at the University of Padova, Italy. He currently holds a researcher osition at the Centre Tecnològic de Telecomunicacions de Catalunya (CTTC, Barcelona, Sain. His research interests lie in the areas of wireless communications, stochastic geometry and random networks, and energy harvesting systems. Martin Haenggi (S95, M99, SM04 is a Professor of Electrical Engineering and a Concurrent Professor of Alied and Comutational Mathematics and Statistics at the University of Notre Dame, Indiana, USA. He received the Dil.-Ing. (M.Sc. and Dr.sc.techn. (Ph.D. degrees in electrical engineering from the Swiss Federal Institute of Technology in Zurich (ETH in 1995 and 1999, resectively. After a ostdoctoral year at the University of California in Berkeley, he joined the University of Notre Dame in In , he sent a Sabbatical Year at the University of California at San Diego (UCSD. For both his M.Sc. and Ph.D. theses, he was awarded the ETH medal, and he received a CAREER award from the U.S. National Science Foundation in 2005 and the 2010 IEEE Communications Society Best Tutorial Paer award. He served an Associate Editor of the Elsevier Journal of Ad Hoc Networks from , of the IEEE Transactions on Mobile Comuting (TMC from , and of the ACM Transactions on Sensor Networks from , and as a Guest Editor for the IEEE Journal on Selected Areas in Communications in and the IEEE Transactions on Vehicular Technology in He also served as a Distinguished Lecturer for the IEEE Circuits and Systems Society in , as a TPC Co-chair of the Communication Theory Symosium of the 2012 IEEE International Conference on Communications (ICC 12, and as a General Co-chair of the 2009 International Worksho on Satial Stochastic Models for Wireless Networks (SaSWiN 09 and the 2012 DIMACS Worksho on Connectivity and Resilience of Large-Scale Networks, and as the Keynote Seaker of SaSWiN 13. Presently he is a Steering Committee Member of TMC. He is a co-author of the monograh Interference in Large Wireless Networks (NOW Publishers, 2009 and the author of the textbook Stochastic Geometry for Wireless Networks (Cambridge University Press, His scientific interests include networking and wireless communications, with an emhasis on ad hoc, cognitive, cellular, sensor, and mesh networks.