Throughput Scaling in Random Wireless Networks: A Non-Hierarchical Multipath Routing Strategy
Awlok Josan, Mingyan Liu, David L. Neuhoff, S. Sandeep Pradhan
aa r X i v : . [ c s . I T ] O c t Throughput Scaling in Random Wireless Networks:A Non-Hierarchical Multipath Routing Strategy
Awlok Josan, Mingyan Liu, David L. Neuhoff and S. Sandeep PradhanElectrical Engineering and Computer Science DepartmentUniversity of Michigan, Ann Arbor, MI 48109
Abstract — Franceschetti et al. [1] have recently shown thatper-node throughput in an extended (i.e., geographically ex-panding), ad hoc wireless network with Θ( n ) randomly dis-tributed nodes and multihop routing can be increased fromthe Ω( √ n log n ) scaling demonstrated in the seminal paper ofGupta and Kumar [2] to Ω( √ n ) . The goal of the present paperis to understand the dependence of this interesting result on theprincipal new features it introduced relative to Gupta-Kumar:(1) a capacity-based formula for link transmission bit-rates interms of received signal-to-interference-and-noise ratio (SINR),instead of the threshold model that positive bit-rate W is attain-able when SINR lies above some threshold, and zero bit-rateotherwise; (2) hierarchical routing from sources to destinationsthrough a system of communal highways, instead of individualdirect routes from each source to the corresponding destination;and (3) cell-based routes constructed by percolation ratherthan by simply interconnecting all cells touched by a straight-line between two end points. The conclusion of the presentpaper is that the improved throughput scaling is principallydue to the percolation-based routing, which enables shorterhops and, consequently, less interference. This is established byshowing that throughput Ω( √ n ) can be attained by a systemthat does not employ highways, but instead uses percolationto establish, for each source-destination pair, a set of Θ(log n ) routes within a narrow routing corridor running from source todestination. As a result, highways are not essential. In addition,it is shown that throughput Ω( √ n ) can be attained with theoriginal threshold transmission bit-rate model, provided thatnode transmission powers are permitted to grow with n . Thus,the benefit of the capacity bit-rate model is simply to permitthe power to remain bounded, even as the network expands. I. I
NTRODUCTION
The problem of asymptotic scalability of throughput inwireless networks has been investigated extensively underdifferent assumptions on the network models. The seminalwork of Gupta and Kumar [2] demonstrated that per-nodethroughput
Ω(1 / √ n ln n ) was achievable as the number ofnodes in the network, n , goes to infinity.Franceschetti et al [1] recently showed that this achievableper-node throughput may be increased. Specifically, theyconsidered an extended (i.e., geographically expanding) net-work with approximately n randomly distributed nodes andmultihop routing, and demonstrated that achievable per-nodethroughput can be increased to Ω( √ n ) .Compared to [2], the construction used in [1] introducedseveral new features. The first is a capacity-based link This work was supported by NSF Grant CCF-0329715. transmission rate formula as a function of the received signal-to-interference noise ratio (SINR), instead of the threshold-based binary rate model used in [2], where a positive bit-rate W is attainable when the SINR is above some threshold, andzero otherwise. (The former requires coding at each hop,while the latter does not.) The second is a routing hierarchyfor data delivery in which data from a source is first delivered(via a single hop) onto a nearby highway – one of a system ofcommunal highways, each with a horizontal and a verticalsegment. The data is then multihopped along the highway(horizontally then vertically), and finally delivered from thehighway to the destination in a single hop. By contrast, themethod used in [2] is a simple shortest path type of routing,where a straight line is drawn connecting the source and thedestination, and nodes along this line are selected to relay thedata, forming an approximately straight line path. The thirddifference introduced in [1] is the use of percolation theory toconstruct the highways that serve as the main routing fabricin the network. Indeed, [1] is the first paper to use percolationtheory to establish network throughput results.The primary interest of the present paper is to understandwhich of the above contribute to the increase in per-nodethroughput in a fundamental way, i.e., to understand thedependence of this new result on the above new features. Theconclusion of this paper is that the improved throughput scal-ing is principally due to the percolation-based routing, whichenables shorter hops and, consequently, less interference.More precisely, the hops along the highways have boundedlengths that do not increase as the network expands. Thiswould not have been possible if one were to use shortest pathrouting, the existence of which then invokes a connectivityrequirement that would force the hop size to increase as thenetwork expands.This conclusion is established by showing that throughput Ω( √ n ) can be attained by a system that does not employhighways, but rather uses percolation to establish, for eachsource-destination (s-d) pair, a set of Θ(log n ) disjoint routeswithin a narrow routing corridor running from source todestination. Thus with this multipath routing structure, high-ways and routing hierarchy are not essential. In addition, itis shown that throughput Ω( √ n ) can be attained with theoriginal threshold transmission bit-rate model, provided thetransmission powers of the nodes are permitted to grow with n . Thus, the benefit of the capacity bit-rate model is simplyto permit the power to remain bounded, even as the networkxpands.The remainder of the paper is organized as follows, Sec-tion II introduces the system and the transmission rate modelswe use. Section III gives our main result and an overviewof the proof. The formal proof follows in sections IV, V, VIand VII, which formalize the path construction, data rates,loading factor and the system scheduling, respectively.II. S YSTEM M ODEL
We consider the random extended network, which consistsof a set of nodes distributed over a disk A n ⊂ R with radius √ n , called the network region. We construct the network byplacing the nodes according to a Poisson point process of unitintensity over R and focusing our attention to the networkregion A n . We denote the location of the i th node by s i .Each node, s i , serves as a source of bits which it wishes tocommunicate to a destination, denoted by d i , which is chosenrandomly from the remaining nodes. Each node may serveas a destination for more than one source. Communicationis done using a multihop relaying scheme under a slottedtime system. There is a transmitter and receiver at eachnode. All transmitters use the same power P , which weget to choose and which may depend upon n . We assumethat node j receives the transmitted signal from node i withpower P η ( d ij ) , where η is a propagation model and d ij isthe Euclidean distance between nodes i and j . We use thepropagation model introduced by Arpacioglu and Haas [3], η ( d ) = 1(1 + d ) α , (1)where α > is a constant depending upon the channelconditions. A. Transmission Rate Models
Let t be a set of simultaneously transmitting nodes. Thenthe SINR ij (signal to interference and noise ratio) at node j when node i is transmitting to it is given bySINR ij = P η ( d ij ) N + P k ∈ tk = i P η ( d kj ) . We use two different transmission rate models.
Model A
In this model, which was used in [1], thetransmission rate is equal to the capacity of the wirelesschannel. That is the rate (in bits/sec) at which node i cantransmit to node j is R ij = 12 W T ln(1 +
SINR ij ) , (2)where W is the bandwidth and T is length of the time slot. Model B
In this model, which has been more commonlyused in throughput analysis of wireless networks [2]–[4] thetransmission rate is R ij = (cid:26) B if SINR ij ≥ τ else , (3)where τ is some pre-determined threshold and B is a numberless than channel capacity. III. M AIN R ESULT
In the following theorem, which is our main result, wedemonstrate the achievability of
Ω(1 / √ n ) throughput forboth transmission rate models, using a non-hierarchical rout-ing strategy, i.e., without the use of highways. Theorem 1:
Under transmission Models A and B, a per-node throughput of
Ω(1 / √ n ) bits/sec is achievable in therandom extended network. Under Model A the throughputis achievable with any constant finite power P at each node,whereas under Model B the throughput is achievable only ifpower P increases to infinity as n → ∞ .We now give an overview of the proof, details of whichare in subsequent sections. For each s-d pair we find withhigh probability Ω(ln n ) disjoint routes (i.e., a sequence ofhops from node to node) from source to destination such that1. each route consists of a draining hop from the source,a path consisting of a sequence of intermediate hops, anda delivery hop ending at the destination,2. the first hop, i.e., the draining hop, has length O (ln n ) and extends from the source to the first node of the path,3. the last hop, i.e. the delivery hop, has length O (ln n ) ,and extends from the last node of the path to thedestination.4. all intermediate hops have lengths bounded by aconstant not depending on n .To make the analysis tractable, we modify these pathsslightly in a way that preserves their distance properties,but does not necessarily preserve their disjointness. We thenshow that for each s-d pair, a rate of Ω(1 / ( √ n ln n )) issustainable on each hop of each of its modified paths. Todo this, we show that the maximum number of source-destination paths on which an intermediate node can lie is O ( √ n ln n ) . From Item 4 above, the intermediate nodes, withthe exception of the delivery node, transmit over a boundeddistance. Theorem 3 of [1] showed that when transmittingover a bounded distance, nodes can maintain a throughput of Ω(1) . Thus for each s-d pair an intermediate node can sustaina throughput of
Ω(1) × /O ( √ n ln n ) = Ω(1 / ( √ n ln n )) .Next, using Theorem 3 of [1] again, we show that a sourcecan transmit data at rate Ω(1 / √ n ) in a way that will bereceived by a node on each of the Ω(ln n ) paths for the s-dpair. Through this node, each path then takes a share of thisrate equal to Ω(1 / ( √ n ln n )) . Therefore, the source is able todrain onto each of the Ω(ln n ) paths at rate Ω(1 / ( √ n ln n )) .Similarly, delivery nodes can deliver data to the destinationat a rate of Ω(1 / ( √ n ln n )) from each path.Combining the above results we see that, for each source-destination pair we have Ω(ln n ) routes, each of which cansustain a rate of Ω(1 / ( √ n ln n )) . Thus the per-node through-put is given by Ω(ln n ) × Ω(1 / ( √ n ln n )) = Ω(1 / √ n ) .IV. P ATH C ONSTRUCTION VIA P ERCOLATION
In this section we show that, with probability approaching1 as n → ∞ , there exist Ω(ln n ) suitable disjoint pathsfor each source-destination pair. Here the probability is with √ cκ ln √ n √ c √ n (a) Tessellation of a rectangular routing corridor with diamonds ofside length c . √ cκ ln √ n √ c √ c √ n (b) Paths crossing the routing corridor from left to right are composedfrom horizontal and vertical edges, shown as dashed lines.Fig. 1. Routing corridor setup for finding paths for a given s-d pair. respect to the Poisson point process for node locations andthe random destination assigned to each source node. Todo this, we use the percolation approach that was used in[1] to establish the existence of suitable highways. Here weapply approach to find a set of suitable paths for each source-destination pair.Since we need to show the existence of paths for everys-d pair, we first need to upper bound the number of nodesin the network region A n , which we denote N n . Lemma 1:
The probability that the number of nodes, N n ,in the network region A n is less than πn goes to 1 as n goes to infinity. Proof:
The number of nodes in the network region, N n ,is a Poisson random variable with mean πn . Applying theChernoff bound gives, Pr( N n > πn ) ≤ e − sπn E [ e sN n ]= e − sπn e πn ( e s − for all s > . Choosing s = 1 gives Pr( N n ≤ πn ) ≥ − e − πn e πn ( e − = 1 − e πn (3 − e ) → as n → ∞ . (cid:3) Next we prove that for a given s-d pair, there are
Ω(ln n ) disjoint paths such that the distance to (from) each pathfrom (to) the source (destination) is O (ln n ) , and that everyintermediate hop along each path is of length O (1) , i.e. itslength is upper bounded by a constant independent of n .To show this, we consider a rectangular routing corridor ofdimensions √ n × √ cκ ln √ n √ c in R that includes both sand d, where c, κ > are constants to be chosen later.Tessellate this routing corridor with diamonds of side c asshown in Figure 1(a). Then for any given diamond, Pr( diamond contains at least one node ) = 1 − e − c , p . If a diamond contains at least one node, it is said to be open ,and closed otherwise. Draw horizontal edges across half thediamonds and vertical edges across the others in the mannershown in Figure 1(b). An edge is considered open if it liesin an open diamond, and closed otherwise. Define a pathas a sequence of connected edges, horizontal or vertical. Apath is said to be open if it contains only open edges. Wewill show that there are
Ω(ln n ) disjoint open paths crossingthe routing corridor lengthwise, i.e. beginning at the left andending at the right side of the routing corridor.Let I m be the event that there exist at least m disjointopen paths that cross the routing corridor lengthwise.The following lemma, whose proof can be found in theproof of Theorem 5 of [1] is based on an important resultfrom percolation theory. Lemma 2:
Given arbitrary constants κ, c > , there existsa strictly positive constant β = β ( c, κ ) such that Pr( I m ) ≥ − (cid:16) n c (cid:17) a (4)where m = βκ ln √ n √ c and a = (cid:0) ( β − κc + κ ln 6 + 1 (cid:1) .We now set up a routing corridor for each s-d pair. Thefollowing theorem demonstrates that when n is large, withhigh probability there are Ω(ln n ) disjoint paths in each oneof those corridors. Theorem 2:
Given κ > and c > ln 6 + 4 /κ , there existsa strictly positive constant β ( c, κ ) such that if for every n we are given at most ⌈ πn ⌉ routing corridors of dimensions √ n ×√ cκ ln √ n √ c in R , then with probability approachingone there exist m = βκ ln √ n √ c disjoint open lengthwisecrossing paths within each of the routing corridors.Observe that when n are large, the routing corridors arequite narrow. Proof:
We prove this theorem using Lemma 2 and the unionbound. It suffices to assume that we have ⌈ πn ⌉ routingcorridors. Then Pr( all ⌈ πn ⌉ routing corridors have m disjoint open paths )= 1 − Pr( at least one routing corridor hasless that m disjoint open paths ) ≥ − ⌈ πn ⌉ X i =1 Pr( ith routing corridor hasless than m disjoint open paths ) ≥ − ⌈ πn ⌉ · Pr( a routing corridor pair hasless than m disjoint open paths )= 1 − ⌈ πn ⌉ (1 − Pr( I m )) ≥ − n · (cid:16) n c (cid:17) a = 1 − c ) a n a +1 where the first inequality follows from the union-bound andthe second inequality uses Lemma 2. Note that the above d √ n √ c ln √ n √ c √ n Fig. 2. For a given s-d pair the orientation of the routing corridor on thenetwork region. expression goes to one as n tends to infinity if a < − .Given κ > and c > ln 6 + 4 /κ , choosing β ( c, κ ) = 1 − ( κ ln 6 + 4) / ( κc ) > results in a < − . (cid:3) Corollary 1:
Given κ > and c > ln 6 + 4 /κ , thereexists a strictly positive constant β ( c, κ ) > such that withprobability approaching one there exist Ω(ln n ) disjoint openpaths for each s-d pair such that the distance of any path fromthe source and destination is less than √ cκ ln( √ n/ √ c ) and every intermediate hop has length less than √ c . Proof:
For any given s-d pair, consider a routing corridor withthe aforementioned dimensions such that it contains bothsource and destination and that the portion of the routingcorridor that intersects the network region is as high aspossible (see Figure 2). According to Lemma 2, with highprobability there are
Ω(ln n ) disjoint open paths that crossthe routing corridor lengthwise. Now consider the part of therouting corridor that lies within the network region. Sincethere are Ω(ln n ) disjoint open paths that cross the routingcorridor lengthwise, there will be Ω(ln n ) disjoint open pathsin the truncated region as well. Also, since the width of therouting corridor is √ cκ ln √ n √ c , the minimum distances ofeach of these paths from the source and the destination isless than √ cκ ln √ n √ c . Also, using a geometric argument, itis easy to see that any intermediate hop has less √ c or less.Theorem 2 shows the existence of paths for a numberof routing corridors no larger than ⌈ πn ⌉ . Using the aboveconstruction for every s-d pair and combining with the factthat the number of s-d pairs is less the πn with highprobability (Lemma 1) completes the proof of the corollary. (cid:3) As suggested earlier, for tractability we need to modifythe paths provided by the corollary. Ignoring the previoustesselations of routing corridors, consider now a tessellationof the entire network region into squares of side c . If a squarehas multiple nodes in it, we designate one node as the relaynode . Now, for every hop of every s-d path, if the node thatis to transmit is not the designated relay node for the square, we replace it with the designated relay node. In this waywe obtain a set of Ω(ln n ) paths for each s-d pair such thateach source (destination) is within O (ln n ) of each of itspaths. Note, however, that now the maximum intermediatehop length has been increased to ( √ √ c . Moreover,the paths corresponding to one s-d pair might no longer bedisjoint. For example, in two originally disjoint paths theremight be a node in one path and a node in the other that arecontained in adjacent diamonds in the original tesselation ofthe routing corridor, but are in the same square of the newtesselation of the entire network region. In this case, the twomodified paths share a common relay node.V. D ATA R ATES
We begin this section by finding a lower bound on theper-node transfer rate when for some
D > every nodehas to send data to all nodes within distance D of itself.This involves setting up a TDMA schedule so as to limit thenumber of simultaneous transmissions taking place, whichin turn limits the interference. Corollaries are then given foruse in the proof of the Theorem 1.For transmission rate Model A, Theorem 3 of [1] can beused. The following extends this theorem to transmission rateModel B. Theorem 3:
Given c > , given a tessellation of the net-work into squares with sides of length c , and given an integer d > there exists a rate R ( d ) = Ω( d − α − ) using Model Aand R ( d ) = Ω( d − ) using Model B such that one node ineach square can successfully transfer data at rate R ( d ) toany node located in any square within Manhattan distance d of the originating square (i.e. d or fewer horizontal and/orvertical steps).The asymptotic behavior of the rate under Model A canbe attained by any fixed finite power at each node. Howeverto achieve the rate under Model B we have to let power P go to infinity as d tends to infinity. Proof:
For Model A the proof is given in [1, Theorem 3],and for the extension to Model B, we now make a similarconstruction. We consider a partition of the network regioninto super-squares, each composed of k smaller squares,for some k to be chosen later. We index the squares ineach super-square starting in the lower left corner, movinghorizontally in the bottom row from left to right, and thenin the row above it from left to right, and so on. We set upa TDMA schedule of k slots such that in the i th slot, fromevery square indexed by i , precisely one node can transmit.Consider a transmitter-receiver pair separated by d squares. Choosing k = x ( d + 1) , where x = max(2 , ⌈ (16 τ γ ) /α (1 + 1 / (2 c )) ⌉ ) , and γ = P ∞ i =1 ( i − / − α +1 , we can see that the closest 8interferers are at least x ( d + 1) − d squares away, the nextclosest 16 interferers are at least x ( d + 1) − d squares away,and so on (Figure 3). The power from interfering nodes can =2(d+1) d Fig. 3. Construction for lower bound on SINR. The shaded square atthe center is the actual signal, all other shaded squares are interferingtransmitters. In the above figure d = 1 . thus be upper bounded as P I ( d ) ≤ ∞ X i =1 iP η ( c ( ix ( d + 1) − d ))= ∞ X i =1 iP (1 + c ( ix ( d + 1) − d )) α ≤ P ∞ X i =1 i ( c ( d + 1)( ix − α = 8 Pc α ( d + 1) α x α ∞ X i =1 i ( i − /x ) − α ≤ Pc α ( d + 1) α x α ∞ X i =1 i − / i − / − α = 16 Pc α ( d + 1) α x α ∞ X i =1 ( i − / − α +1 = 16 P γc α ( d + 1) α x α . Next we lower bound the signal power at the receiver. TheEuclidean distance between the transmitter and receiver is atmost c ( d + 1) . Thus the signal power, P S ( d ) , satisfies P S ( d ) ≥ P η ( c ( d + 1))= P (1 + c ( d + 1)) α . Using the above two bounds we obtain a bound on theSINR:SINR ( d ) = P S ( d ) N + P I ( d ) ≥ P (1 + c ( d + 1)) − α N + 16 P γ (2 c ) − α x − α = (cid:18)(cid:18) c ( d + 1) (cid:19) α N P + (cid:18) c ( d + 1) (cid:19) α γx α (cid:19) − . It can be easily shown that the second term in the aboveequation is less than /τ . Choosing P large enough thatthe sum of two terms still remains less than /τ results inSINR > τ . In this case according to Model B, one node ineach square can transmit at rate 1 in such a way that allnodes within Manhattan distance d will successfully receivethe transmissions. Since each square is allowed to have atransmitting node once every k = x ( d + 1) time slots,to get the asymptotic behavior we need to divide the abovetransfer rate by d . Thus under Model B, R ( d ) = Ω( d − ) isattainable. (cid:3) We now give a corollary to the above theorem that willbe used to show an achievable data delivery rate to thedestination.
Corollary 2:
Given c > , given a tessellation of thenetwork into squares with sides of length c , and given aninteger d > there exists a rate R ( d ) = Ω( d − α − ) forModel A and R ( d ) = Ω( d − ) for Model B such that onenode in each square can receive data at rate R ( d ) from atransmitter located in any square within Manhattan distance d of the receiving square (i.e. d or fewer horizontal and/orvertical steps). Proof
The proof is obtained by switching the role of trans-mitters and receivers in the proof of the previous theorem. (cid:3)
We conclude this section with three corollaries that useTheorem 3 to establish rates at which, respectively, draining,delivery and transmission along the intermediate hops canproceed.
Corollary 3:
With probability approaching one, everysource node in the network can transmit to every one ofthe
Ω(ln n ) paths in its corresponding routing corridor ata rate Ω((ln n ) − α − ) under transmission Model A, and Ω((ln n ) − ) under Model B. Proof:
First, for Model A, consider the tessellation of A n intosquares of side length c . Consider also any one source node.Since the Manhattan distance from this source to each of itspaths is less than φ ln n , for some φ > , if this node is theonly node within its square then Theorem 3 with d = φ ln n implies it can transmit data that is successfully received bya node on each of its paths at rate R ( φ ln n ) = Ω((ln n ) − α − ) . It is therefore decided that nodes will transmit at rate
Θ((ln n ) − α − ) , and since each path takes responsibility forrelaying an equal share of this data, each path is responsibleto relay Θ((ln n ) − α − ) . When n is large, with high proba-bility the number of nodes in a square of size c is O (ln n ) [1, Lemma 1]. Every node can actually transmit data at rateof Θ((ln n ) − α − ) . The proof for Model B follows similararguments. (cid:3) orollary 4: With probability approaching one, every des-tination node in the network can receive data from every oneof the
Ω(ln n ) paths in its corresponding routing corridor ata rate Ω(ln n ) − α − ) under Model A, and Ω((ln n ) − ) underModel B. Proof:
First, for Model A, consider a tessellation of A n into squares of side length c . Consider any one destinationnode and one of the source nodes that corresponds to thatdestination. Since the distance to the destination from eachof its paths is less than φ ln n , for some φ > , if this nodeis the only node within its square then Corollary 3 impliesthat data can be successfully received by the destination atrate R ( φ ln n ) = Ω((ln n ) − α − ) . It is therefore decided thatnodes delivering data to this destination will transmit at rate Θ((ln n ) − α − ) . Using the Chernoff bound we can easily seethat the number of sources that choose any given node asits destination is O (ln n ) with high probability. Setting up aTDMA scheme in which each epoch consisting of O ((ln n ) ) slots would allow the destination to receive from every pathof every source that selects the given node as its destinationat least once in every epoch. Thus a destination can receive atrate Ω((ln n ) α − ) . When n is large with high probability thenumber of nodes in a square of size c is O (ln n ) [1, Lemma1]. Thus every node can receive data at rate Ω((ln n ) − α − ) .The proof for Model B follows similar arguments. (cid:3) Corollary 5:
Given c > , and a tessellation of A n intosquares of side length c , one node in every square cantransmit to every node located within distance O (1) , i.e.,distance is upper bounded by a constant that does not dependupon n , at a constant rate that does not depend upon n . Proof:
First consider Model A. From Theorem 3 we knowthat one node in every square can achieve a rate of Ω( d − α − ) while transmitting to every node located within Manhattandistance d of the originating square. For transmissions overdistance that is upper bounded by a constant not dependingupon n , d would be a constant. Hence rate Ω(1) is achievableover constant distance. The proof for Model B followssimilar arguments. (cid:3)
VI. L
OADING F ACTOR
The loading factor of a designated relay node is thenumber of s-d paths on which it lies. We also consider itto be the loading factor of the square containing the relaynode. In this section we find a probabilistic upper bound tothe maximum loading factor among all squares, which thenupper bounds the maximum loading factor of all relay nodes.Let L i ( n ) represent the loading factor of the i th square,and let L ( n ) = max i L i ( n ) . We observe that if an s-d paircontributes a path or paths to the L i ( n ) , then it must be thatthe corresponding routing corridor intersects the i th square.Now, we observe that if the i th square intersects a given s-drouting corridor, it can, at most, intersect 9 diamonds of therouting corridor tessellation. Recall that the tentative pathsfor a given s-d pair are disjoint, i.e. a diamond of the s-drouting corridor can lie on only one tentative path. Thus, if s r c √ c/ √ √ r − ( c/ √ ( r + √ n ) √ n Fig. 4. The i th square lies on a s-d path only if the destination lies in thestriped region. the i th square intersects the s-d routing corridor it may haveto service at most 9 paths corresponding to that s-d pair.Therefore as an upper bound to L ( n ) , we upper boundthe number of s-d routing corridors that intersect any givensquare and multiply that number by 9. Theorem 4:
For a tessellation of the network region intosquares of side c , there exists a constant δ such that Pr( L ( n ) ≤ δ √ n ln n ) → as n → ∞ . Proof:
Pr( L ( n ) ≤ δ √ n ln n ) = Pr(max i L i ( n ) ≤ δ √ n ln n ) ≥ − M n X i =1 Pr( L i ( n ) > δ √ n ln n ) (5)where M n ≈ πnc is the number of squares in the networkregion. We have L i ≤ P N n j =1 A ij where A ij = 1 if the i thsquare intersects the routing corridor corresponding to the j th s-d pair and A ij = 0 otherwise. Note that for a given i , A i , A i ... are independent and identically distributed.However the L i ’s are not identically distributed. Instead L i will generally have a higher value for squares near the centerof A n than its boundary. The following lemma, which givesa uniform upper bound to p n,i , Pr( A ij = 1) , will be usedto find a lower bound to the term Pr( L i > δ √ n ln n ) thatappears in (5). Lemma 3:
Given c > / √ there exists µ such that p n,i ≤ p n , µ ln n/ √ n , for all n, i . (6) Proof:
We setup a polar coordinate system such that theorigin lies at the center of the network region. As theprobability of intersection of a square by a random s-d pairrouting corridor is highest at the center, we consider the i thsquare to lie at the center of the network region, i.e., tocontain the origin. Since such a square of side c is completelycontained in a circle of radius c/ √ , we upper bound p n,i bythe probability of a random s-d routing corridor intersectinga circle of radius c/ √ centered at the origin.or a source located at ( r, θ ) , the probability that square i is intersected by the s-d pair routing corridor is upperbounded by the probability of the destination lying in thestriped regions of Figure 4. Since the diameter of the networkregion is √ n , the area of the horizontally striped regions canbe upper bounded by · √ n · √ cκ ln √ n √ c . Since c > / √ the upper bound can be relaxed to √ n · √ cκ ln n . Also,the area of the vertically striped portion is ( c/ √ √ r − ( c/ √ ( r + √ n ) . Therefore Pr ( s-d routing corridor intersects square i | s = ( r, θ )) ≤ ( if r ≤ c √ κcπ ln n √ n + ( c/ √ √ r − ( c/ √ ( r + √ n ) πn otherwiseSince the joint probability density of the polar coordinatelocations is p ( r, θ ) = rn π , we have p n,i = Z π Z √ n Pr (cid:16) s-d routing corridorintersects square i (cid:12)(cid:12)(cid:12) s = ( r, θ ) (cid:17) p ( r, θ ) drdθ ≤ Z c rn dr + Z √ nc √ κcπ ln n √ n +( c/ √ q r − ( c/ √ ( r + √ n ) πn rn dr ≤ c n + 2 √ κcπ ln n √ n + ( c/ √
2) ln n √ n = c √ n ln n + 2 √ κπ + 1 √ ! c ln n √ n ≤ µ ln n √ n , p n where µ = c (2 + (2 √ κ ) /π ) . (cid:3) Returning to the proof of Theorem 4, since L i ≤ P N n j =1 A ij , we have E [ L i ] ≤ E [ N n X j =1 A ij ] = 9 E [ N n ] E [ A ij ] = 9 πnp n,i ≤ πnp n . Applying the Chernoff bound [4, Lemma C3],
Pr( L i > πnp n ) ≤ exp (cid:26) − πnp n ln 3 e (cid:27) . Substituting the above into (5) and choosing δ = 27 π gives Pr( L ( n ) ≤ δ √ n ln n ) ≥ − M n X i =1 exp (cid:18) − πnp n ln 3 e (cid:19) = 1 − exp (cid:18) − πnp n ln 3 e + ln M n (cid:19) = 1 − exp (cid:18) − πn µ ln n √ n ln 3 e +ln πnc (cid:17) → − as n → ∞ , which concludes the proof of Theorem 4. (cid:3) VII. S
YSTEM S CHEDULING
In this section we explain a system protocol that achievesa per-node throughput of
Ω(1 / √ n ) and complete the proofof Theorem 1.For every path corresponding to an s-d pair we designatethe node on the path that is closest to the source (destination)as the draining (delivery) node. We cycle among threedifferent categories of time slots: draining, relaying anddelivery. In draining slots, the source transmits its packetsto the designated draining nodes. In the relaying slots, therelaying nodes transmit the data towards the destination.Finally in the delivery slots, the delivery nodes transmitsthe data to the destination.Theorem 4 shows that the maximum number of s-d pathsthat a relaying node may have to serve is O ( √ n ln n ) . Sinceall relaying nodes can transmit at rate Ω(1) (Corollary 5),the relaying node can maintain a throughput of
Ω(1 / √ n ln n ) per path.From Corollaries 3 and 4, it is easy to see that a rate of Ω(1 / ( √ n ln n )) per path can be maintained in the drainingand the delivery phase.Thus every s-d pair can achieve a rate of Ω (cid:18) √ n ln n (cid:19) bits/sec/path × Ω(ln n ) paths = Ω (cid:18) √ n (cid:19) bits/sec(7)which completes the proof of Theorem 1.R EFERENCES[1] M. Franceschetti, O. Dousse, D. Tse and P. Thiran, “On the throught-put capacity of random wireless networks,” to appear in
IEEE Trans.on Inform. Theory .[2] P. Gupta and P. R. Kumar, “The capacity of wireless networks,”
IEEETrans. Inform. Theory , vol. 46, pp. 388-404, Mar. 2000.[3] O. Arpacioglu and Z. Haas “On the scalability and capacity of wirelessnetworks with omnidirectional antennas,”
IPSN , Berkeley, Apr. 2004.[4] E. Duarte-Melo, A. Josan, M. Liu, D. L. Neuhoff and S. Pradhan, ”Theeffect of node density and propagation model on throughput scalingof wireless networks,”, Berkeley, Apr. 2004.[4] E. Duarte-Melo, A. Josan, M. Liu, D. L. Neuhoff and S. Pradhan, ”Theeffect of node density and propagation model on throughput scalingof wireless networks,”