The User Capacity of Barrage Relay Networks

Transcription

1 The User Capacity of Barrage Relay Networks Thomas R. Halford TrellisWare Techologies, Ic Via Esprillo, Suite 300 Sa Diego, CA Thomas A. Courtade Ceter for Sciece of Iformatio Staford Uiversity Staford, CA Kurt A. Turck Networkig Techology Brach Air Force Research Laboratory Rome, NY Abstract Barrage relay etworks (BRNs) are a class of mobile ad hoc etworks based o a autoomous cooperative commuicatios scheme that affords a distributed, rapid, ad robust broadcast mechaism. BRN-based radios are curretly beig used operatioally; uderstadig scalig laws for BRNs ca thus shed light o how future systems ought be desiged to address a wider rage of military missios. It has previously bee show that BRNs scale optimally for broadcast traffic (i terms of sum throughput ad latecy). Furthermore, experimetal evidece supports the scalability of BRNs i practice: a 215-ode etwork of TrellisWare s BRN-based CheetahNet radios was demostrated i 2010 as part of the Rim of the Pacific (RIMPAC) exercise. I this work, we study scalig laws for uicast i BRNs. The spatial extet of uicast trasmissios i BRNs ca be cotaied via cotrolled barrage regios (CBRs), while barrage access cotrol protocols which are a type of path-orieted medium access cotrol (PO-MAC) protocol ca be used to schedule CBRs i space ad time. Traditioal techiques for scalig aalysis support either autoomous cooperatio or PO- MAC protocols; we therefore study the fudametal limits of CBR-based protocols through the les of user capacity, which we defie as the maximum umber of costat-rate, delay-optimal uicast flows that a -ode etwork ca simultaeously support. The user capacity of dese etworks operatig uder a CBR protocol is show to scale as Θ( ) whe source-destiatio pairs are chose radomly. BRNs are thus able to trasport Θ( / log ) bit-meters/sec, which is order optimal (i the Gupta-Kumar sese). If the source-destiatio pairs are istead chose so that their separatio (i hops) is a geometric radom variable, the the user capacity scales as Θ(/ log ). Thus, i a wireless etwork domiated by localized traffic, BRNs ca achieve throughput that is almost liear i the umber of users. Fially, it is show via software simulatio that distributed, greedy algorithms provide order optimal schedulig of CBRs for both the radom ad localized traffic models. This suggests that scalable protocols for uicast i BRNs ca be realized i practice. I. INTRODUCTION I respose to the limited success of traditioal, strictly layered approaches to mobile ad hoc etwork (MANET) architectures for military applicatios (cf., [1 3]), barrage relay etworks were desiged from the groud up to meet the demads of commuicatios at the tactical edge [4]. As Approved for Public Release; Distributio Ulimited: 88ABW dated 20 Jul ACKNOWLEDGMENT OF SUPPORT AND DIS- CLAIMER: (a) Cotractor ackowledges Govermet s support i the publicatio of this paper. This material is based upo work fuded by AFRL uder AFRL Cotract No. FA C (b) Ay opiios, fidigs ad coclusios or recommedatios expressed i this material are those of the author(s) ad do ot ecessarily reflect the views of AFRL. detailed i [5], BRNs utilize time divisio multiple access (TDMA) ad cooperative commuicatios as the basis of a efficiet broadcast protocol wherei packets ripple out from a source i pipelied spatial waves. This broadcast mechaism is the fudametal physical layer resource i a BRN; barrage access cotrol (BAC) protocols operatig at Layer 2 coordiate which ode is the broadcast source o ay give time slot, rather tha which poit-to-poit liks are active. It was show i [5] that BRNs provide optimal broadcast scalig: i a etwork where each ode ca trasmit W bits/sec, the sum throughput that ca be achieved by a arbitrary umber of broadcast sources scales as 1 Θ(W ). BRNs were proposed iitially for platoo-sized missios domiated by latecy-critical, broadcast ad multicast traffic (e.g., push-to-talk (PTT) voice ad real-time streamig video). Icreasigly, BRN-based radios are beig used to support larger missios ad richer classes of data services. For example, the Marie Corps Warfightig Laboratory deployed a 215- ode etwork of TrellisWare s CheetahNet radios as part of the Limited Objective Experimet 4 (LOE-4) exercise at RIMPAC Durig this exercise, applicatios that are iheretly uicast (e.g., IP chat) were supported i additio to broadcast services such as PTT voice ad positio locatio iformatio (PLI) moitorig. Such practical use cases motivate the study of efficiet protocols for uicast trasport i BRNs. The buildig blocks for uicast protocol desig i BRNs have bee described i [5 8]. Briefly, the spatial extet of trasmissios i BRNs ca be be cotaied via cotrolled barrage regios (CBRs). BAC protocols ca be used to schedule CBRs that are multiplexed i both space ad time (i.e., more tha oe uicast source is active o ay time slot). This work studies the fudametal limits that gover ay protocol employig CBRs for uicast trasmissio i BRNs. The assumptios made i traditioal scalig studies [9] fail to capture two saliet features of BRNs: cooperative commuicatios ad path-orieted medium access cotrol techiques that reserve etire multihop routes at Layer 2. We therefore study scalig laws for BRNs through the les of user capacity, which we defie as the maximum umber of costat-rate, delay-optimal uicast flows that a -ode 1 A fuctio f() =Ω(g()) if costats 0 ad c such that f() cg(), > 0. Similarly, f() =O (g()) if f() cg(), > 0.Iff() is Ω(g()) ad O (g()), the we say f() = Θ(g()). Throughout, o(1) deotes a quatity that vaishes as /12/$ IEEE

2 etwork ca simultaeously support. I the cotext of BAC protocols, user capacity ca be iterpreted as the maximum umber of CBRs that ca be accessed at ay give time. I a tactical sceario, user capacity measures, for example, how may streamig video feeds ca be simultaeously supported, a measure ot readily derived from the aalysis of [9]. Usig several ew results that are derived i a appedix, we demostrate i Sectio III that the user capacity of BRNs employig CBRs scales as Θ( ) whe the source ad destiatio odes are paired radomly. BRNs are thus capable of trasportig Θ( / log ) bit-meters/sec ad ca achieve the trasport capacity defied by Gupta ad Kumar [9]. Ituitively, this result idicates that although a give CBR cotais more odes tha is strictly required for uicast trasport the CBR cotais all possible shortest paths from the source to the destiatio, rather tha a sigle shortest path this does ot affect asymptotic scalability (all the while dramatically icreasig the reliability [10] ad stability [8] of multihop data trasport). It is further demostrated i Sectio III that the user capacity of BRNs employig CBRs scales as Θ(/ log ) whe the distaces betwee uicast source/destiatio pairs is geometrically distributed. Thus, i a wireless etwork domiated by localized traffic, BRNs ca achieve throughput that is almost liear i the umber of users. Fially, it is show via software simulatio i Sectio III that distributed, greedy algorithms provide order optimal schedulig of CBRs for both the radom ad localized traffic models. This suggests that scalable protocols for uicast trasport i BRNs ca be realized i practice. This paper cocludes i Sectio IV with a discussio of the desig challeges associated with such practical protocols. A. Network Model II. MODELS AND METRICS We adopt the stadard radom etwork model i this work wherei odes are distributed radomly o the uit square ad commuicatio at a rate of W bits/sec is possible betwee two odes if ad oly if they are separated by less tha the trasmissio rage r(). A key result i geometric radom graph theory asserts that the threshold o r() required for such a etwork to be coected scales as [11, 12]: ( ) log r() Θ. (1) This critical desity is assumed throughout. Whe simulatig radom etworks i Sectio III, a trasmissio rage of r () = 3 log (2) 2 π is assumed to assure coectivity with high probability 2. 2 A evet E occurs with high probability if α>1 there is a appropriate choice of costats for which E occurs with probability at least 1 O(1/ α ). B. Cotrolled Barrage Regios ad CBR Protocols Due to space limitatios, the reader is referred to [5] for a full descriptio of BRNs. It suffices presetly to describe CBRs. Briefly, cotrolled barrage regios ca be established by a reactive protocol as follows. Cotrol messages broadcast by the source (s) ad destiatio (d) of a uicast flow iform odes of their respective distace to s ad d i hops, as well as the distace of the shortest path from s to d. Nodes the use these three distaces to ascertai whether they are o ay of the shortest paths from s to d, i which case they are iterior to the CBR ad relay messages from s per the barrage cooperative decode-ad-forward protocol. Nodes that are ot iterior to the CBR, but which are eighbors of iterior odes, suppress their relay fuctio so that exteral packets do ot propagate ito the CBR, or do iteral packets propagate to the rest of the etwork. I this way, multiple uicast trasmissios may be established i the etwork. Defiitio 1 (CBR Protocol): A CBR protocol is a mechaism that establishes a cotrolled barrage regio such that a ode is iterior to the CBR if ad oly if it lies o ay shortest path coectig the source ad destiatio. C. Path-Orieted Medium Access Cotrol Although we focus primarily o BRNs, the results preseted herei ca be iterpreted i the cotext of ay PO-MAC protocol that reserves etire source-destiatio (s-d) paths at Layer 2 (e.g., [1]). Observe that a istataeous throughput of W/3 bits/sec ca be achieved alog what is effectively a multihop lie etwork uder such protocols. Furthermore, this trasmissio is delay-optimal: the ed-to-ed latecy depeds oly o the average s-d hop distace ad ot o, for example, the umber of two-hop eighbors of ay relay ode. Our goal i this paper is to compare the CBR protocol to the performace attaiable by ay etwork that employs a PO- MAC protocol. To this ed, we make the followig defiitio: Defiitio 2 (PO-MAC User Capacity): The PO-MAC user capacity is the maximum umber of mutually o-iterferig uicast flows that ca be simultaeously supported by a wireless etwork usig a PO-MAC protocol. Observe that these flows support Θ(W )-throughput, delay optimal data trasport provided the PO-MAC reserves etire routes. We show i the Sectio III that CBR protocols attai order optimal scalig with respect to the PO-MAC user capacity. D. Uicast Traffic Models Fially, we cosider two differet types of traffic model which represet the extremes of global ad local traffic i this paper. I a realistic sceario, the traffic geeratio process would likely be a mixture of these two extremes. Defiitio 3 (Uiform Traffic): Uicast traffic is said to be geerated accordig to the uiform traffic model if /2 distict source-destiatio pairs are chose radomly from the etwork odes. Note that the uiform traffic model is equivalet to radomly placig /2 source-destiatio pairs i the plae ad geeratig the iduced radom geometric graph.

3 Defiitio 4 (Localized Traffic): Uicast traffic is said to be geerated accordig to the localized traffic model with mea β if /2 distict sources are chose from the set of vertices, ad for each source s, a destiatio is chose uiformly from the set of vertices that are H(u) hops away from s. Here, H(u) is a geometric radom variable with mea β. I the localized traffic model, it is possible that a ode is a source for oe uicast flow ad a destiatio for aother distict albeit mutually iterferig flow. III. MAIN RESULTS AND DISCUSSION A. Results All of our results assume that the uderlyig etwork is modeled as a dese radom geometric graph with coectivity radius r() = Θ( log /) chose sufficietly large to esure coectivity with high probability (w.h.p.). All results hold with high probability i. With the exceptio of the first result, we delay proofs util the appedix. Mai Result 1: Uder ay traffic model, a etwork operatig uder a PO-MAC protocol supports at most O(/ log ) simultaeous uicast flows. Proof: A PO-MAC protocol reserves paths i a etwork, oe per uicast flow, ad the paths must be mutually oiterferig. Each path creates a iterferece regio of area greater tha πr() 2 sice it must cotai at least oe trasmittig ode. Dividig the area of the uit square by πr() 2 yields the desired upper boud. This first mai result places a upper limit o the user capacity of ay etwork employig a PO-MAC protocol, ad hece a upper limit o the umber of CBRs that ca be accessed simultaeously i a BRN. Ituitively, this upper boud results from the iterferece footprit of the source odes aloe. Our secod mai result idicates that this upper boud is i fact achievable i BRNs whe traffic is localized: Mai Result 2: A BRN etwork usig a CBR protocol supports Θ(/ log ) simultaeous flows uder the assumptios of the radom localized traffic model. Note that Result 1 does ot imply that ay Θ(/ log )- sized subset of the /2 flows i a realizatio of the localized traffic model ca be supported. Istead, it says that the size of the largest simultaeously supportable subset scales as Θ(/ log ). Result 1 ca be combied with the lower boud implicit i Result 2 i.e., if a CBR protocol achieves Ω(/ log ) user capacity the at least oe path-orieted MAC protocol has bee show to do so to yield a scalig law that characterizes the user capacity for etworks operatig with ay PO-MAC protocol uder the localized traffic model. Mai Result 3: Uder the localized traffic model, the PO- MAC user capacity scales as Θ(/ log ). Results 2 ad 3 together are iterestig as they demostrate that CBR protocols scale as well as ay path-orieted MAC protocol. That is to say, the additioal relay odes reserved i a CBR beyod the shortest path route odes do ot affect asymptotic scalability whe traffic is localized. It was show i [10] ad [8] that these additioal relay odes serve to ehace the reliability ad stability, respectively, of multihop data trasport, eve over 2- ad 3-hop CBRs. Our last two aalytical results cocer the trasport capacity of BRNs ad the umber of uicast flows that ca be supported by a BRN uder the assumptios of the uiform traffic model. Mai Result 4: Uder the uiform traffic model, a BRN employig a CBR protocol supports Θ( ) simultaeous uicast flows. Mai Result 5: Uder the uiform traffic model, a BRN employig a CBR protocol is capable of trasportig 3 Θ( / log ) bit-meters/sec ad therefore ca achieve the trasport capacity i the sese of [9]. Result 5 shows that, up to some costat factor that is idepedet of the etwork size, BRNs ca trasport as much data as arbitrary wireless etworks architectures. Sice shortest paths betwee vertices i a sufficietly dese radom geometric graph ca be approximated by straight-lie segmets, the results derived i the ext sectio suggest that uder the uiform traffic model, a etwork operatig uder a PO-MAC protocol ca support at most O( ) simultaeous traffic flows. This leads to the followig cojecture: Cojecture 1: A barrage relay etwork employig a CBR protocol achieves the PO-MAC user capacity (i the sese of order-optimal scalig) for the radom uiform traffic model. The proofs of these results hit that a greedy CBR formatio protocol ca be employed to achieve the user capacity. To this ed, we have simulated a greedy CBR formatio protocol uder each traffic model. The umerical results strogly idicate that the performace of a greedy CBR formatio protocol coicides with the optimal scalig results give i the above results. Simulatio results are give i Figure 1. B. Discussio By their very ature, BRNs embrace the fudametal broadcast characteristic of the the wireless medium rather tha try to cotrol it via a collisio avoidace mechaism. It was demostrated i [5] that a simple Layer 2 protocol achieves order optimal scalig for broadcast traffic. However, if the rapid broadcast mechaism is left uchecked, BRNs suffer from scalability issues for uicast ad multicast traffic sice a sigle multicast iteded for a small group of receivers ca moopolize the etire etwork for the duratio of the trasmissio. Employig a protocol that uses CBRs to cotai the spatial extet of BRN broadcasts is oe method of copig with this scalability issue. While the CBR protocol is a atural extesio of the traditioal barrage framework, it was ukow util ow whether it would scale i a optimal maer for uicast traffic. The results of the previous subsectio idicate that the aswer to this questio is affirmative for the localized traffic model, ad umerical ad aalytical evidece strogly idicate that this is the case for the uiform traffic 3 A bit-meter correspods to the trasport of 1 bit of iformatio 1 meter closer to its iteded destiatio.

4 Number of Simultaeous CBRs / log 0.46 Localized Traffic Model Uiform Traffic Model Number of Nodes i Network () Fig. 1. Simulatio results for a distributed, greedy implemetatio of the CBR protocol. The solid blue lies represet the umber of simultaeous mutually o-iterferig CBRs supported by a greedy implemetatio of the CBR protocol. The results were averaged over 100 idepedet trials for each etwork size. The vertical blue lies idicate ±1 stadard deviatio. model as well. Thus, BRNs employig a CBR protocol ejoy the simplicity ad beefits of a traditioal BRN (cf., [5]), but they also support scalig to hudreds or thousads of odes. IV. CONCLUSION I this paper we proved that BRNs employig a CBR protocol ca support Θ(/ log ) ad Θ( ) simultaeous uicast sessios uder localized ad uiform traffic models, respectively. These results prove that order-optimal throughput scalig is achieved for the localized traffic model, ad the umerical ad aalytical results strogly suggest the same statemet is true for the uiform traffic model. We have also show that the CBR protocol ca achieve the trasport capacity for the uiform traffic model. These results resolve the scalig issue for dese BRNs; future work will cosider sparse etworks from a theoretical perspective. Fially, we also demostrate that a greedy, distributed implemetatio of the CBR protocol achieves the order-optimal scalig predicted by the theoretical results. This suggests that scalable barrage relay etwork architectures ca ideed be realized i practice. We have focused oly o data scalability i this paper i.e., how efficietly a etwork uses the RF spectrum whe trasmittig Layer 4 data. The Layer 2 ad 3 cotrol overhead that is eglected i such aalysis ca be a sigificat, eve limitig, factor i practical etworks. Ideed, the desig of practical BAC protocols for barrage relay etworks requires that the overhead used to multiplex CBRs i space ad time be cotaied. While the mathematical tools to address etworkig scalig i terms of both data ad cotrol are as yet ascset (cf., [3]), the developmet of a theoretical framework for uderstadig cotrol scalability i wireless etworks will provide valuable metrics agaist which practical protocol implemeters ca measure their desigs. V. ACKNOWLEDGEMENTS We wish to thak Mike Neely of the Uiversity of Souther Califoria for suggestig the proof techique for Theorem 3, as well as his cojecture that Claim 2 ought be provable. APPENDIX The mai results preseted i Sectio III ca be immediately deduced from Theorems 1-3, all of which assume the dese radom graph etwork model described i Sectio II. Theorem 1: Uder a localized traffic model, ay PO-MAC protocol that reserves all the shortest paths betwee sourcedestiatio (S-D) pairs (e.g., a CBR protocol) ca support Ω(/ log ) simultaeous traffic flows w.h.p. Proof: Let H(u) be the (radom) umber of hops separatig a source-destiatio pair. By Markov s iequality, Pr(H(u) 2β) 1/2, where β = E[H(u)], ad thus (w.h.p.) there exists /8 S-D pairs that are at most 2β hops apart. Each of these S-D pairs, ad all the shortest paths coectig the source ad destiatio, ca each be eclosed withi a ball of radius 2βr cetered at the respective source. Now, tessellate the uit square by boxes with side legth 2 8log/. By Lemma 5.7 i [13], each of these boxes cotais at least oe source vertex w.h.p. Now, partitio the boxes ito classes of equal size so that ay two boxes i the same class are at least distace 5βr apart. Sice each box has dimesio proportioal to r, a fiite umber of classes suffices (depedig o β, but ot ). The sources cotaied i a sigle class, the correspodig destiatios, ad the correspodig shortest paths are mutually o-iterferig by costructio. Thus, ay PO-MAC protocol that reserves shortest paths betwee source-destiatio pairs ca support Ω(/ log ) simultaeous traffic flows w.h.p i. By combiig Theorem 1 with Mai Result 1 of Sectio III, we obtai the followig result: Corollary 1: Uder the localized traffic model, the PO- MAC user capacity scales as Θ(/ log ). Moreover, ay PO- MAC protocol that reserves shortest paths betwee S-D pairs scales order-optimally with respect to PO-MAC user capacity. Now, we tur our attetio toward the task of showig that BRNs with a CBR protocol ca support Θ( ) simultaeous flows whe the source-destiatio pairs are chose radomly. Before we ca accomplish this, we eed two auxiliary lemmas. The first lemma is a combiatorial result that characterizes the umber of o-crossig lie segmets i the uit square. This result is proved via the probabilistic method (cf., [14]). The secod lemma relates lie segmets to paths i a radom geometric graph. We begi with some defiitios. Let Q =[0, 1) [0, 1) deote the uit square ad let L be a set of lie segmets i Q. Defie the distace betwee two lie segmets l 1,l 2 L as the miimum distace betwee ay poit i l 1 ad ay poit i l 2. Formally, d(l 1,l 2 ) := if x y, x l 1,y l 2

5 where x y is the Euclidea distace betwee poits x, y Q. A set U of lie segmets is said to be d-disjoit if all pairs of lie segmets i U are at least distace d apart. The N d (L ):= max U L { U : d(l 1,l 2 ) d for all l 1,l 2 U} is the size of the largest d-disjoit subset of L. I what follows, we will distiguish betwee left ad right edpoits of a lie segmet. This assigmet is a artificial oe, however, it simplifies the expositio cosiderably. Defiitio 5: A lie segmet is said to be draw accordig to a uiform process if the left ad right edpoits are chose idepedetly ad uiformly from the poits i Q. Lemma 1: If L is a set of lie segmets geerated accordig to a uiform process, the with probability approachig 1 as, N d() (L ) = Θ( ) for ay d() =O( 1/4 ). Proof: We prove Lemma 1 by showig that N d() (L )= Ω( ) ad N d() (L )=O( ). Claim 1: With probability tedig to 1 as, N d() (L )=Ω( ) for ay d() =O( 1/4 ). Proof of Claim 1: Partitio Q ito disjoit squares, each of size 1/4 1/4. Note that if a lie segmet l is cotaied i a sigle square, the it will ot itersect lie segmets cotaied i ay other square. Let Y be the umber of squares that cotai lie segmets. By assumptio, there exists a iteger m such that d() m 1/4 for all. Therefore, by the pigeo-hole priciple, we ca fid at least Y (1 o(1))/(m +1) 2 squares cotaiig lie segmets that are d()-disjoit. Thus, it remais to be show that Y c with high probability for some absolute costat c. Sice the area of each square is 1/, the probability that a edpoit falls ito a particular square j is 1/. Sice edpoits are chose idepedetly, the probability that a give lie l falls etirely iside that square l is 1/, ad the probability that square j does ot cotai ay lie segmets is (1 1/) 1/e. Further, ote that: Pr [{lie l ot i square i} {lie l ot i square j}] =1 Pr [{lie l i square i} {lie l i square j}] =1 (Pr [lie l i square i]+pr[lie l i square j]) =1 2/, where we used the fact that the evets {lie l i square i} ad {lie l i square j} are disjoit. The for ay pair of squares (i, j), the probability that either square i or square j cotais ay lie segmets is (1 2/) 1/e 2. Let X i be the idicator radom variable takig value 1 if square i cotais o lie segmets ad 0 otherwise. Note that EX i e 1,Var(X i ) e 1 (1 e 1 ), ad Cov(X i,x j )=E[X i X j ] E [X i ] E [X j ] =(1 2/) (1 1/) 2 e 2 e 2 + o(1) = o(1). The, lettig X = i=1 X i be the umber of squares that do t cotai ay lie segmets, ad otig that Var(X) = Var(X i )+ Cov(X i,x j ) i i j ( ( ) ) + o(1) + ( 1)o(1) e e + o(1), Chebyshev s iequality yields: [ Pr X EX 1 ] 100 Var(X) 10 + o(1) Therefore, with probability tedig to 1, ( X 1+ 1 ) 10 + o(1) e 2. Hece, Y = X /2 with probability tedig to 1. This proves the claim. Claim 2: With probability tedig to 1 as, N d() (L )=O( ) for ay d() =O( 1/4 ). Proof of Claim 2: First, observe that it suffices to prove that N 0 (L ) = O( ) sice N d (L ) is oicreasig i d. To this ed, from [15], there exists a absolute costat c such that for ay 2k poits i the plae, the umber of o-crossig left-right 4 perfect matchigs is upper-bouded by c 29 k. Cosider ay realizatio of lie segmets i the plae ad further cosider the 2k (k left ad k right) edpoits correspodig to ay subset S cosistig of k lie segmets. Every left-right perfect matchig of these 2k poits is equally likely, ad thus the probability that these k segmets are ocrossig is upper bouded by c 29 k /k! sice there are k! leftright perfect matchigs o the 2k edpoits. Upper boudig the right had side of this expressio usig Stirlig s formula ad recallig the crude upper boud ( ) ( k e ) k k yields the followig upper boud: ( ) c 29 k Pr[ k o-crossig segmets] k k! ( 29 e 2 o(1) Lettig k =15,wehave Pr[ 15 ( 29 e 2 o-crossig segmets] o(1) 15 2 k 2 ) k ) 15 o(1) (.96) 15 0, thus completig the proofs of the claim ad Lemma 1. 4 A matchig i a graph is a set of pairwise o-adjacet edges i.e., o two edges share a commo vertex while a perfect matchig is oe that matches every vertex i the graph. A left-right perfect matchig simply distiguishes betwee left edpoits ad right edpoits i edges. I other words, a edge is oly allowed to match a left edpoit to a right edpoit.

6 Now, we state a result from [16] that relates lie-segmets to shortest paths i a radom geometric graph. Lemma 2 ([16] p. 426): For ay two vertices v 1,v 2 i a dese radom graph G, w.h.p., d G (v 1,v 2 ) K v 1 v 2 r() where d G (v 1,v 2 ) is the legth of the shortest path coectig vertices v 1 ad v 2 ad r() is the coectivity radius. Lemma 2 essetially states that the legth of a shortest path betwee two vertices is proportioal to the Euclidea distace betwee the two correspodig poits i the plae. Therefore, we ca use this lemma to coclude that the shortest paths coectig two vertices must be well-behaved. With the above lemmas i had, we ca ow prove Mai Result 4. Theorem 2: Cosider a set of /2 radomly chose distict source-destiatio pairs i a dese radom geometric graph G. With probability tedig to 1 as, there exist Ω( ) S-D pairs which geerate mutually o-iterferig CBRs. Proof: For a source-destiatio pair (s, d), deote the straight-lie segmet coectig s to d as l s,d. Sice the vertices of G are uiformly chose from Q ad the S-D pairs are chose radomly, the set of segmets coectig S-D pairs is geerated accordig to a uiform process. Defie { } T (s, d, α) = x Q : if x y α y l s,d to be the set of poits i Q that are withi distace α of l s,d. Lemma 1 states that, with high probability, there exist Θ( ) distict source-destiatio pairs such that T (s, d, 1/4 ) T (s,d, 1/4 )= for distict pairs (s, d), (s,d ). Deote this set of source-destiatio pairs as D. Lemma 2 implies (via a simple geometric exercise) that all shortest paths coectig (s, d) lie etirely i T (s, d, 0.5 1/4 ). Thus, o shortest paths coectig differet S-D pairs i D are adjacet. Sice a CBR cosists of a S-D pair ad all shortest paths coectig the source to the destiatio, the S-D pairs i D geerate mutually o-iterferig CBRs. Sice paths are closely approximated by lie segmets i a sufficietly dese radom geometric graph, Lemma 1 suggests that a etwork operatig uder ay PO-MAC ca support at most Θ( ) simultaeous flows. Thus, we cojecture that the user capacity of a etwork scales as Θ( ). To verify this cojecture, oe would eed to show that the maximum set of mutually o-iterferig routes selected usig ay metric i.e., o-shortest paths caot scale faster tha O( ). Theorem 3: Cosider a set of /2 radomly chose distict S-D pairs i dese radom geometric graph. With probability tedig to 1 as, there exist Ω( / log ) S-D pairs which geerate mutually o-iterferig CBRs. Moreover, these S-D pairs have average distace separatio Θ(1). Proof: Repeat the costructive part of the proof of Lemma 1, except divide Q ito horizotal strips of height 1/ ad width 1. With probability tedig to 1, we ca fid /2 distict strips that cotai lie segmets. By pruig the umber of strips ad keepig every ( r()) th strip, we are left with a set of Θ( / log ) lie segmets that are r()- disjoit. Call this set of lie segmets U. Cosider a sigle lie segmet l U, ad deote its edpoits i Cartesia form as: (x 1,y 1 ) ad (x 2,y 2 ). The l x 1 x 2 := L l, where the radom variable L l is idepedet from the evet {l U}which depeds oly o y 1,y 2. It is readily show that E[L l ]=Θ(1), which proves the theorem. If X i Q is the locatio of ode i i a etwork, the the trasport capacity T () is defied as the supremum of i j λ i,j X i X j, where {λ i,j : 1 i, j } is a achievable rate vector [9]. Although ot explicitly stated i the literature, T () =Θ( / log ) for the Gupta-Kumar protocol i which the trasmissio radius r() =Θ( log /) is idetical for all odes. To see that T () =O( / log ), oe ca repeat the argumet i Sectio 2.5 of [13], ad ote that the maximum umber of simultaeous trasmissios is Θ(1/r() 2 ) istead of /2. 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