Dynamic Multiple-Message Broadcast: Bounding Throughput in the Affectance Model

Transcription

1 Dynamic Muliple-Message Broadcas: Bounding Throughpu in he Affecance Model Dariusz R. Kowalski Universiy of Liverpool Dep. of Compuer Science Liverpool, UK Miguel A. Moseiro Kean Universiy Dep. of Compuer Science Union, NJ Tevin Rouse Kean Universiy Dep. of Compuer Science Union, NJ ABSTRACT We sudy a dynamic version of he Muliple-Message Broadcas problem, where packes are coninuously injeced in nework nodes for disseminaion hroughou he nework. Our performance meric is he raio of he hroughpu of such proocol agains he opimal one, for any sufficienly long period of ime since sarup. We presen and analyze a dynamic Muliple-Message Broadcas proocol ha works under an affecance model, which parameerizes he inerference ha oher nodes inroduce in he communicaion beween a given pair of nodes. As an algorihmic ool, we develop an efficien algorihm o schedule a broadcas along a BFS ree under he affecance model. To provide a rigorous and accurae analysis, we define wo novel nework characerisics based on he nework opology, he affecance funcion and he chosen BFS ree. The combinaion of hese characerisics influence he performance of broadcasing wih affecance (modulo a polylogarihmic funcion). We also carry ou simulaions of our proocol insaniaing affecance in he Radio Nework model. To he bes of our knowledge, his is he firs dynamic Muliple-Message Broadcas proocol ha provides hroughpu guaranees for coninuous injecion of messages and works under he affecance model. Caegories and Subjec Descripors F.2.2 [Analysis of Algorihms and Problem Complexiy]: Nonnumerical Algorihms and Problems sequencing and scheduling Keywords Muliple-Message Broadcas, Radio Nework, Affecance. INTRODUCTION We sudy he dynamic Muliple-Message Broadcas problem in wireless neworks under he affecance model. This model subsumes many communicaion-inerference models Permission o make digial or hard copies of all or par of his work for personal or classroom use is graned wihou fee provided ha copies are no made or disribued for profi or commercial advanage and ha copies bear his noice and he full ciaion on he firs page. Copyrighs for componens of his work owned by ohers han ACM mus be honored. Absracing wih credi is permied. To copy oherwise, or republish, o pos on servers or o redisribue o liss, requires prior specific permission and/or a fee. Reques permissions from permissions@acm.org. FOMC 4, Augus, 204, Philadelphia, PA, USA. Copyrigh 204 ACM /4/08...$5.00. hp://dx.doi.org/0.45/ sudied in he lieraure, such as Radio Nework (cf., [6]) and models based on he Signal o Inerference and Noise Raio (SINR) (cf. [9, 30]). The noion of affecance was firs inroduced in [9] in he conex of link scheduling in he more resriced SINR model of wireless neworks, in an aemp o formalize he combinaion of inerferences from a subse of links o a seleced link under he SINR model. Laer on, oher realizaions of affecance were defined and absraced as an independen model of inerference in wireless neworks [23,24]. The concepual idea of his model is o parameerize he inerference ha ransmiing nodes inroduce in he communicaion beween a given pair of nodes. Our resuls. In he dynamic Muliple-Message Broadcas problem considered in his work, packes arrive a nodes in an online fashion and need o be delivered o all nodes in he nework. We are ineresed in he hroughpu, i.e., he number of packes delivered in a given period of ime. In paricular, we measure compeiive hroughpu of deerminisic disribued algorihms for he dynamic Muliple-Message Broadcas problem. We analyse our algorihms in he (general) affecance model, in which here is a given undireced communicaion graph G of n nodes and diameer D, ogeher wih he affecance funcion a( ) of nodes of disance a leas 2 on each of he communicaion links. The affecance funcion has a degradaion parameer α, being a disance afer which he affecance is negligible. Our conribuion is wo fold. Firs, we inroduce new model characerisics based on he underlying communicaion nework, he affecance funcion, and a chosen BFS ree called maximum average ree-layer affecance (denoed by K) and maximum fas-pahs affecance (denoed by M), see Secion 2 for he definiions, and show how hey influence he ime complexiy of broadcas. More precisely, if one uses a specific BFS ree, called GBST (cf., [6]), ha minimizes he produc M (K + M) of he wo above characerisics, hen a single broadcas can be done in ime D + O(M(K + M) log 3 n), cf., Corollary 3 in Secion 3. Second, we exend his mehod of analysis o a dynamic packe arrival model and he Muliple-Message Broadcas problem, and design a new algorihm reaching compeiive hroughpu of Ω(/(αK log n)). In paricular, in he Radio Nework model i implies a compeiive hroughpu of Ω(/(log 2 n)). For deails, see Secion 4. Our deerminisic resuls are exisenial, ha is, we show he exisence of a deerminisic schedule by applying a probabilisic ar- Throughou, we denoe log 2 simply as log, unless oherwise saed. 39

2 gumen o a proocol ha includes a randomized subrouine for layer o layer disseminaion. Given ha we measure compeiive hroughpu in he limi, preprocessing (communicaion infrasrucure seup, opology informaion disseminaion, ec.) can be carried ou iniially wihou asympoic impac. Thus, he proocol presened is disribued, and i works for every nework afer learning is opology. The proocol can also be applied o mobile neworks, if he movemen is slow enough o recompue he srucure. Our rigorous asympoic analysis is furher complemened by simulaions done for he Radio Nework model, c.f., Secion 5. To he bes of our knowledge, ours is he firs work on he dynamic Muliple-Message Broadcas problem in wireless neworks under he general affecance model. Previous and relaed work. There is a rich hisory of research on broadcasing dynamically arriving packes on a single-hop radio nework, also called a muliple access channel. Mos of he research focused on sochasic arrivals, cf., a survey by Chlebus [8]. In he remainder of his paragraph, we focus on he on-line adversarial packe arrival seing. Bender e al. [5] sudied sabiliy, undersood as hroughpu being no smaller han he packe arrival rae, of randomized backoff proocols on muliple access channels in he queuefree model, in which every packe is handled independenly as if i has been a sandalone saion (hus avoiding queuing problems). Kowalski [26] considered a dynamic broadcas on he channel in he seing where packes could be combined in a single message, which again avoids various imporan issues relaed wih queuing. Ananharamu e al. [3] sudied packe laency of deerminisic dynamic broadcas proocols for arrival raes smaller han. Sabiliy, undersood as bounded queues, of dynamic deerminisic broadcas on muliple access channels agains adversaries bounded by arrival rae was sudied by Chlebus e al. [0], and for arrival raes smaller han by Chlebus e al. []. In paricular, in [0] a proocol Move-big-o-fron (MBTF) was designed, achieving sabiliy bu no fairness (as boh hese properies are impossible o achieve simulaneously); we use his algorihm as a subrouine in our dynamic Muliple-Message Broadcas proocol. In muli-hop Radio Neworks, he previous research concenraed on ime complexiy of single insances (i.e., from a single source) of broadcas and muli-message broadcas. For direced neworks, he bes deerminisic soluion is a combinaion of he O(n log n log log n)-ime algorihm by De Marco [5] and he O(n log 2 D)-ime algorihm by Czumaj and Ryer [3]. In undireced neworks, he bes up o dae deerminisic broadcas in O(n log(n/d)) rounds was given by Kowalski [26]. The lower bounds for deerminisic broadcas in direced and undireced radio neworks are Ω(n log(n/d)) [2] and Ω(n log D n) [27], respecively. Deerminisic muli-message broadcas, group communicaion and gossip were also considered (again, in a single insance). Chlebus e al. [9] showed a O(k log 3 n + n log 4 n) ime deerminisic muli-broadcas algorihm for k packes in undireced radio neworks. Single broadcas can be done opimally in Θ(D log(n/d) + log 2 n), as proved in [2, 29] (lower bounds) and in [3,27] (maching upper bound). Bar- Yehuda e al. [4], and recenly Khabbazian and Kowalski [25] and Ghaffari e al. [8], sudied randomized muli-broadcas proocols; he bes resuls obained for k-sources singleinsance muli-broadcas is he amorized O(log ) rounds per packe w.h.p. in [25], where is he maximum node degree, and O(D + k log n + log 2 n) w.h.p. o broadcas he k packes, for seings wih known opology in [8]. For he same problem, Ghaffari e al. showed a hroughpu upper bound of O(/ log n) for any algorihm in [7]. Alhough his bound is wors-case, i can be compared wih our /O(αK log n) ha applies even under affecance. Chlebus e al. [0] gave various deerminisic and randomized algorihms for group communicaion, all of hem being only a small polylogarihm away of he corresponding lower bounds on ime complexiy. In he SINR model, single-hop insances of broadcas in he ad-hoc seing were sudied by Jurdzinski e al. [2, 22] and Daum e al. [4], who gave several deerminisic and randomized algorihms working in ime proporional o he diameer muliplied by a polylogarihmic facor of some model parameers. In he SINR model wih resriced sensiiviy, so called weak-sensiiviy device model, Jurdzinski and Kowalski [20] designed an algorihm spanning an efficien backbone sub-nework, ha migh be used for efficien implemenaion of muli-broadcas. The generalized affecance model was inroduced and used only in he conex of one-hop communicaion, more specifically, o link scheduling by Kesselheim [23]. He also showed how o use i for dynamic link scheduling in baches. This model was inspired by he affecance parameer inroduced in he more resriced SINR seing [9]. They give a characerisic of a se of links, based on affecance, ha influence he ime of successful scheduling hese links under he SINR model. In our paper, we generalize his characerisic, called he maximum average ree-layer affecance, o be applicable o muli-hop communicaion asks such as broadcas, ogeher wih anoher characerisic, called he maximum fas-pahs affecance. For deails see Secion PRELIMINARIES Model. We sudy a model of nework consising of n nodes, where communicaion is carried ou hrough radio ransmissions in a shared channel. Time is discreized in a sequence of ime slos, 2,..., which we call he global ime. The nework is modeled by he underlying conneciviy graph G = {V, E}, where V is he se of nodes and E he se of links among nodes. Le a link l E beween wo nodes u, v V be he se {u, v}. The nework is assumed o be conneced bu mulihop. Tha is, no all possible links are presen in E, bu any pair of nodes may communicae, possibly hrough muliple hops. Messages o be broadcas o he nework hrough radio ransmissions are called packes. Packes are injeced a nodes a he beginning of ime slos, and each ime slo is long enough o ransmi a packe o a neighboring node. Any given node can eiher ransmi or lisen (in order o receive, if possible) in a ime slo. Two or more ransmissions received a a hird node simulaneously are garbled. This even is called a collision. Nodes canno disinguish beween a collision and he background noise in he channel, ha is, collisions canno be deeced. Addiional inerference on a link due o ransmissions a more han one hop is modeled as affecance. We use a model of affecance ha subsumes oher communicaioninerference models, such as he Radio Nework model (c.f., [6]) and he SINR model (c.f., [9]). Specifically, we realize affecance as a value a i (j) ha quanifies he inerference ha a ransmiing node i inroduces o he 40

3 communicaion hrough link j. We do no resric ourselves o any paricular affecance funcion, as long as is effec is addiive. Tha is, denoing a V (j) as he affecance of a se of nodes on a link, for any V V and j E, i is a V (j) = P i V a i (j). For a link (u, v), where u is he ransmier, we define a u ((u, v)) = 0 and a v ((u, v)) =, o model he posiive (resp. negaive) impac of a ransmission from he ransmier (resp. receiver). Also, for N(v) being he se of neighbors of v, we define a w((u, v)) = for each w u such ha w N(v). Under he affecance model, we define a successful ransmission as follows. For any pair of nodes u, v V such ha {u, v} E, a ransmission from u is received a v in a ime slo if and only if: u ransmis and v lisens in ime slo, and a T (u, v) <, where T is he se of nodes ransmiing in ime slo. We also denoe he affecance of a se of nodes V on a se of links E as a V (E ), for any V V and E E. Communicaion ask. Under he above model, we sudy he following Muliple-Message Broadcas problem. Saring a ime slo, packes are being dynamically injeced ino source nodes for disseminaion hroughou he nework. The se of all source nodes is denoed as S V. Afer a packe has been received by all he nodes in he nework, we say ha he packe was delivered. The injecions are adversarial, ha is, packes can be injeced a any ime slo a any source node, bu he injecions are limied o be feasible. We say ha an injecion is feasible if here exiss an opimal algorihm OPT such ha he laency (i.e., he ime elapsed from injecion o delivery) of each packe is bounded for OPT. Given ha a mos one packe may be received by a node in each ime slo, and ha all nodes mus receive he packe in order he packe o be delivered, his assumpion limis he adversarial injecion rae o a mos packe per ime slo for all nodes. The goal is o find a broadcasing schedule, ha is, a emporal sequence of ransmi/no-ransmi saes for each node, so ha packes are delivered. We denoe he period of ime since a packe is ransmied from he source unil i is delivered he lengh of he schedule. Performance meric. We evaluae he raio of he performance of a disribued online algorihm ALG agains an opimal algorihm OPT. For one hop neworks i is known [0] ha no proocol is boh sable (i.e., bounded number of packes in he sysem a any ime) and fair (i.e., every packe is evenually delivered). For mulihop neworks he same resul holds as a naural exension of he single hop model. Thus, insead of furher limiing he adversary (beyond feasibiliy) o achieve sabiliy or bounded laency, our goal is o prove a lower bound on he compeiive hroughpu, for any sufficienly long prefix of ime slos since global ime. Specifically, we wan o prove ha here exiss a funcion f, possibly depending on nework parameers, such ha lim d ALG ()/d OP T () Ω(f), where d X () is he number of packes delivered o all nodes by algorihm X unil ime slo. Nework characerizaion. We characerize a nework by is affecance degradaion disance, which is he number of hops α such ha he affecance of nodes of disance bigger ha α in he nework G o a given link is negligible, ha is, zero. Addiionally, we characerize he nework wih wo measures of affecance based on broadcas rees, as follows. Given a nework wih a se of nodes V including a source node s, consider a gahering-broadcas spanning ree (GBST) [6] rooed a s. A GBST is a breadh-firs-search ree wih a specific node ranking, saisfying he propery ha no wo links of senders and receivers wih he same rank creae collisions (i.e., he receivers are differen and here is no cross link beween he sender in one link and receiver in he oher). We define a node-se pariion (slighly differen han he pariion in [6] for convergecas) based on ha ranking and he disance o he source. Specifically, for a GBST ree T, he se of nodes V is pariioned in ses F r d (T ) and ses S d (T ). A node of rank r a (shores) disance d from he source is in se F r d (T ) if i has a child of he same rank (so called fas nodes), or i is in se S d (T ) oherwise (so called slow nodes). Le V d (T ) = F d (T ) S d (T ), where F d (T ) = S r F r d (T ). Tha is, V d (T ) is he se of all nodes a disance d from he roo. Based on his pariion, we define he maximum average ree-layer affecance K(T, s) = max d max V V d (T ) L(V ) a V (L(V )), where L(V ) is he se of GBST links beween V and nodes a disance d + of he source. Addiionally, we define he maximum fas-pahs affecance M(T, s) = max d,r max a l F d r(t ) F d r(t )\l (l). Given a GBST ree, he former characerisic says wha is he maximum average affecance of a subse of nodes in he same layer on he links o heir children in he ree, while he laer characerisic says wha is he maximum affecance of fas links of he same rank and originaed in he same layer o one of hem. Inuiively, he former characerisic indicaes wha migh be he wors affecance o overcome when rying o broadcas from one layer o anoher, while he laer one indicaes wha is he wors affecance when rying o pipeline a packe via fas links. In he res of he paper, he specific ree and source node s will be omied when clear from he conex. 3. A BROADCAST TREE In his secion, we show a broadcasing schedule ha, under he affecance model, disseminaes a packe held a a source node o all oher nodes. The schedule is defined consrucively wih a proocol ha uses randomizaion, hus providing only sochasic guaranees. Given ha he proocol is Las Vegas, he consrucion also proves he exisence of a deerminisic broadcasing schedule. Firs, we deail he consrucion of a ranked ree spanning he nework rooed a he source node ha will be used o define he broadcasing schedule ha we deail aferwards. The following noaion will be used. Given a ree T (s) E rooed a s V, spanning a se of nework nodes V wih se of links E, le d(v) be he disance in hops from a node v V o he roo of T (s), le p(l) and c(l) be he paren and child nodes of link l T (s) respecively, and le D(T (s)) be he maximum disance in T (s) from any node o he roo s. Addiionally, a rank (a number in N) will be assigned o each node. Le r(u) be he rank of node u V, le R(T (s)) be he maximum rank in he ree, and le F r d = {u u V d(u) = d v V : v = c(u) r(v) = r(u) = r}, ha is, he se of nodes of rank r a disance d from he roo ha have a child wih he same 4

4 rank. In he above noaion, he specific ree parameer and/or source node will be omied when clear from he conex. Then, given a graph G and a source node s S, consider he following consrucion of a Low-Affecance Broadcas Spanning Tree (LABST). Le T min be he GBST ha minimizes he following polynomial on he affecance measures. Leing T be he class of all GBSTs ha can be defined wih source s, i is T T : M(T min, s)(m(t min, s) + K(T min, s)) M(T, s)(m(t, s) + K(T, s)). Then, using Algorihm, ransform T min ino a LABST T ha avoids links beween nodes of he same rank wih big affecance. Algorihm : LABST consrucion. T T min 2 foreach rank r = R(T ), R(T ),..., 2, do 3 r r M(T ) //now i is R(T ) = R(T min) M(T ) 4 updae all ses Fd r. 5 foreach disance d = D(T ),..., 2, do 6 foreach rank r =, 2,..., R(T ) do 7 foreach link l such ha p(l) Fd r do 8 if a F r d \l(l) hen r(p(l)) r + 9 updae all ses F r d. The broadcasing schedule is defined using he LABST T obained. Being a radio-broadcas nework, ransmissions migh be received using oher links or ime slos, bu he LABST and broadcasing schedule defined provide he communicaion guaranees. Each node follows cerain broadcasing schedule, bu using only ime slos reserved for iself. Specifically, le a node v V be called fas if i belongs o he se F r(v) d(v) (T ), and slow oherwise. Then, for each node v V, if v is fas, i uses each ime slo such ha d(v) + 2h(R(T ) r(v)) (mod 2hR(T )), where h = max{3, α} and α is he affecance degradaion disance. Oherwise, if v is slow, i uses each ime slo such ha d(v)+h (mod 2h). (The reason for his paricular choice of reserved slos will become clear in Theorem 2.) The broadcasing schedule for fas nodes is simple: upon receiving a packe for disseminaion, ransmi in he nex ime slo reserved. For slow nodes, he schedule is deermined by a randomized conenion resoluion proocol ha can be run in he reserved ime slos. The proocol is simple: upon receiving a packe for disseminaion, each slow node ransmis repeaedly wih probabiliy /(4K(T min, s)), unil he packe is delivered. In he res of his secion, we bound he lengh of he broadcasing schedule. The following upper bound will be used. Lemma. The maximum rank of a LABST on a nework of n nodes wih source node s is R(T ) log n M(T min, s). Proof. Consider he consrucion of a LABST T. The iniial GBST T min guaranees ha he maximum rank is R(T min ) log n (cf. [6]). Consider Algorihm, afer Line 3, i is R(T ) log n M(T min). We show here ha such overhead is enough for all he updaes in Line 8. Consider any pah p from roo o leaf in T min defined by is se of links in he pah (he order is implici). Le p p be he se of all links in a maximal subpah of p where all nodes have he same rank. The maximum number of ranksıneeded P for he updaes Line 8 is a l p r(p(l)) F. The d(p(l)) \l(l)/ p bound holds because each ime ha a link is removed from such pah, a value is reduced from he oal affecance of he pah, and fas nodes coninue being fas (possibly in a differen se) even afer updaing he rank. Also, because fas nodes are sill fas afer he updae, no new collisions appear and he links do no need o be updaed. Given ha ı P a r(p(l)) F \l(l) M(T min), i is a d(p(l)) l p r(p(l)) F d(p(l)) l \l(l)/ p P l p M(T min )/ p m = M(T min ). Thus, he rank overhead wih respec o T min is enough. Theorem 2. For any given nework of n nodes wih a source node, diameer D, and affecance degradaion disance α, here exiss a broadcasing schedule of lengh D + 2h log n 2 ` M(T min ) M(T min ) K(T min ), where h = max{3, α}. Proof. Firs we show ha he broadcasing schedule is correc. Consider any pair of nodes u, v V ransmiing in he same ime slo. If d(u) = d(v) and hey are boh fas nodes wih he same rank, he affecance on each oher s links is low by definiion of he LABST. If d(u) = d(v) and hey are boh slow nodes, he conenion resoluion proocol will disseminae he packe o he nex layer. Oherwise, given he slo reservaion, d(u) d(v) h. Given ha h α, he affecance on each oher s links is negligible, and given ha h 3, here are no collisions beween heir ransmissions. To prove he schedule lengh, consider any pah p from roo o leaf in he LABST T. The pah p can be pariioned ino consecuive maximal subpahs according o rank. In each maximal subpah p p of consecuive nodes of he same rank, he firs node may have o wai up o 2hR(T ) slos for he nex reserved ime slo, bu afer ha all nodes excep he las one ransmi in consecuive ime slos. Given ha here are a mos R(T ) such maximal subpahs and ha heir aggregaed lengh is a mos D(T ), he schedule lengh in he fas nodes of pah p is a mos D(T ) + 2hR(T ) 2 D + 2hR(T ) 2, where he laer inequaliy holds because T is a BFS ree. Consider now any link l p where he rank changes, ha is r(p(l)) r(c(l)) and p(l) S d(p(l)) V d(p(l)). Recall ha he schedule in such link is defined by a randomized conenion resoluion proocol where each node ransmis wih probabiliy /(4K(T min)), where K(T min ) = max d max V V d (T min ) L(V ) a V (L(V )), where L(V ) is he se of GBST links beween V and nodes a disance d + of he source, and V d (T min) is he se of nodes a disance d from he source in T min. For a probabiliy of ransmission q 4 max S Vd(p(l)) a S(L(S))/ L(S), i was proved in [24] ha he probabiliy ha here is sill some link in S where no ransmission was successful afer 4c ln V d(p(l)) /q ime slos running Algorihm in [24], is a 42

5 mos V d(p(l)) c, c >. Given ha /(4K(T min )) verifies such condiion, we know ha afer 6cK(T min ) ln V d(p(l)) 6cK(T min ) ln n (reserved) ime slos, he ransmission in link l has been successful wih posiive probabiliy. Given ha here are a mos R(T ) links where he rank changes, using he union bound, we know ha afer (R(T ) )6cK(T min ) ln n (reserved) ime slos all slow nodes have delivered heir packes wih some posiive probabiliy, which shows he exisence of a deerminisic schedule of such lengh 2. The ime slos reserved for slow nodes appear wih a frequency of 2h. Thus, he schedule lengh in he slow nodes of pah p is a mos 2h(R(T ) )6cK(T min) ln n 32hR(T )K(T min) ln n, for c = R(T )/(R(T ) ). Adding boh schedule lenghs we have D + 2hR(T ) hR(T )K(T min) ln n Replacing he bound on R(T ) in Lemma, he claim follows. For neworks wih affecance degradaion disance log n, Theorem 2 yields he following corollary. Corollary 3. For any given nework of n 8 nodes, diameer D, and affecance degradaion disance log n, here exiss a broadcasing schedule of lengh D + O(log 3 n(m(t min )(M(T min ) + K(T min )))). For comparison, for less conenious neworks where affecance is no presen (Radio Nework model), using a GBST a broadcas schedule of lengh D + O(log 3 n) was shown in [6] and of lengh O(D + log 2 n) was proved in [28]. 4. A DYNAMIC Muliple-Message Broadcas PROTOCOL In his secion, we presen our Muliple-Message Broadcas proocol and we bound is compeiive hroughpu. The proocol uses he LABST 3 presened in Secion 3. 4 The inuiion of he proocol is he following. Each source node has a (possibly empy) queue of packes ha have been injeced for disseminaion. Then, saring wih an arbirary source node s S wih large enough number of packes in is queue, packes are disseminaed hrough a LABST rooed a s. If he number of packes in he queue of s becomes small, s sops sending packes and, afer some delay o clear he nework, anoher source node s S sars disseminaing packes hrough a LABST rooed a s. The procedure is repeaed following he order of a lis of source nodes, which is dynamically updaed according o queue sizes o guaranee good hroughpu. Packes from any given source are pipelined wih some delay o avoid collisions and affecance. Being a radio broadcas nework, packes migh be received earlier han expeced using links or ime slos 2 In seings wih collision deecion and where he affecance on any given link is O(n), a big enough consan c > yields a randomized proocol ha succeeds wih probabiliy /n. 3 We refer o he ree and he broadcas schedule indisincively. 4 Any broadcas schedule ha works under he affecance model could be used. oher han hose defined by he LABST. If ha is he case, o guaranee he pipelining, nodes ignore hose packes. The following noaion will be also used. The LABST rooed a s S is denoed as T (s). We denoe he lengh of he broadcas schedule (ime o deliver o all nodes) from s as (s), and = max s S (s). Le he pipeline delay (he ime separaion needed beween consecuive packes o avoid collisions and affecance) from s be δ(s), and δ = max s S δ(s). Given a node i S and ime slo, he lengh of he queue of i is denoed l(i, ). Le he lengh of all queues a ime be l() = P i S l(i, ). We say ha, a ime, a node i is empy if l(i, ) <, small if l(i, ) < n, and big if l(i, ) n. Consider he following Muliple-Message Broadcas Proocol.. For each source node s S define a LABST rooed a s. 2. Define a Move-big-o-fron (MBTF) lis [0] of source nodes, iniially in any order. According o his lis, source nodes circulae a oken. While being disseminaed, he oken has a ime-o-live couner of, mainained by all nodes relaying he oken. A source node s receiving he oken has o wai for he oken couner o reach zero before saring a new ransmission. Le he ime slo when he couner reaches zero be. Then, node s does he following depending on he lengh of is queue. (a) If s is empy a, i passes he oken o he nex node in he lis. We call his even a silen round. (b) If s is small a, i broadcass packes pipelining hem in inervals of δ slos. Afer δ more slos, i passes he oken o he nex node in he lis. (c) If s is big a, i moves iself o he fron of he lis. We call his even a discovery. Then, s broadcass packes pipelining hem in inervals of δ slos as long as i is big, bu a minimum of packes. Wih he firs of hese packes s broadcass he changes in he lis. δ more slos afer ransmiing hese packes, i passes he oken o he nex node in he lis. The following heorem shows an upper bound on he number of packes in he sysem a any ime, which allows o prove he compeiive hroughpu of our proocol. The proof srucure is similar o he proof in [0] for MBTF, bu many deails have been redone o adap i o a mulihop nework. Theorem 4. For any given nework of n nodes, a any given ime slo of he execuion of he Muliple-Message Broadcas proocol defined, he overall number of packes in queues is l() < (δ/( + δ)) + 2 n 2. Proof. For he sake of conradicion, assume ha here exiss a ime such ha he overall number of packes in he sysem is l() (δ/( + δ)) + 2 n 2. The number of packes in queues a he end of any given period of ime is a mos he number of packes in queues a he beginning of such period, plus he number of ime slos when no packe is delivered, given ha a mos one packe is injeced in each ime slo. We arrive o a conradicion by upper bounding 43

6 he number of ime slos when no packe is delivered wihin a convenienly defined period before. Consider he period of ime T such ha l( T ) n 2 ( T )δ + + δ () [ T, ] : l( ) n 2 (2) l() (δ/( + δ)) + 2 n 2 (3) From now on, he analysis refers o he period of ime T. We omi o specify i for clariy. Le C S be he se of nodes ha are big a some poin. Due o he pigeonhole principle and Equaion (2), we know ha for each ime slo here is a leas one big source node. In oher words, he oken canno be passed hroughou he whole lis wihou a leas one discovery. As a wors case, assume ha only nodes in C have packes o ransmi. For each node i C, he oken has o be passed hrough a mos S \C n C nodes ha are no in C before i is discovered, because afer i is discovered no node in S \ C will be before i in he lis. Hence, here are a mos C (n C ) silen rounds, each of lengh for oken pass. So, due o passing he oken hrough nodes in S \ C, here are a mos C (n C ) ime slos when no packe is delivered. We bound now he ime slos when no packe is delivered due o passing he oken hrough nodes in C before being discovered for he firs ime. Consider any given node i C. The argumen is similar o he previous case. Any oher node j C ha is discovered before i is moved o he fron of he lis. If i is going o be before j in he lis laer, i is no going o happen before i is discovered for he firs ime. Then, before i is discovered, i may hold he oken a mos C imes. As a wors case, assume ha for each of hese imes i is empy. Hence, here are a mos C ( C ) silen rounds, each of lengh for oken pass. So, due o passing he oken hrough nodes in C before being discovered, here are a mos C ( C ) ime slos when no packe is delivered. I remains o bound he ime slos when no packe is delivered due o pipelining and passing he oken hrough nodes in C afer being discovered. Consider any given node i C afer being discovered. If i is big during he res of T, i broadcass packes pipelining hem in inervals of δ slos. If insead i becomes small during T, i will have packes o ransmi for a leas n imes ha holds he oken aferwards before becoming empy, because righ afer becoming small i has a leas (n ) packes in queue. And here are a mos n nodes in C ha will no be behind i in he lis unil i becomes big again. Hence, i always has packes o ransmi afer being discovered he firs ime. Afer becoming small, i has o pass he oken o he nex node in he lis inroducing a delay of. As a wors case scenario, we assume ha upon each discovery of each node i C, only packes are broadcas before passing he oken. Then, for each packes delivered, here are a mos + (δ ) = δ ime slos when no packe is delivered, over a period of + δ = ( + δ) ime slos. Because C is he se of nodes ha are discovered in T, we can bound he number of baches of packes delivered in T by T/( ( + δ)) T/( ( + δ)). Then, here are a mos T δ/( ( + δ)) = T δ/( + δ) ime slos when no packe is delivered due o nodes in C afer being discovered. Combining hese bounds wih Equaion (), we have ha here are a mos n 2 + ( T )δ + δ + C (n C ) + C ( C ) + T δ + δ = n 2 + δ + C (n ) + δ < δ + δ + 2 n2 ime slos when no packe is delivered. Which is a conradicion. Lemma 5. There exiss a Muliple-Message Broadcas proocol ha achieves a compeiive hroughpu of a leas lim + δ 2 n2. Proof. A packe is delivered when i has been received by all nodes. The opimal algorihm delivers a mos one packe per ime slo, since any given node can receive a mos one packe per ime slo. Addiionally, he injecion is limied o be feasible, ha is, here mus exis an opimal algorihm OPT such ha he laency of each packe is bounded for OPT. Thus, a mos one packe may be injeced in each ime slo. Then, he compeiive hroughpu is a leas d ALG() lim d OP T () lim nd ALG(), where nd ALG() is he max number of packes ha could no be delivered by ALG by ime. Using he bound in Theorem 4 we have ha d ALG () lim d OP T () lim (δ/( + δ)) 2 n 2 lim + δ 2 n2. The following heorem shows our main resul. Theorem 6. For any given nework of n nodes, diameer D, and affecance degradaion disance α, here exiss a Muliple-Message Broadcas proocol ha achieves a compeiive hroughpu of a leas Where lim + δ 2 n2. D + 2 max{3, α} log n 2 ` M(Tmin ) M(T min ) K(T min ), K = max s S K(T min(s), s), M = max M(Tmin(s), s), s S δ = max{3, α}6k ln n. Proof. The lengh (s) of he broadcas schedule in a LABST rooed a s is given in Theorem 2. Wih respec o δ(s), as explained in he proof of Theorem 2, slow nodes a disance d from he roo deliver a packe o he nex node in a pah of a LABST T (s) wihin 6cK(T min (s)) ln V d wih posiive probabiliy for any c >. This shows he exisence 44

7 of a deerminisic schedule of ha lengh. Addiionally, packes mus be separaed by a leas max{3, α} o avoid collisions and affecance from nodes a differen disances from he source (see he proof of Theorem 2 for furher deails). Then, i is δ(s) = max{3, α}6k(t min (s)) ln n, for c = ln n/ ln V d. Replacing, he claim follows. The above heorem yields he following corollary ha provides inuiion. Corollary 7. For any given nework of n nodes, diameer D, and affecance degradaion disance α, here exiss a Muliple-Message Broadcas proocol such ha he compeiive hroughpu converges o O(αK log n), where K = max s S K(T min (s), s). To evaluae hese resuls, i is imporan o noice ha he compeiive hroughpu bound was compued agains a heoreical opimal proocol ha delivers one packe per ime slo, which is no possible in pracice in a muli-hop nework. For comparison, insaniaing our inerference model in he Radio Nework model (no affecance), using he WEB proocol [7] for slow ransmissions our Muliple-Message Broadcas proocol can be shown o converge o /O(log 2 n). Furhermore, for single-insance muli-broadcas in Radio Nework, Ghaffari e al. showed in [7] a hroughpu upper bound of O(/ log n) for any algorihm. Alhough his bound is wors-case, i can be compared wih our /O(αK log n) ha applies even under affecance. We evaluae he Radio Nework case hrough simulaions of our proocol in he following secion. 5. SIMULATIONS For simpliciy, we carried ou simulaions of he Muliple- Message Broadcas proocol assuming he Radio Nework model. Tha is, inerference is due o collisions only. In absence of affecance, he LABST consrucion is simply a GBST. Furhermore, he affecance measures are zero and he broadcas ree becomes any GBST as defined in [6]. We simulaed he ree broadcas schedule specified in Secion 3, excep for he proocol for small nodes ransmissions from layer o layer, which in [6] is he deerminisic schedule of he WEB proocol [7]. Regarding he delay δ and he schedule lengh, using a GBST and he WEB proocol hey are δ = ln 2 n (cf. Lemma 6.2 in [7]) and = max s S D(s) + 6r max (s) 2 + r max (s) ln 2 n (cf. [6]), where r max is he maximum rank in he GBST rooed on he source node s. Using such broadcas rees from each source, we simulaed he proocol in Secion 4 for nework sizes n = 8, 6, 32, 64, 28, 256, and 52. The inpu neworks were random graphs G(n, p), where p = /5, and each node was chosen o be a source a random wih probabiliy /3. The injecion a a rae of one packe per ime slo was also random wih uniform disribuion on he nodes. The packe queues of he source nodes were iniialized o packes. Tha is, iniially all source nodes were small inroducing overhead due o oken passing. The resuls of he simulaions are illusraed by he plo in Figure. I can be seen ha, afer an iniial phase, for any of he nework sizes sudied, he compeiive hroughpu converges o a consan (wih respec o ime). Furhermore, excep for he small neworks, for bigger values of n i can be seen ha he value of convergence decreases linearly alhough n grows exponenially, showing ha he convergence value is approximaely inverse logarihmic (wih respec o n) as expeced from replacing he value of δ in Lemma 5. I is imporan o noice ha he compeiive hroughpu was compued agains a heoreical opimal proocol ha delivers one packe per ime slo, which is no possible in pracice in a muli-hop nework. 6. ACKNOWLEDGMENTS This research was suppored in par by he Naional Science Foundaion (CCF 4930, CCF ) and Kean Universiy UFRI gran. 7. REFERENCES [] Y. Afek, edior. Disribued Compuing - 27h Inernaional Symposium, DISC 203, Jerusalem, Israel, Ocober 4-8, 203. Proceedings, volume 8205 of Lecure Noes in Compuer Science. 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