Optimal Control for Generalized Network-Flow Problems

Transcription

1 Opmal Conrol for Generalzed Nework-Flow Problems Abhshek Snha, Eyan Modano Laboraory for Informaon and Decson Sysems, Massachuses Insue of Technology, Cambrdge, MA 0239 Emal: Absrac We consder he problem of hroughpu-opmal packe dssemnaon, n he presence of an arbrary mx of uncas, broadcas, mulcas and anycas raffc, n a general wreless nework. We propose an onlne dynamc polcy, called Unversal Max-Wegh (UMW), whch solves he above problem effcenly. To he bes of our knowledge, UMW s he frs hroughpu-opmal algorhm of such versaly n he conex of generalzed nework flow problems. Concepually, he UMW polcy s derved by relaxng he precedence consrans assocaed wh mul-hop roung, and hen solvng a mn-cos roung and max-wegh schedulng problem on a vrual nework of queues. When specalzed o he uncas seng, he UMW polcy yelds a hroughpu-opmal cycle-free roung and lnk schedulng polcy. Ths s n conras o he well-known hroughpu-opmal Back- Pressure (BP) polcy whch allows for packe cyclng, resulng n excessve laency. Exensve smulaon resuls show ha he proposed UMW polcy ncurs a subsanally smaller delay as compared o he BP polcy. The proof of hroughpu-opmaly of he UMW polcy combnes deas from sochasc Lyapunov heory wh a sample pah argumen from adversaral queueng heory and may be of ndependen heorecal neres. Index Terms Throughpu-opmal polces, Generalzed flows, Queueng heory I. INTRODUCTION The Generalzed Nework Flow problem nvolves effcen ransporaon of messages, generaed a source node(s), o a se of desgnaed desnaon node(s) over a mul-hop nework. Dependng on he number of desnaon nodes assocaed wh each source node, he problem s known eher as uncas (sngle desnaon node), broadcas (all node are desnaon nodes), mulcas (some nodes are desnaon nodes) or anycas (several choces for a sngle desnaon node). Over he las few decades, a remendous amoun of research effor has been dreced o address each of he above problems n dfferen neworkng conexs. However, despe he ncreasngly dverse mx of nerne raffc, o he bes of our knowledge, here exss no unversal soluon o he general problem, only solaed soluons ha do no neroperae and are ofen subopmal. In hs paper, we provde he frs such unversal soluon: A hroughpu opmal dynamc conrol polcy for he generalzed nework flow problem. We sar wh a bref dscusson of he above neworkng problems and hen survey he relevan leraure. In he Broadcas problem, packes generaed a a source need o be dsrbued among all nodes n he nework. In Par of he paper wll appear n he proceedngs of INFOCOM, 207, IEEE []. he classc paper of Edmonds [2], he broadcas capacy of a wred nework s derved and an algorhm s proposed o compue he maxmum number of edge-dsjon spannng rees, whch ogeher acheve he maxmum broadcas hroughpu. The algorhm n [2] s combnaoral n naure and does no have a wreless counerpar, wh assocaed nerferencefree edge acvaons. Followng Edmonds work, a varey of dfferen broadcas algorhms have been proposed n he leraure, each one argeed o opmze dfferen mercs such as delay [3], energy consumpon [4] and faul-olerance [5]. In he conex of opmzng hroughpu, [6] proposes a randomzed broadcas polcy, whch s opmal for wred neworks. However, exendng hs algorhm o he wreless seng proves o be dffcul [7]. The auhors of [8] propose an opmal broadcas algorhm for a general wreless nework, albe wh exponenal complexy. In a recen seres of papers [9] [0], a smple hroughpu-opmal broadcas algorhm has been proposed for wreless neworks wh an underlyng DAG opology. However, hs algorhm does no exend o non-dag neworks. The Mulcas problem s a generalzaon of he broadcas problem, n whch he packes generaed a a source node needs o be effcenly dsrbued o a subse of nodes n he nework. In s combnaoral verson, he mulcas problem reduces o fndng he maxmum number of edge-dsjon rees, spannng he source node and desnaon nodes. Ths problem s known as he Sener Tree Packng problem, whch s NP-hard []. Numerous algorhms have been proposed n he leraure for solvng he mulcas problem. In [2] [3], back-pressure ype algorhms are proposed for mulcasng over wred and wreless neworks respecvely. These algorhms forward packe over a se of pre-compued dsrbuon rees, and are lmed o he hroughpu obanable by hese rees. Moreover, compung and mananng hese rees s mpraccal n large and me-varyng neworks. We noe ha because of he need for packe duplcaons, he Mulcas and Broadcas problems do no sasfy sandard flow conservaon consrans, and hus he desgn of hroughpu-opmal algorhms s non-rval. The Uncas problem nvolves a sngle source and a sngle desnaon. The celebraed Back-Pressure (BP) algorhm [4] was proposed for he uncas problem. In hs algorhm, he roung and schedulng decsons are aken based on local queue lengh dfferences. As a resul, BP explores all possble pahs for roung and usually akes a long me for convergence, resulng n consderable laency, especally n lghly loaded neworks. Subsequenly, a number of refnemens have

2 2 been proposed o mprove he delay characerscs of he BP algorhm. In [5] BP s combned wh hop lengh based shores pah roung for faser roue dscovery, and [6] proposes a second order algorhm usng he Hessan marx o mprove delay. The Anycas problem nvolves roung from a sngle source o any one of he several gven desnaons. Anycas s ncreasngly used n Conen-Dsrbuon Neworks (CDNs) for opmally dsrbung geo-replcaed conens [7]. Our proposed soluon uses a vrual nework of queues - one vrual queue per lnk n he nework. We solve he roung problem dynamcally usng a smple weghed-shores-roue compuaon on he vrual nework and usng he correspondng roue on he physcal nework. Opmal lnk schedulng s performed by a max-wegh compuaon, also n he vrual nework, and hen usng he resulng acvaon n he physcal nework. The overall algorhm s dynamc, cycle-free, and solves he generalzed roung and schedulng problem opmally (.e., maxmally sable or hroughpu opmal). In addon o hs, he proposed UMW polcy has he followng advanages: ) Generalzed Soluon: Unlke he BP polcy, whch solves only he uncas problem, he proposed UMW polcy effcenly addresses all of he aforemenoned nework flow problems n boh wred and wreless neworks n a very general seng. 2) Delay Reducon: Alhough he celebraed BP polcy s hroughpu-opmal, s average delay performance s known o be poor due o he occurrence of packecyclng n he nework [5] [8]. In our proposed UMW polcy, each packe raverses a dynamcally seleced acyclc roue, whch drascally reduces he average laency. 3) Sae-Complexy Reducon: Unlke he BP polcy, whch manans per-flow queues a each node, he proposed UMW polcy manans only a vrual-queue couner and a prory queue per lnk, rrespecve of he number and ype of flows n he nework. Ths reduces he amoun of overhead ha needs o be mananed for effcen operaon. 4) Effcen Implemenaon: In he BP polcy, roung decsons are made hop-by-hop by he nermedae nodes. Ths pus a consderable amoun of compuaonal overhead on he ndvdual nodes. In conras, n he proposed UMW polcy, he enre roue of he packes s deermned a he source (smlar o he dynamc source roung [9]). Hence, he enre compuaonal requremen s ransferred o he source, whch ofen has hgher compuaonal/energy resources han he nodes n he res of he nework (e.g., wreless sensor neworks). The res of he paper s organzed as follows: In secon II we dscuss he basc sysem model and formulae he problem. In secon III we gve a bref overvew of he proposed UMW polcy. In secon IV we dscuss he srucure and dynamcs of he vrual queues, on whch UMW s based. In secon V we prove s sably propery n he mul-hop physcal nework. In secon VI we dscuss mplemenaon deals. In (a) a wreless nework acvaon vecor s (b) acvaon vecor s Fg. : A wreless nework and s wo maxmal feasble lnk acvaons under he prmary nerference consran. secon VII we provde exensve smulaon resuls, comparng UMW wh oher compeng algorhms. In secon VIII we conclude he paper wh a few drecons for furher research. II. SYSTEM MODEL AND PROBLEM FORMULATION A. Nework Model We consder a wreless nework wh arbrary opology, represened by he graph G(V,E). The nework consss of V = n nodes and E = m lnks. Tme s sloed. A lnk, f acvaed, can ransm one packe per slo. Due o wreless nerference consrans, only ceran subses of lnks may be acvaed ogeher a any slo. The se of all admssble lnk acvaons s known as he acvaon se and s denoed by M 2 E. We do no mpose any resrcon on he srucure of he acvaon se M. As an example, n he case of nodeexclusve or prmary nerference consran [20], he acvaon se M prmary consss of he se of all machngs [2] n he graph G(V,E). Wred neworks are a specal case of he above model, where he acvaon se M wred =2 E. In oher words, n wred neworks, packes can be ransmed over all lnks smulaneously. See Fgure for an example of a wreless nework wh prmary nerference consrans. For smplcy n exposon, n he followng, we assume ha he nework opology s sac. However, he proposed polcy and s analyss apply even f he nework s mevaryng. Moreover, an aracve feaure of our polcy s ha me-varyng neworks do no ncur any exra compuaonal overhead n mplemenaon. The performance of he proposed polcy n me-varyng neworks s evaluaed numercally n Secon VII. B. Traffc Model In hs paper, we consder he Generalzed Nework Flow problem, where ncomng packes a a source node are o be dsrbued among an arbrary se of desnaon nodes n

3 3 a mul-hop fashon. Formally, he se of all dsnc classes of ncomng raffc s denoed by C. A class c raffc s denfed by s source node s 2 V and he se of s requred desnaon nodes D V. As explaned below, by varyng he srucure of he desnaon se D of class c, hs general framework yelds he followng four fundamenal flow problems as specal cases: UNICAST: All class c packes, arrvng a a source node s, are requred o be delvered o a sngle desnaon node D = { }. BROADCAST: All class c packes, arrvng a a source node s, are requred o be delvered o all nodes n he nework,.e., D = V. MULTICAST: All class c packes, arrvng a a source node s, are requred o be delvered o a proper subse of nodes D = {, 2,..., k } ( V. ANYCAST: A Packe of class c, arrvng a a source node s, s requred o be delvered o any one of a gven se of k nodes D = 2... k. Thus he anycas problem s smlar o he uncas problem, wh all desnaons formng a sngle super desnaon node. Arrvals are..d. a every slo, wh A () packes from class c arrvng a he source node s a slo. The mean rae of arrval for class c s EA () =. The arrval rae o he nework s characerzed by he vecor = {,c2c}. The oal number of exernal packe arrvals o he enre nework a any slo s assumed o be bounded by a fne number A max. C. Polcy-Space An admssble polcy for he generalzed nework flow problem execues he followng wo acons a every slo : LINK ACTIVATIONS: Acvang a subse of nerferencefree lnks s() from he acvaon se M. PACKET DUPLICATIONS AND FORWARDING: Possbly duplcang and forwardng packes over he acvaed lnks. Due o he lnk capacy consran, a mos one packe may be ransmed over an acve lnk per slo. The se of all admssble polces s denoed by. The se s unconsraned oherwse and ncludes polces whch may use all pas and fuure packe arrval nformaon. A polcy 2 s sad o suppor an arrval rae-vecor f, under he acon of he polcy, he desnaon nodes of any class c receve dsnc class c packes a he rae,c 2C. Formally, le R () denoe he number of dsnc class-c packes, receved n common by all desnaon nodes 2D 2, under he acon of he polcy, up o me. In order o ransm a packe over mulple downsream lnks (e.g. n Broadcas or Mulcas), he sender mus duplcae he packe and send he copes o he respecve downsream lnk buffers. 2 To be precse, he super-desnaon node n case of Anycas. Defnon. [Polcy Supporng Rae-Vecor ]: A polcy 2 s sad o suppor an arrval rae vecor f lm nf! R () =, 8c 2C, w.p. () The nework-layer capacy regon (G, C) 3 s defned o be he se of all supporable raes,.e., (G, C) def = { 2 R C + : 9 2 supporng } (2) Clearly, he se (G, C) s convex (usng he usual me-sharng argumen). A polcy 2, whch suppors any arrval rae n he neror of he capacy regon (G, C), s called a hroughpu-opmal polcy. D. Admssble Roues of Packes We wll desgn a hroughpu-opmal polcy, whch delvers a packe p o any node n he nework a mos once. 4 Ths mmedaely mples ha he se of all admssble roues T for packes of any class c, n general, comprses of rees rooed a he correspondng source node s. In parcular, dependng on he ype of class c raffc, he opology of he admssble roues T akes he followng specal forms: UNICAST TRAFFIC: T = se of all s pahs n he graph G. BROADCAST TRAFFIC: T = se of all spannng rees n he graph G, rooed a s. MULTICAST TRAFFIC: T = se of all Sener rees [] n G, rooed a s and spannng he verces D = {, 2,..., k }. ANYCAST TRAFFIC: T = unon of all s pahs n he graph G, =, 2,...,k. E. Characerzaon of he Nework-Layer Capacy Regon Consder any arrval vecor 2 (G, C). By defnon, here exss an admssble polcy 2, whch suppors he arrval rae by means of sorng, duplcang and forwardng packes effcenly. Takng me-averages over he acons of he polcy, s clear ha here exs a randomzed flowdecomposon and schedulng polcy o roue he packes such ha none of he edges n he nework s overloaded. Indeed, n he followng heorem, we show ha for every 2 (G, C), here exs non-negave scalars { }, ndexed by he admssble roues T 2T and a convex combnaon of he lnk acvaon vecors µ 2 conv(m) such ha, =, 8c 2C (3) T 2T 3 Noe ha, Nework-layer capacy regon s, n general (e.g. mulcas), dfferen from he Informaon-Theorec capacy regon [22]. 4 Ths should be conrased wh he popular hroughpu-opmal uncas polcy Back-Pressure [4], whch does no sasfy hs consran and may delver he same packe o a node mulple mes, hus poenally degradng s delay performance.

4 4 e (def.) = (,c):e2t,t 2T apple µ e, 8e 2 E. (4) Eqn. (3) denoes decomposon of he average ncomng flows no dfferen admssble roues and Eqn. (4) denoes he fac ha none of he edges n he nework s overloaded,.e. arrval rae of packes o any edge e under he polcy s a mos he rae allocaed by he polcy o he edge e o serve packes. To sae he resul precsely, defne he se o be he se of all arrval vecors 2 R C +, for whch here exss a randomzed acvaon vecor µ 2 conv(m) and a non-negave flow decomposon { }, such ha Eqns. (3) and (4) are sasfed. We have he followng heorem: Theorem. The nework-layer capacy regon (G, C) s characerzed by he se, up o s boundary. Proof of Theorem consss of wo pars: converse and achevably. Proof of he converse s gven n Appendx I-A, where we show ha all supporable arrval raes mus belong o he se. The man resul of hs paper, as developed n he subsequen secons, s he consrucon of an effcen admssble polcy, called Unversal Max-Wegh (UMW), whch acheves any arrval rae n he neror of he se. III. OVERVIEW OF THE UMW POLICY In hs secon, we presen a bref overvew of our hroughpu-opmal UMW polcy, desgned and analyzed n he subsequen secons. Cenral o he UMW polcy s a global sae vecor, called vrual queues Q(), used for packe roung and lnk acvaons. Each componen of he vrual queues s updaed a every slo accordng o a onehop queueng (Lndley) recurson, correspondng o a relaxed nework, descrbed n deal n secon IV. Unlke he wellknown Back-Pressure algorhm for he uncas problem [4], n whch packe roung decsons are made hop-by-hop usng physcal queue lenghs Q(), he UMW polcy prescrbes an admssble roue o each ncomng packe mmedaely upon s arrval (dynamc source roung). Ths roue selecon decson s dynamcally made by solvng a suable mn-cos roung problem (e.g., shores pah, MST ec.) a he source wh edge coss gven by he curren vrual-queue vecor Q(). Lnk acvaon decsons a each slo are made by a Max- Wegh algorhm wh lnk-weghs se equal o Q(). Havng fxed he roung and acvaon polcy as above, n secon V we desgn a packe schedulng algorhm for he physcal nework, whch effcenly resolves conenon among mulple packes ha wa o cross he same (acve) edge a he same slo. We show ha he overall polcy s hroughpu-opmal. One sgnfcanly new feaure of our algorhm s ha s enrely oblvous o he lengh of he physcal queues of he nework and ulzes he auxlary vrual-queue sae varables for sablzng he former. Our proof of hroughpu-opmaly of UMW leverages deas from deermnsc adversaral queueng heory and combnes effecvely wh he sochasc Lyapunov-drf based echnques and may be of ndependen heorecal neres. IV. GLOBAL VIRTUAL QUEUES: STRUCTURES, ALGORITHMS, AND STABILITY In hs secon, we nroduce he noon of vrual queues 5, whch s obaned by relaxng he dynamcs of he physcal queues of he nework n he followng nuve fashon. A. Precedence Consrans In a mul-hop nework, f a packe p s beng roued along he pah T = l l 2... l k, where l 2 E s he h lnk on s pah, hen by he prncple of causaly, he packe p canno be physcally ransmed over he j h lnk l j f has no already been ransmed by he frs j lnks l,l 2,...,l j. Ths consran s known as he precedence consran n he nework schedulng leraure [24]. In he followng, we make a radcal deparure by relaxng hs consran o oban a smpler sngle-hop vrual sysem, whch wll play a key role n desgnng our polcy and s opmaly analyss. B. The Vrual Queue Process { Q()} The Vrual queue process Q() = Q e (),e 2 E s an E = m dmensonal conrolled sochasc process, mang a fcous queueng nework whou he precedence consrans. In parcular, when a packe p of class c arrves a he source node s, a dynamc polcy prescrbes a suable roue T () 2T o he packe. Denong he se of all edges n he roue T () by {l,l 2,...,l k }, hs ncomng packe nduces a vrual arrval smulaneously a each of he vrual queues Ql, =, 2,...,k, rgh upon s arrval o he source. Snce he vrual nework s assumed o be relaxed wh no precedence consrans, any packe presen n he vrual queue s elgble for servce. See Fgure 2 for an llusraon. The (conrolled) servce process allocaed o he vrual queues s denoed by {µ ()}. We requre he servce process o sasfy he same acvaon consrans as n he orgnal sysem,.e., µ () 2M, 8. Le A e () s he oal number of vrual packe arrval (from all classes) o he queue Q e a me under he acon of he polcy,.e., A e () = c2c A () e 2 T (), 8e 2 E. (5) Hence, we have he followng one-sep evoluon (Lndley recurson) of he vrual queue process { Q e ()} : Q e ( + ) = Q e ()+A e () µ e () +, 8e 2 E, (6) We emphasze ha A e () s a funcon of he roung ree T () ha he polcy chooses a me, from he se of all admssble roues T. Ths s dscussed n he followng. 5 Noe ha our noon of vrual queues s compleely dfferen from and unrelaed o he noon of shadow-queues proposed earler n [8], [3] and vrual queues proposed n [23].

5 5 p 2 3 A Mulhop Nework G 4 Q 2 () p Q 23 () p Q 24 () Q 3 () Q 34 () p T p = {{, 2}, {2, 3}, {3, 4}} Vrual Queues µ 2 () µ 23 () µ 24 () µ 3 () µ 34 () Fg. 2: Illusraon of he vrual queue sysem for he four-node nework G. Upon arrval, he ncomng packe p, belongng o a uncas sesson from node o 4, s prescrbed a pah T p = {{, 2}, {2, 3}, {3, 4}}. Relaxng he precedence consrans, he packe p s couned as an arrval o he vrual queues Q 2 and Q 23 and Q 34 smulaneously a he same slo. In he physcal sysem, he packe p may ake a whle before reachng any edge n s pah, dependng on he conrol polcy. C. Dynamc Conrol and Sably of he Vrual Queues Nex we desgn a dynamc roung and lnk acvaon polcy for he vrual nework, whch sablzes he vrual queue process { Q()}, for all arrval rae-vecors 2 n( ). Ths polcy s obaned by mnmzng he one-sep drf of a quadrac Lyapunov-funcon of he vrual queue lenghs (as opposed o he real queue lenghs used n he Back-Pressure polcy [4]). In he followng secon, we wll show ha when hs dynamc polcy s used n conjuncon wh a suable packe schedulng polcy n he physcal nework, he overall polcy s hroughpu-opmal for he physcal nework. To derve a sablzng polcy for he vrual nework, consder a quadrac Lyapunov funcon L( Q()) defned n erms of he vrual queue lenghs: L( Q()) = Q 2 e() From he one-sep dynamcs of he vrual queues (6), we have: Q e ( + ) 2 apple ( Q e () µ e ()+A e ()) 2 = Q 2 e()+(a e ()) 2 +(µ e ()) 2 +2 Q e ()A e () 2 Q e ()µ e () 2µ e ()A e () Snce µ e () 0 and A e () 0, we have Q 2 e( + ) Q2 e () apple (A e ()) 2 +(µ e ()) Q e ()A e () 2 Q e ()µ e () Hence, he one-sep Lyapunov drf (), condonal on he curren vrual queue lenghs Q(), under he operaon of any admssble Markovan polcy 2 s upper-bounded by () def = E L( Q( + )) L( Q()) Q() apple B +2 Q e ()E A e () Q() 2 Q e ()E µ e () Q() (7) where B s a consan, bounded by P e (E(A e ()) 2 + E(µ e ()) 2 ) apple A 2 max + m. The upper-bound on he drf, gven by (7), holds good for any admssble polcy n he vrual nework. In parcular, by mnmzng he upper-bound pon wse, and explong he separable naure of he objecve, we derve he followng decoupled dynamc roung and lnk acvaon polcy for he vrual nework: Dynamc Roung Polcy: The opmal roue for each class c, over he se of all admssble roues, s seleced by mnmzng he followng cos funcon, appearng n he mddle of Eqn. (7) RoungCos Q e ()A e (), where we remnd he reader ha A e () are he roung polcy dependen arrvals o he vrual queue correspondng o he lnk e a me. Usng Eqn. (5), we may rewre he objecve-funcon as RoungCos = A () Q e () e 2 T ()) (8) c2c Usng he separably of he objecve (8), he above opmzaon problem decomposes no followng mn-cos roueselecon problem T op() for each class c: T op() 2 arg mn T 2T Q e () e 2 T ) (9) Dependng on he ype of flow of class c, he opmal roueselecon problem (9) s equvalen o one of he followng well-known combnaoral problems on he graph G, wh s edges weghed by he vrual queue lengh vecor Q: UNICAST TRAFFIC: T op() =The shores s pah n he weghed-graph G. BROADCAST TRAFFIC: T op() = The mnmum wegh spannng ree rooed a he source s, n he weghed-graph G. MULTICAST TRAFFIC: T op() = The mnmum wegh Sener ree rooed a he source s and spannng he desnaons D = 2,..., k }, n he weghed-graph G. ANYCAST TRAFFIC: T op() =The shores of he k shores s pahs, =, 2,...,k n he weghed-graph G. {, Thus, he roues are seleced accordng o a dynamc source roung polcy [9]. Apar from he mnmum wegh Sener ree problem for he mulcas raffc (whch s NP-hard wh several known effcen approxmaon algorhms [25]), all of he above roung problems on he weghed vrual graph may be solved effcenly usng sandard algorhms [26].

6 6 Dynamc Lnk Acvaon Polcy: A feasble lnk acvaon schedule µ () 2Ms dynamcally chosen a each slo by maxmzng he las erm n he upper-bound of he drfexpresson (7), gven as follows: µ () 2 arg max µ2m Q e ()µ e (0) Ths s he well-known max-wegh schedulng polcy, whch can be solved effcenly under varous nerference models (e.g., Prmary or node-exclusve model [27]). In solvng he above roung and schedulng problems, we acly made he assumpon ha he vrual queue vecor Q() s globally known a each slo. We wll dscuss praccal dsrbued mplemenaon of our algorhm n secon VI. Nex, we esablsh sably of he vrual queues under he above polcy, whch wll be nsrumenal for provng hroughpu-opmaly of he overall UMW polcy: Theorem 2. Under he above dynamc roung and lnk schedulng polcy, he vrual queue process { Q()} 0 s srongly sable for any arrval rae 2 n( ),.e., lm sup T! T T E( Q e ()) <. =0 queue Q e s gven by m+ Eµ RAND e () = p s (e) =µ e (4) = Snce our Max-Wegh polcy, UMW, maxmzes he RHS of he drf expresson n Eqn. (7) from he se of all feasble polces, we can wre UMW () apple B +2 2 (a) = B +2 Q e () EA RAND (b) = B +2 Q e () e µ e apple B 2 Q e (), Q e ()E A RAND e () Q() Q e ()E µ RAND e () Q() e () Eµ RAND e () where (a) follows from he fac ha he randomzed polcy RAND s memoryless and hence, ndependen of he vrual queues Q(), (b) follows from Eqns. (3) and (4) and fnally follows from Eqn. (). Takng expecaon of boh sdes w.r.. he vrual queue lenghs Q(), we bound he expeced drf a slo as EL Q( + ) EL Q() apple B 2 E( Q e ()) (5) Proof. Consder an arrval rae vecor 2 n( ). Thus, from Eqns. (3) and (4), follows ha here exss a scalar >0 and a vecor µ 2 conv(m), such ha we can decompose he oal arrval for each class c 2Cno a fne number of roues, such ha (def.) e = apple µ e, 8e 2 E () (,c):e2t,t 2T By Caraheodory s heorem [28], we can wre µ = m+ = p s, (2) for some acvaon vecors s 2M, 8 and some probably dsrbuon p. Now consder he followng auxlary saonary randomzed roung and lnk acvaon polcy RAND 2 for he vrual queue sysem { Q()}, whch wll be useful n our proof. The randomzed polcy RAND randomly selecs he acvaon vecor s j wh probably p j,j =, 2,...,m+ and roues he ncomng packe of class c along he roue T 2T, wh probably, 8, c. Hence he oal expeced arrval rae o he vrual queue Q e a me slo, due o he acon of he saonary randomzed polcy RAND s gven by EA RAND e () = e = (,c):e2t,t 2T, 8e 2 E (3) and he expeced oal servce rae o he vrual server for he Summng Eqn. (5) from =0o T and rememberng ha L(Q(T )) 0 and L( Q(0)) = 0, we conclude ha T T E( Q e ()) apple B 2 =0 Takng lm sup of boh sdes proves he clam. (6) As a consequence of he srong sably of he vrual queues { Q e (),e 2 E}, we have he followng sample-pah resul, whch wll be he key o our subsequen analyss: Lemma. Under he acon of he above polcy, we have for any 2 n( ): Q e () lm =0, 8e 2 E, w.p..! In oher words, he vrual queues are rae-sable [29]. Proof. See Appendx I-C. The sample pah resul of Lemma may be nerpreed as follows: For any gven realzaon! of he underlyng sample space, defne he funcon F (!, ) = max Q e (!, ).

7 7 Noe ha, for any 2 Z +, due o he boundedness of arrvals per slo, he funcon F (!, ) s well-defned and fne. In vew of hs, Lemma () saes ha under he acon of he UMW polcy, F (!, ) = o() almos surely. 6 Ths resul wll be used n our sample pahwse sably analyss of he physcal queueng process {Q()} 0. D. Consequence of he Sably of he Vrual Queues I s apparen from he vrual queue evoluon equaon (6), ha he sably of he vrual queues under he UMW polcy mples ha he arrval rae a each vrual queue s a mos he servce rae offered o under he UMW roung and schedulng polcy. In oher words, effecve load of each edge e n he vrual sysem s a mos uny. Ths s a necessary condon for sably of he physcal queues when he same roung and lnk acvaon polcy s used for he mul-hop physcal nework. In he followng, we make he noon of effecve-load mahemacally precse. Skorokhod Mappng: Ierang on he sysem equaon (6), we oban he followng well-known dscree me Skorokhod- Map represenaon [30] of he vrual queue dynamcs + Q e () = A e (,) Se (,), (7) sup apple apple where A e (, 2 ) def = P 2 = A e ( ), s he oal number of arrvals o he vrual queue Q e n he me nerval [, 2 ) and Se (, 2 ) def = P 2 = µ e ( ), s he oal amoun of servce allocaed o he vrual queue Q e n he nerval [, 2 ). For reference, we provde a proof of Eqn. (7) n Appendx I-B. Combnng Equaon (7) wh Lemma, we conclude ha under he UMW polcy, almos surely for any sample pah! 2, for each edge e 2 E and any 0 <, we have A e (!; 0,) apple S e (!; 0,)+F (!, ), (8) where F (!, ) =o(). Implcaons for he Physcal Nework: Noe ha, every packe arrval o a vrual queue Q e a me corresponds o a packe n he physcal nework, ha wll evenually cross he edge e. Hence he loadng condon (8) mples ha under he UMW polcy, he oal number of packes njeced durng any me nerval ( 0,], wllng o cross he edge e, s less han he oal amoun of servce allocaed o he edge e n ha me nerval up o an addve erm of o(). Thus nformally, he effecve load of any edge e 2 E s a mos uny. By ulzng he sample-pah resul n Eqn. (8), n he followng secon we show ha here exss a smple packe schedulng scheme for he physcal nework, whch guaranees he sably of he physcal queues, and consequenly, hroughpuopmaly. V. OPTIMAL CONTROL OF THE PHYSICAL NETWORK Wh he help of he vrual queue srucure as defned above, we nex focus our aenon on desgnng a hroughpuopmal conrol polcy for he mul-hop physcal nework. 6 g() =o() f lm! g() =0. As dscussed n Secon II, a conrol polcy for he physcal nework consss of hree componens, namely () Roung, (2) Lnk acvaons and (3) Packe schedulng. In he proposed UMW polcy, he () Roung and (2) Lnk acvaons for he physcal nework s done exacly n he same way as n he vrual nework, based on he curren values of he vrual queue sae varables Q(), descrbed n Secon IV-C. I s o be noed ha, n he parcular case of wreless neworks, s possble ha a parcular edge wh posve vrual queue lengh s scheduled for ransmsson a a slo, even hough he edge does no have any packe o ransm n s physcal queue. The surprsng fac, ha follows from Theorem 4 s ha hs knd of wased ransmssons are rare and does no affec he hroughpu. There exs many possbles for he hrd componen, namely he packe scheduler, whch effcenly resolves conenon when mulple packes aemp o cross an acve edge e a he same me-slo. Popular choces for he packe scheduler nclude FIFO, LIFO ec. In hs paper, we focus on a parcular schedulng polcy whch has s orgn n he conex of adversaral queueng heory [3]. In parcular, we exend he Neares To Orgn (NTO) polcy o he generalzed nework flow seng, where a packe may be duplcaed. Ths polcy was proposed n [32] n he conex of wred neworks for he uncas problem. We appropraely exend hs polcy for use n generalzed flow problems, ncludng mulcas, broadcas, and anycas, even n wreless neworks. Our proposed schedulng polcy s called Exended NTO (ENTO) and s defned as follows: Defnon 2 (Exended NTO). If mulple packes aemp o cross an acve edge e a he same me slo, he Exended Neares To Orgn (ENTO) polcy gves prory o he packe whch has raversed he leas number of hops along s pah from s orgn up o he edge e. The Exended NTO polcy may be easly mplemened by mananng a sngle prory queue [26] per edge. The nal prory of each ncomng packe a he source s se o zero. Upon ransmsson by any edge, he prory of a ransmed packe s decreased by one. The ransmed packe s hen coped no he nex-hop prory queue(s) (f any) accordng o s assgned roue. See Fgure 3 for an llusraon. The pseudo code for he full UMW algorhm s provded n Algorhm. We nex sae he followng heorem whch proves sably of he physcal queues under he ENTO polcy: Theorem 3. Under he acon of he UMW polcy wh ENTO packe schedulng, he physcal queues are raesable [29] for any arrval vecor 2 n( ),.e., P lm Q e() =0, w.p.!

8 8 prory[p ] e3 = 2 prory[p 2 ] e3 = Proof. Ths heorem s proved by exendng he argumen of Gamarnk [32] and combnng wh he sample pah loadng condon n Eqn. (8). See Appendx I-D for he dealed argumen. As a drec consequence of Theorem 3, we have he man resul of hs paper: p e 5 Q e3 p p 2 Theorem 4. The UMW polcy s hroughpu-opmal. S e e 2 e 3 Fg. 3: A schemac dagram showng he schedulng polcy ENTO n acon. The packes p and p 2 orgnae from he sources S and S 2. Par of her assgned roues are shown n blue and red respecvely. The packes conend for crossng he acve edge e 3 a he same me slo. Accordng o he ENTO polcy, he packe p 2 has hgher prory (havng crossed a sngle edge e 4 from s source) han p (havng crossed wo edges e and e 2 from s source) for crossng he edge e 3. Noe ha, alhough a copy of p mgh have already crossed he edge e 5, hs edge does no fall n he pah connecng he source S o he edge e 3 and hence does no ener no prory calculaons. e 4 S 2 p 2 Proof. For any class c 2C, he number of packes R (), receved by all nodes 2D may be bounded as follows: A (0,) Q e () ( ) apple R () apple A (0,), (9) where he lower-bound ( ) follows from he smple observaon ha f a packe p of class c has no reached all desnaon nodes D, hen a leas one copy of mus be presen n some physcal queue. Dvdng boh sdes of Eqn. (9) by, akng lms and usng SLLN and Theorem 3, we conclude ha w.p. R () lm =! Hence from he defnon (), we conclude ha UMW s hroughpu-opmal. Algorhm Unversal Max-Wegh Algorhm (UMW) a slo for he Generalzed Flow Problem n a Wreless Nework Requre: Graph G(V,E), Vrual queue lenghs { Q e (),e 2 E} a he slo. : [Edge-Wegh Assgnmen] Assgn each edge of he graph e 2 E a wegh W e () equal o Q e (),.e. W () Q() 2: [Roue Assgnmen] Compue a Mnmum Wegh Roue T () 2T () for a class c ncomng packe n he weghed graph G(V,E), accordng o Eqn. (9). 3: [Lnk Acvaon] Choose he acvaon µ() from he se of all feasble acvaons M, whch maxmzes he oal acvaed lnk-weghs,.e. µ() arg max s2m s W () 4: [Packe Forwardng] Forward physcal packes from he physcal queues over he acvaed lnks accordng o he ENTO schedulng polcy. 5: [Vrual Queue Couner Updae] Updae he vrual queues assumng a precedence-relaxed sysem,.e., + Q e ( + ) Q e ()+A e () µ e (), 8e 2 E VI. DISTRIBUTED IMPLEMENTATION The UMW polcy n s orgnal form, as gven n Algorhm, s cenralzed n naure. Ths s because he sources need o know he opology of he nework and he curren value of he vrual queues Q() o solve he shores roue and he Max-Wegh problems a seps (2) and (3) of he algorhm. Alhough he opology of he nework may be obaned effcenly by opology dscovery algorhms [33], keepng rack of he vrual queue evoluon (Eqn. (6)) s suble. Noe ha, n he specal case where all packes arrve only a a sngle source node, no nformaon exchange s necessary and he vrual queue updaes (Sep 5) may be mplemened a he source locally. In he general case wh mulple sources, s necessary o perodcally exchange packe arrval nformaon among he sources o mplemen Sep 5 exacly. To crcumven hs ssue, we propose he followng class of heursc UMW polces: [Heursc UMW] Assgn he edge weghs o be he Physcal queue lenghs Q(), nsead of he vrual queue lenghs Q(), n eher sep (2) or sep (3) or boh n he orgnal UMW Algorhm. Roung based on physcal queue lenghs sll requres he exchange of queue lengh nformaon. However, hs can be mplemened effcenly usng he sandard dsrbued Bellman-Ford algorhm. The smulaon resuls n secon VII-B show ha he heursc polcy works well n pracce and

9 9 s =) s 2 (= 2 6 pahs o roue packes o her desnaons. In he low-load regme, he packes may also cycle n he nework ndefnely, whch ncreases her laency. The UMW polcy, on he oher hand, ransms all packes along opmal acyclc roues. Ths resuls n subsanal reducon n end-o-end delay & Fg. 4: The wred nework opology used for uncas smulaon Avg. Queue Lenghs UMW (opmal) BP UMW (heursc) Fg. 5: Comparson of he delay performances of he BP and UMW (opmal and heursc) polces n he uncas seng of Fg. 4. s delay performance s subsanally beer han he vrual queue based opmal UMW polcy n wreless neworks. The affrmave smulaon resuls shown n he nex secon mmedaely promps us o make he followng conjecure: Conjecure. The Heursc UMW polcy s hroughpuopmal. VII. NUMERICAL SIMULATION A. Delay Improvemen Compared o he Back Pressure Polcy - he Uncas Seng To emprcally demonsrae superor delay performance of he UMW polcy over he Back-Pressure polcy n he uncas seng, we smulae boh polces n he wred nework shown n Fgure 4. All lnks have a un capacy. We consder wo concurren uncas sessons wh source-desnaon pars gven by (s =, = 8) and (s 2 = 5, 2 = 2) respecvely. I s easy o see ha Max-Flow(s! )=2and Max-Flow(s 2! 2 )=and here exs muually dsjon pahs o acheve he opmal rae-par (, 2) = (2, ). Assumng Posson arrvals a he sources s and s 2 wh nenses =2 and 2 =, 0 apple apple, where denoes he load facor, Fgure 5 shows a plo of oal average queue lenghs as a funcon of he load facor under he operaon of he BP, UMW (opmal) and UMW (heursc) polcy. From he plo, we conclude ha boh he opmal and heursc UMW polces ouperforms he BP polcy n erms of average queue lenghs, and hence (by Lle s Law), end-o-end delay, especally n low-o-moderae load regme. The prmary reason beng, he BP polcy, n prncple, explores all possble B. Usng he Heursc UMW polcy for Improved Laency n he Wreless Neworks - he Broadcas Seng Nex, we emprcally demonsrae ha he heursc UMW polcy ha uses physcal queue lenghs Q() (nsead of vrual queues Q() as n he opmal UMW polcy) no only acheves he full broadcas capacy bu yelds beer delay performance n hs parcular wreless nework. As dscussed earler, he heursc polcy s praccally easer o mplemen n a dsrbued fashon. We smulae a 3 3 wreless grd nework shown n Fgure 6, wh prmary nerference consrans [20]. The broadcas capacy of he nework s known o be = 2 5 [8]. The ENTO polcy s used for packe schedulng. The average queue lengh s ploed n Fgure 7 as a funcon of he packe arrval rae under he operaon of he (a) UMW (opmal) and (b) UMW (heursc) polces. The plo shows ha he heursc polcy resuls n much smaller queue lenghs han he opmal polcy. The reason beng ha physcal queues capure he nework congeson more accuraely for proper lnk acvaons. Source =) r a b c d e f g h Fg. 6: The wreless opology used for broadcas smulaon Avg. Queue lenghs (a) UMW (opmal) (b) UMW (heursc) Arrval rae ( ) Fg. 7: Comparson of he Avg. Queue lenghs as a funcon of he arrval rae for he opmal (n blue) and he heursc (n red) UMW Polcy for he grd nework n Fgure 6 n he broadcas seng. C. Performance of he Opmal and Heursc UMW Polcy n Tme-varyng neworks- he Broadcas Seng In hs secon, we ake a closer look a he wreless grd nework n Fgure 6 by numercally evaluang he broadcas-

10 0 Avg. Queue Lenghs p ON =0.2 p ON = Fg. 8: Comparson of he me-averaged oal queue lenghs under he opmal (sold lne) and heursc (dashed lne) UMW polcy n he me-varyng grd nework (wh parameer p ON), for he broadcas problem. ng performance of he proposed polces, when he nework s me-varyng. In parcular, we assume ha a each slo a lnk s ON wh probably p ON, and s OFF w.p. p ON, ndependen of everyhng else. Packes can be ransmed only over he ON lnks a a gven slo. Usng smlar analyss ha we dd for he sac nework, can be easly shown ha he proposed UMW polcy remans hroughpu-opmal when he Max-Wegh lnk acvaon a each slo s done wh respec o he ON lnks a ha slo. The packe roung polcy remans he same as n he orgnal UMW polcy. The performance of he opmal and heursc UMW polcy s shown n Fgure 8 for wo dfferen values of he parameer p ON. I can be seen from he plo ha he heursc polcy ncurs subsanally smaller queue lenghs, compared o he opmal polcy, especally n he low-load regme. Also, from he nearly dencal vercal asympoes n he queue lengh vs arrval rae plos, we conclude ha he heursc polcy s also hroughpu-opmal n hs case. VIII. CONCLUSION In hs paper, we have proposed a new, effcen and hroughpu-opmal polcy, named Unversal Max-Wegh (UMW), for he Generalzed Nework Flow problem. The UMW polcy can smulaneously handle a mx of Uncas, Broadcas, Mulcas and Anycas raffc n arbrary neworks and s emprcally shown o have superor performance compared o he exsng polces. The nex sep would be o nvesgae wheher he UMW polcy sll reans s opmaly when mplemened wh physcal queue lenghs, nsead of he vrual queue lenghs. An affrmave answer o hs queson would mply a more effcen mplemenaon of he polcy. REFERENCES [] A. Snha and E. Modano, Opmal conrol for generalzed nework-flow problems, n Compuer Communcaons (INFOCOM), 207 IEEE Conference on, (To appear). [Onlne]. Avalable: hp: //b.ly/nfocom7snha [2] R. Rusn, Combnaoral Algorhms. Algorhmcs Press, 973. [3] A. Czumaj and W. 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11 [27] L.. Bu, S. Sanghav, and R. Srkan, Dsrbued lnk schedulng wh consan overhead, Neworkng, IEEE/ACM Transacons on, vol. 7, no. 5, [28] J. Maoušek, Lecures on dscree geomery. Sprnger New York, 2002, vol. 08. [29] M. J. Neely, Sochasc nework opmzaon wh applcaon o communcaon and queueng sysems, Synhess Lecures on Communcaon Neworks, vol. 3, no., pp. 2, 200. [30] S. Meyn, Conrol echnques for complex neworks. Cambrdge Unversy Press, [3] M. Andrews, B. Awerbuch, A. Fernández, T. Leghon, Z. Lu, and J. Klenberg, Unversal-sably resuls and performance bounds for greedy conenon-resoluon proocols, Journal of he ACM (JACM), vol. 48, no., pp , 200. [32] D. Gamarnk, Sably of adapve and non-adapve packe roung polces n adversaral queueng neworks, n Proceedngs of he hryfrs annual ACM symposum on Theory of compung. ACM, 999. [33] R. Chandra, C. Fezer, and K. Hogsed, A mesh-based robus opology dscovery algorhm for hybrd wreless neworks, n Proceedngs of AD-HOC Neworks and Wreless, [34] R. Durre, Probably: heory and examples. Cambrdge unversy press, 200. [35] N. Glck, Breakng records and breakng boards, Amercan Mahemacal Monhly, pp. 2 26, 978. I. APPENDI A. Proof of Converse of Theorem Proof. Consder any admssble arrval rae vecor 2 (G, C). By defnon, here exss an admssble polcy 2 whch suppors he arrval vecor n he sense of Eqn. (). Whou any loss of generaly, we may assume he polcy o be saonary and he assocaed DTMC o be ergodc. Le () denoe he oal number of packes from class c ha have fnshed her roung along he roue T 2T up o A me. Noe ha, each packe s roued along one admssble roue only. Hence, f he oal number of arrval o he source s of class c up o me s denoed by he random varable A (), we have A () (a) T 2T A () (b) = R (). (20) In he above, he nequaly (a) follows from he observaon ha any packe p whch has fnshed s roung along some roue T 2T by he me, mus have arrved a he source by he me. The equaly (b) follows from he observaon ha any packe p whch has fnshed s roung by me along some roue T 2T, has reached all of he desnaon nodes D of class c by me and vce versa. Dvdng boh sdes of equaon (20) by and akng lm as!, we have w.p. (d) = lm! A () lm nf! = lmnf! (f) =, T R () 2T A where equaly (d) follows from he SLLN, and equaly (f) follows from he Defnon (). () From he above nequales, we conclude ha w.p. lm! () =, 8c 2C (2) T 2T A Now we use he fac ha he polcy s saonary and he assocaed DTMC s ergodc. Thus he me-average lms exs and hey are consan a.s.. For all T 2T c and c 2C, defne def = lm! Hence, from he above, we ge = A T 2T () (22) (23) Now consder any edge e 2 E n he graph G. Snce he varable A () denoes he oal number of packes from class c, ha have compleely raversed along he ree T, he followng nequaly holds good for any me A () apple µ e ( ), (24) (,c):e2t,t 2T = where he lef-hand sde denoes a lower-bound on he number of packes ha have crossed he edge e and he rgh hand sde denoes he amoun of servce ha have been provded o edge e up o me by he polcy. Dvdng boh sdes by and akng lms of boh sde, and nong ha he lm on he lef-hand sde exss w.p., we have apple µ e, (25) (,c):e2t,t 2T c P where µ =lm! = µ( ). Snce µ( ) 2M, 8 and he se conv(m) s closed, we conclude ha µ 2 conv(m). Eqns. (23) and (25) concludes he proof of he heorem. B. Proof of he Skorokhod Map Represenaon n Eqn. (7) Proof. From he dynamcs of he vrual queues n Eqn. (6), we have for any Q e () Qe ( ) + A e ( ) µ e ( ). (26) Ierang (26) mes apple apple, we oban Q e () Qe ( )+A e (,) S e (,), where A e (, 2 ) = P 2 = A e ( ) and S e (, 2 ) = P 2 = µ e ( ), as defned before. Snce each of he vrualqueue componens are non-negave a all mes (vz. (6)), we have Q e ( ) 0. Thus, Q e () A e (,) S e (,). Snce he above holds for any me apple apple and he queues are always non-negave, we oban Q e () A e (,) S e (,) (27) sup apple apple +

12 2 To show ha Eqn. (27) holds wh equaly, we consder wo cases. Case I: Q e () =0 Snce he RHS of Eqn. (27) s non-negave, we mmedaely oban equaly hroughou n Eqn (27). Case II: Q e () > 0 Consder he laes me 0, apple 0 apple, pror o, a whch Q e ( 0 ) = 0. Such a me 0 exss because we assumed he sysem o sar wh empy queues a me =0. Hence Q e (z) > 0 hroughou he me nerval z 2 [ 0 +,]. As a resul, n hs me nerval he sysem dynamcs for he vrual-queues (6) akes he followng form Q e (z) = Q e (z ) + A e (z ) µ e (z ), Ierang he above recurson n he nerval z 2 [ we oban 0 +,], Q e () =A e ( 0,) S e ( 0,) (28) We conclude he proof upon combnng Eqns. (27) and (28). C. Proof of Lemma Proof. We wll esablsh hs resul by appealng o he Srong Sably Theorem (Theorem 2.8) of [29]. For hs, we frs consder an assocaed sysem { ˆQ()} 0 wh a slghly dfferen queueng recurson, as consdered n [29] (Eqn. 2., pp-5). For a gven sequence {A(), µ()} 0, defne he followng recurson for all e 2 E, ˆQ e ( + ) = ( ˆQ e () µ e ()) + + A e (), (29) ˆQ e (0) = 0. Recall he dynamcs of he vrual queues (Eqn. (6)): Q e ( + ) = ( Q e ()+A e () µ e ()) +, (30) Q e (0) = 0. We nex prove he followng proposon: Proposon 5. For all e 2 E A max + Q e () ( ) ˆQe () ( ) Qe (), 8 0. Proof. We frs prove he second nequaly (**) by nducng on me. Base Sep =0: Holds wh equaly snce ˆQe (0) = Q e (0) = 0. Inducon Sep: Assume ha ˆQe () Qe () for some 0. From he dynamcs (29), we can wre ˆQ e ( + ) = max ˆQ e () µ e ()+A e (),A e () (a) (b) max ˆQ e () µ e ()+A e (), 0 max Q e () µ e ()+A e (), 0 = Q e ( + ), where Eqn. (a) follows from he fac ha A e () 0, Eqn. (b) follows from he nducon assumpon and Eqn. follows from he dynamcs (30). Ths complees he nducon sep and he proof of he second nequaly (**) of he proposon. Proof of he frs nequaly (*) may also be carred ou smlarly. Takng expecaon hroughou he frs nequaly (*) of Proposon 5 for any e 2 E, we have for each 0 Thus, lm sup T! T T =0 E( ˆQ e ()) apple E( Q e ()) + A max E( ˆQ e ()) apple lm sup T! (a) <, T T =0 E( Q e ()) + A max where (a) follows from he srong sably of he vrual queues under UMW. Ths shows ha, he assocaed queue process { ˆQ()} 0 s also srongly sable under UMW. Snce he oal exernal arrval A() = P e A e() a slo s assumed o be bounded w.p., applyng Theorem 2.8, par (b) of [29], we conclude ha for any e 2 E ˆQ e () lm =0, w.p.! Usng he second nequaly (**) of Proposon 5 and he nonnegavy of he vrual queues, we conclude ha for any e 2 E Q e () lm =0, w.p.! Fnally, usng he unon bound we conclude ha lm! D. Proof of Theorem 3 Q e () =0, 8e 2 E w.p. Throughou hs proof, we wll fx a sample pon! 2, gvng rse o a sample pah sasfyng he condon (8). All random processes 7 wll be evaluaed a hs sample pah. For he sake of noaonal smplcy, we wll drop he argumen! for evaluang any random varable a he sample pon!, e.g., he deermnsc sample-pah (!, ) wll be smply denoed by (). We now esablsh a smple analycal resul whch wll be useful n he man proof of he heorem: Lemma 2. Consder a non-negave funcon {F (), } defned on he se of naural numbers, such ha F () = o(). Defne M() =sup 0apple apple F ( ). Then. M() s non-decreasng n. 2. M() =o() 7 Recall ha, a dscree-me neger-valued random process (!; ) s a measurable map from he sample space o he se of all neger-sequences Z [34],.e., :! Z.

13 3 Proof. Tha M() s non-decreasng follows drecly from he defnon of M() =sup 0apple apple F (). We now prove he clam (2). Case I: The funcon F () s bounded In hs case, he funcon M() s also bounded and he clam follows mmedaely. Case II: The funcon F () s unbounded Defne he subsequence {r k } k, correspondng o he me of maxmums of he funcon M() up o me. Formally he sequence {r k } k s defned recursvely as follows, r = (3) r k = {mn >r k : F () > max F ( )} (32) apple Snce he funcon F () s assumed o be unbounded, we have r k! as k!. In he leraure [35], he sequence {r k } s also known as he sequence of records of he funcon F (). Wh hs defnon, for any and for r k apple correspondng o he laes record up o me, we readly have Hence, M() M() =F (r k ) (33) = F (r k) (a) apple F (r k), (34) r k where Eqn. (a) follows from he fac ha r k apple. Thus for any sequence of naural numbers { }, we have a correspondng sequence {r k } = such ha for each, we have Ths mples, lm sup! M( ) = F (r k ) (a) apple F (r k ) r k M() apple lm sup! F () (b) =0, (35) where Eqn (b) follows from our hypohess on he funcon F (). Also snce M() F (), from Eqn. (35) we conclude ha M() lm =0 (36)! As a drec consequence of Lemma 2 and he propery of he sample-pon! under consderaon, we have: A e ( 0,) apple S e ( 0,)+M(), 8e 2 E, 8 0 apple (37) for some non-decreasng non-negave funcon M() = o(). Equpped wh Eqn. (37), we reurn o he proof of he Theorem 3. Proposon 6. ENTO s rae-sable. Proof. We generalze he argumen by Gamarnk [32] o prove he proposon. We remnd he reader ha we are analyzng he me-evoluon of a fxed sample pon! 2, whch sasfes Eqn. (37). Le R e (0) denoe he oal number of packes wang o cross he edge e a me =0. Also, le R k () denoe he oal number of packes a me, whch are exacly k hops away from her respecve sources. Such packes wll be called layer k packes n he sequel. If a packe s duplcaed along s assgned roue T (whch s, n general, a ree), each copy of he packe s couned separaely n he varable R k (),.e., R k () = T 2T R (e T k,t )(), (38) where he varable R (e,t ) () denoes he number of packes followng he roung ree T, ha are wang o cross he edge e 2 T a me. The edge e T k s an edge locaed kh hop away from he source n he ree T. If here are more han one such edge (because he ree T has more han one branch), we nclude all hese edges n he summaon (38). We show by nducon ha R k () s almos surely bounded by a funcon, whch s o(). Base Sep k =0: Fx an edge e and me. Le 0 apple be he larges me a whch no packes of layer 0 (packes whch have no crossed any edge ye) were wang o cross e. If no such me exss, se 0 =0. Hence, he oal number of layer 0 packes wang o cross he edge e a me 0 s a mos Q e (0). Durng he me nerval [ 0,], as a consequence of he UMW conrol polcy (37), a mos S e ( 0,)+M() exernal packes have been admed o he nework, ha wan o cross he edge e n fuure. Also, by he choce of he me 0, he edge e was always havng packes o ransm durng he enre me nerval [ 0,]. Snce ENTO schedulng polcy s followed, layer 0 packes have prory over all oher packes. Hence, follows ha he oal number of packes a he edge e a me sasfes R (e,t ) () apple R e (0) + S e ( 0,)+M() S e ( 0,) T :e2e T 0 apple R e (0) + M() (39) As a resul, we have R 0 () apple P e R e(0) + E M(), for all. Le B 0 () def = P e R e(0) + E M(). Snce M() =o(), we have B 0 () =o(). Noe ha, snce M() s monooncally non-decreasng by defnon, so s B 0 (). Inducon Sep: Suppose ha, for some monooncally nondecreasng funcons B j () = o(),j = 0,, 2,...,k, we have R j () apple B j (), for all me. We nex show ha R k () apple B k () for all, where B k () =o(). Agan, fx an edge e and an arbrary me. Le 0 apple denoe he larges me before, such ha here were no layer k packes wang o cross he edge e. Se 0 =0f no such me exss. Hence he edge e was always havng packes o ransm durng he me nerval [ 0,] (packes n layer