606 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 27, NO. 5, JUNE Abdallah Khreishah, Chih-Chun Wang, and Ness B.

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1 66 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL 7, NO 5, JUNE 9 Cross-Layer Optmzaton for Wreess Muthop Networks wth Parwse Intersesson Network Codng Abdaah Khreshah, Chh-Chun Wang, and Ness B Shroff Abstract For wreess mut-hop networks wth uncast sessons, most codng opportuntes nvove ony two or three sessons as codng across many sessons requres greater transmsson power to broadcast the coded symbo to many recevers, whch enhances nterference Ths work shows that wth a new fowbased characterzaton of parwse ntersesson network codng (codng across two uncast sessons), an optma jont codng, schedung, and rate-contro scheme can be devsed and mpemented usng ony the bnary XOR operaton The new schedung/rate-contro scheme demonstrates provaby gracefu throughput degradaton wth mperfect schedung, whch factates the desgn tradeoff between the throughput optmaty and computatona compexty of dfferent schedung schemes Our resuts show that parwse ntersesson network codng mproves the throughput of non-codng soutons regardess of whether perfect/mperfect schedung s used Both the determnstc and stochastc packet arrvas and departures are consdered Ths work shows a strkng resembance between parwse ntersesson network codng and non-coded soutons, and thus advocates extensons of non-codng wsdoms to ther network codng counterpart Index Terms Network codng, parwse ntersesson network codng, mperfect schedung, cross-ayer optmzaton I INTRODUCTION THE INTERFERENCE-HEAVY nature of wreess meda presents a great chaenge for desgnng hgh-throughput wreess mut-hop networks Recenty, many technques have been deveoped for enhancng the throughput of wreess mut-hop networks, among whch at east two technques demonstrate promsng throughput mprovements The frst method s but around the exstng routng (non-networkcodng) concept and focuses on cross-ayer desgn that consders the correspondng network utty maxmzaton probem (see [8], [9] and the reference theren) The second method s network codng, whch aows ntermedate nodes to perform codng operatons n addton to pure packet forwardng Advantages of network codng are shown both theoretcay Manuscrpt receved August 8; revsed 9 February 9 Ths work was supported n part by the ARO MURI award W9NF-8--38, AFOSR award FA , and by the NSF CAREER Award CCF Ths paper was presented n part at the 4nd Conference on Informaton Scences and Systems, Prnceton, New Jersey, USA, March 9-, 8 A Khreshah and C-C Wang are wth Center for Wreess Systems and Appcatons (CWSA), Schoo of Eectrca and Computer Engneerng, Purdue Unversty (e-mas: {akhresh,chhw}@purdueedu) NB Shroff s wth the Departments of Eectrca and Computer Engneerng and Computer Scence and Engneerng, The Oho State Unversty (e-ma: shroff@eceosuedu) Dgta Object Identfer 9/JSAC /9/$5 c 9 IEEE [], [5], [], [5] and emprcay [5], [7], [7], [33], [36], especay when the broadcast nature of wreess meda s propery expoted Network codng can be further cassfed nto two dfferent sub-categores: ntrasesson and ntersesson network codng, the former of whch focuses on a snge mutcast sesson and codng s performed on packets from the same sesson The atter consders mutpe coexstng sessons and codng s performed on packets across dfferent sessons Intrasesson network codng s we-understood as the achevabe mutcast rate s characterzed by the mn-cut maxfow theorem [] Thsfow-based characterzaton eads to a natura extenson of the cross-ayer optmzaton framework to ntrasesson network codng, ncudng throughput, utty, and energy optmzaton as n [8], [3], [35], [4], [43] Unfortunatey, for the most frequent scenaro n whch ony uncast sessons are present, the throughput benefts of ntrasesson network codng vansh Intersesson network codng provdes performance mprovement even when ony uncast sessons are present Athough ts throughput beneft s ceary demonstrated n the butterfy structure [7], [5], the much needed compete characterzaton of ntersesson network codng s ess understood and some theoretc studes can be found n [], [8], [4], [6], [3] Most research hence focuses on captazng two partcuar network substructures that admt the ntersesson codng benefts: the butterfy structure of any szes [38] and the wreess one-hop codng opportunty (aso known as the wreess cross topoogy) [7] For the butterfy-based approaches, the achevabe rate regon was studed n [34], [38] and the assocated back-pressure agorthm was studed n [], [4] The one-hop codng opportuntes were frst studed and expoted by the COPE protoco n [7] The smpe one-hop nature and the emprca success of COPE has snce motvated numerous subsequent works Some exampes ncude the centrazed computaton of the achevabe rates wth schedung [36], the energy effcent schedung wth opportunstc codng [9], the power and throughput tradeoff between mutcastng and uncastng [6], and the maxmum number of overhearng opportuntes under practca wreess settngs [3] By takng advantage of both the oca butterfy structure and the one-hop codng opportuntes, a hybrd, practca scheme has been proposed to further mprove the throughput of wreess mut-hop networks [33] In COPE [7], codng can be performed among sessons especay when opportunstc stenng and codng Authorzed censed use mted to: Purdue Unversty Downoaded on June, 9 at : from IEEE Xpore Restrctons appy

2 KHREISHAH et a: CROSS-LAYER OPTIMIZATION FOR WIRELESS MULTIHOP NETWORKS WITH PAIRWISE INTERSESSION NETWORK CODING 67 s used In ts emprca study usng 8 [7], 5% of the codng operatons are performed over ony two symbos In a smar but energy-aware settng wth schedung [9], ess than % codng operatons s used to combne 3 symbos The ntuton behnd s that codng more symbos together requres greater transmsson power to broadcast the coded symbo to more recevers, whch enhances nterference and affects negatvey the throughput of other traffc As a resut, n ths work we consder parwse ntersesson network codng (PINC) that aows network codng ony between pars of coexstng sessons In [39], [4], [4], a new necessary and suffcent condton s estabshed for PINC Wth a fow-based form smar to the mn-cut max-fow theorem for mut-path routng and for ntrasesson network codng, the new characterzaton of PINC prompts tghter ntegraton of cross-ayer optmzaton and network codng The man contrbuton of ths work s twofod () Based the characterzaton of PINC, a new dstrbuted, optma, jont codng, schedung, and rate-contro scheme s devsed, whch uses ony bnary XOR operatons, admts fuy decouped rate-contro/schedung, and acheves the optma rates by dentfyng a parwse codng opportuntes that ncude the one-hop opportunty and the butterfy structure as speca cases () Optma schedung s computatonay expensve to acheve even n a pure non-codng paradgm, et aone wth network codng The PINC-based scheme demonstrates provaby gracefu throughput degradaton for mperfect schedung [8] that factates the desgn tradeoff between the throughput optmaty and computatona compexty of dfferent schedung schemes Our resuts show that PINC mproves the throughput of routng-based soutons regardess of whether perfect/mperfect schedung s used The strkng resembance between PINC and non-codng communcatons thus advocates extensons of non-codng wsdoms to ther network codng counterpart The characterzaton probem of parwse ntersesson network codng s studed n [39], [4], [4] for acycc ([4], [4]) and cycc networks ([39]) A new necessary and suffcent condton has been estabshed, and extensve dscussons have been made from codng, nformaton-theoretc, and graph-theoretc (topoogca) perspectves n [39] The new theoretc fndng of [4], [4] s apped to the ratecontro probem for wrene networks and a decoubed optma rate-contro agorthm s obtaned n [9], [] for statc packet arrvas In ths paper we take further advantage of the PINC characterzaton resuts and consder the correspondng cross-ayer schedung and rate-contro probem for mut-hop wreess networks A genera wreess settng s used n ths work, ncudng the statc and dynamc arrvas of the packets We aso quantfy the mpact of mperfect schedung on parwse ntersesson network coded wreess mut-hop traffc for the frst tme Ths work further bounds the compexty of the commony-used wreess-to-wrene converson technque that modes wreess broadcast channe (frst proposed for ntrasesson network codng n [3], [43]) n the context of parwse ntersesson network codng for mut-hop networks An XOR-based codng scheme that s more sutabe for wreess networks than the one n [9], [] s aso deveoped n ths paper Intersesson network codng for wreess networks s aso consdered n [9] by convertng the probem nto fndng ntermedate nodes such that ntrasesson mutcast s performed wth these nodes beng the sources and snks of the mucast sessons Compared to the approach n ths work, [9] does not consder a ntersesson network codng opportuntes but on the other hand permts reencodng the decoded packets at the ntermedate nodes whch s not consdered n ths paper To obtan a ow compexty agorthm, [9] further mts the encodng and decodng nodes to be one hop away and assumes the node excusve mode The resutng one-hop agorthm n [9] s a back-pressure agorthm, whch generay ncurs more deay and takes more tme to converge than the pathbased approach used n ths work as observed n [], [37] For comparson, our path based approach fasctates anayzng the mpact of mperfect schedung for both the determnstc and stochastc arrvas, whch s not consdered n [9] On the other hand back-pressure technques aow better modeng of ossy nks as n [9] The remander of ths paper s organzed as foows Secton II ntroduces the mode of wreess networks and the fow-based characterzaton for PINC Concrete exampes are provded to ustrate the benefts of PINC For streamnng the dscusson, a new dstrbuted PINC code desgn s reegated to Secton V, whch s based on bnary XOR operatons An optma jont schedung/rate-contro scheme s provded n Secton III, whch admts decouped mpementatons Secton IV studes the mpact of mperfect schedung on the proposed PINC souton for both determnstc and stochastc modes of packet arrvas and departures The throughput mprovement of PINC and ts performance under mperfect schedung are verfed by smuaton n Secton VI Secton VII concudes the paper II PRELIMINARY RESULTS A Parwse Intersesson Network Codng (PINC) For Wrene Networks Consder drected cycc/acycc wrene network G = (V,E), n whch each edge s abe to carry one GF(q) symbo per unt tme (say a second) and the propagaton deay s aso one second Hgh-rate nks are modeed by parae edges and ong-deay nks are modeed by ong paths wth added auxary ntermedate nodes A par of coexstng uncast sessons (s,d ) and (s,d ) woud ke to transmt two strngs of ndependenty dstrbuted GF(q) symbos X,,X T and Y,,Y T (one strng for each sesson) smutaneousy over a gven duraton of T seconds Parwse ntersesson network codng (PINC) s aowed and packets of these two strngs {X t,y t : t =,,T} can be arbtrary mxed n a near or non-near fashon We say a PINC souton exsts for transmttng two rate- strngs of packets (over the gven unt-edge-capacty network), f gven any ɛ>, there exsts a suffcenty arge T such that T I([X]T ;[M d ] T ) > ( ɛ) og(q) and T I([Y ]T ;[M d ] T ) > ( ɛ) og(q), Authorzed censed use mted to: Purdue Unversty Downoaded on June, 9 at : from IEEE Xpore Restrctons appy

3 68 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL 7, NO 5, JUNE 9 s Y t Y t v X t X t + Y t X t v X t + Y t d Fg An acycc butterfy network that satsfes condton of Theorem and the correspondng codng operatons Δ where [X] T = {X,,X T }, [Y ] T = {Y,,Y T }, I( ; ) s the mutua nformaton and [M d ] T s the symbos receved by destnaton d for =, In [39], a graph-theoretc characterzaton for the exstence of a PINC souton has been estabshed for drected cycc networks (Dscusson for acycc networks can be found n [4], [4]) For the foowng, we use P u,v to represent a path connectng nodes u and v Theorem : A PINC souton exsts f and ony f one of the foowng two condtons hods Condton : There exst two edge-dsjont paths P s,d and P s,d Condton : There exst sx paths grouped nto two sets P = {P s,d,p s,d,p s,d } and Q = {Q s,d,q s,d,q s,d } such that for a e E, {e Ps,d } + {e Ps,d } + {e Ps,d } and {e Qs,d } + {e Qs,d } + {e Qs,d }, where { } s the ndcator functon If condton of the theorem s satsfed, a non-codng souton s suffcent for the probem On the other hand, f ony condton s satsfed, network codng s necessary to acheve smutaneous rate- transmsson For exampe, the acycc network n Fg satsfes condton by choosng P s,d = s v v d, P s,d = s d, P s,d = s v v d, Q s,d = s v v d, Q s,d = s d,andq s,d = s v v d A unt rate can be supported between (s,d )and(s,d ), f the codng scheme represented n the fgure s used Fg contans a cycc network that satsfes condton of the theorem by choosng P s,d = s v 7 v 6 v 5 v v 3 v 4 d, P s,d = s v v 4 d, P s,d = s v v v 3 v 6 v 5 v 8 d, Q s,d = s v 7 v 6 v 5 v v 3 v 4 d, Q s,d = s v 7 v 8 d,andq s,d = s v v v 3 v 6 v 5 v 8 d The correspondng coded symbos carred by each edge are aso ustrated n Fg, and rate- s sustanabe for both uncast sessons (s,d )and(s,d ) Note that the (ɛ, T ) s essenta to take nto account the deay of each edge as seen n Fgs and Theorem shows that as the exstence of noncodng soutons s equvaent to fndng edge-dsjont paths, the exstence of PINC soutons s equvaent to fndng paths wth controed edge overap The ntuton s that when the packets/paths are not overy usng any botteneck edge (those edges used by three paths), codng enabes the nformaton to be transmtted smutaneousy for both sessons The subgraph G nduced by any sx paths satsfyng condton of s d Δ s d X t X t Y t 6 v M v Y t 7 v v X t 3 v 6 M M 4 M 3 v M 5 v 8 v 4 M 4 M = X t 4 + Y t M = X t 5 + Y t 3 Y t Y t X t 6 s d M 3 = X t + Y t 4 M 4 = X t 3 + Y t 5 Fg A cycc butterfy network that satsfes condton of Theorem and the correspondng codng operatons Theorem w be referred as a parwse ntersesson codng confguraton (PICC) In a broad sense, a path s the smaest graph unt for non-codng mutpe sesson communcatons whe a PICC s the smaest graph unt when codng across two sessons s permtted Ths work w bud an optma decouped codng/schedung/rate-contro scheme based on ths new PICC unt and study ts performance degradaton when mperfect schedung s used B Anaytca Framework for Wreess Mut-hop Networks A Wreess to Wrene Converson An mportant feature of wreess mut-hop network s the broadcast nature of wreess meda, whch s termed the wreess mutcast advantage (WMA) The WMA can be modeed as foows (see [3], [43] for detas) For each node u wth k neghbors {v,,v k }, ntroduce k auxary nodes such that each auxary node corresponds to a non-empty eement of the powerset of {v,,v k }Add k drected edges connectng u and each of the auxary nodes For each auxary node, add drected edges from the auxary node to each node n the correspondng subset of neghbors Fg 3 ustrates one such converson for a node wth three neghbors In a wreess network, every tme a packet s about to be sent, the sender u chooses the target recever(s) of the packet, whch s equvaent to choosng the correspondng auxary node/nk for transmsson Therefore, desgnng a wreess transmsson scheme that expots the advantages of the WMA s equvaent to desgnng a good routng/schedung agorthm on ts wrene counterpart wth the addtona node-excusve schedung constrants that auxary nodes correspondng to the same u cannot be actve smutaneousy Ths framework takes nto account the WMA and maps the wreess schedung probem to a wrene schedung probem whe the underyng nterference mode for the former s absorbed as schedung constrants Θ for the atter probem, whch w be cear n the ater secton C Parwse Intersesson Network Codng for Wreess Networks Based on a necessary and suffcent condton, the sxpath-based PICC captures a parwse codng opportuntes once the aforementoned wreess to wrene converson s propery expoted, whch ncude the wdey studed butterfy structure [], [4], [33], [34], [38] and the one-hop codng Authorzed censed use mted to: Purdue Unversty Downoaded on June, 9 at : from IEEE Xpore Restrctons appy

4 KHREISHAH et a: CROSS-LAYER OPTIMIZATION FOR WIRELESS MULTIHOP NETWORKS WITH PAIRWISE INTERSESSION NETWORK CODING 69 u v v v 3 (a) Wreess broadcast advantage Fg 3 Modeng the wreess mutcast advantage A X + Y X B 3 Y X + Y C (a) A B C u v v v 3 (b) The correspondng wrene counterpart (b) (c) (d) Fg 4 A smpe wreess one-hop codng opportunty wth two sessons: Sesson : A X C; Sesson : C Y A (a) The sot-by-sot wreess transmsson (b) Wreess to wrene converson (c) Paths P s,d, P s,d, and P s,d (d)pathsq s,d, Q s,d,andq s,d opportuntes [9], [6], [7], [3], [36] as speca cases when codng s permtted ony between two sessons Fg 4(a) s a cassc exampe of the one-hop ntersesson codng opportunty for wreess networks Node A woud ke to send symbo X to node C whe C ntends to send Y to A Fg 4(a) depcts how to send two symbos n three tme sots The and n the sma boxes ndcate that A sends X to B n the frst tme sot whe C sends Y to B n the second tme sot In the thrd tme sot, B broadcasts coded symbo X + Y to A and C usng the WMA If we foow the wreess to wrene converson, Fg 4(a) s transformed to Fg 4(b) By notcng the exstence of the P and Q paths wth controed edge-overap as n Fgs 4(c) and 4(d), ths one-hop codng opportunty for codng across two sessons s captured by Theorem and corresponds to an nstance of PICC Note that Fg 4(b) aso ndcates that under a node excusve mode at east three tme sots are necessary as the three nvoved auxary nodes cannot be schedued smutaneousy Smary Fg 5(a) descrbes the cassc wreess crossfows n whch symbos X and Y can be sent from A to E and from B to D n three tme sots D and E use the overheard packets X and Y for decodng Fgs 5(b) to 5(d) depct the correspondng wreess to wrene converson and show that the wreess cross fows can agan be captured as a speca nstance of PICC (whch s actuay a butterfy n the correspondng wrene network) Snce the path-based characterzaton of PINC does not requre that encodng and decodng happen at nodes that are -hop apart from each other, our formuaton naturay takes nto account codng opportuntes over subgraphs of dfferent szes, eg -hop butterfes used n [33] Capturng a parwse codng opportuntes, the PICCs aso prompt new wreess ntersesson codng opportuntes dfferent from Fgs 4 and 5 For exampe, n Fg 6(a), a new type of wreess cross fowssdentfed, for whch A sends symbo X to F whe C sends symbo Y to D Ths exampe s not captured by the tradtona one-hop codng opportunty as node D does not overhear the orgna B A Y X Y X C E X + Y 3 (a) D B E C (b) A D s s d d (c) s s d d Fg 5 The wreess cross fows wth two sessons: Sesson : A X E; Sesson : B Y D (a) The sot-by-sot wreess transmsson (b) Wreess to wrene converson (c) Paths P s,d, P s,d, and P s,d (d) Paths Q s,d, Q s,d,andq s,d A X B X + Y Y D 3 C X Y 4 E X F (a) D A B E F (b) C Fg 6 A new type of wreess cross fows wth two sessons: Sesson : A X F ; Sesson : C Y D (a) The sot-by-sot wreess transmsson (b) Wreess to wrene converson (c) Paths P s,d, P s,d,andp s,d (d) Paths Q s,d, Q s,d,andq s,d symbo X sent by A but overhears the reconstructed symbo X decoded and sent by E Fgs 6(c) and 6(d) ustrate the correspondng P and Q paths, whch verfy that ths new type of wreess cross fows s a speca nstance of PICC (that s dfferent than the cassc butterfy structure) By ncudng the exstng codng opportuntes as speca cases and capturng addtona ones, our PICC-based souton w enhance further the achevabe capacty regon of ntersesson network codng III OPTIMAL JOINT SCHEDULING/RATE-CONTROL WITH PINC Foowng Secton II-B, we mode a wreess network by ts wrene counterpart denoted by G =(V,E) where V s the set of network nodes pus auxary nodes and E s the edge set Consder sotted transmsson, a schedung pocy Θ s a coecton of actve edges and the assocated power eves Under a gven nterference mode, we use re Θ to denote the rate that can be supported on edge e under the schedung pocy Θ, and we often use r Θ for the coectve rate vector Let Θ denote the coecton of a poces and et R = Δ {r Θ : Θ Θ} denote the correspondng rates Any rate vector r Co(R), the convex hu of R, can be acheved va tme sharng Wthout oss of generaty, we assume the rate regon s bounded There are N coexstng uncast sessons usng the network to send data from source s to destnaton d where =,,N The utty functon U (x) for each sesson s strcty concave and monotoncay ncreasng, where x s the end-to-end data rate for the sesson The utty optmzaton for mutpe uncast sessons usng PINC can be cast as foows d s d (c) s d (d) s d (d) s Authorzed censed use mted to: Purdue Unversty Downoaded on June, 9 at : from IEEE Xpore Restrctons appy

5 6 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL 7, NO 5, JUNE 9 N max U x M X,r Co(R) = subject to P x k + k= P N H k (e)x k = k= + (,j): j PICC j = j:j PICC j = H j (e)x j x j () r e, e E () x j = x j, (, j) :<j, (3) where P s the coecton of paths from s to d aong whch packets w be routed wthout any codng operatons and x k s the rate assgned for the k-th path PICC j s the coecton of PICCs between sessons and j on whch ntersesson network codng w be performed and x j s the packet rate of source s that w be network coded usng the -th PICC of PICC j Wthout oss of generaty, we further assume the ndces of PICC j and of PICC j are consstent Namey, for a, the -th PICC of PICC j s aso the -th PICC of PICC j Snce n PINC, packets from s and s j are coded bjectvey wth each other, the system requres the equa-rate constrant (3) Wthout oss of generaty, we aso assume that the rate vector x s bounded by a fntey arge constant M X In (), H k (e) s the ndcator functon whether the k-th path n P uses edge e Hj (e) s the ndcator functon whether the -th PICC n PICC j uses edge e PINC ensures that the two packet fows (wth rates x j and x j respectvey) jonty use ony the max rate max(hj (e)x j,h j (e)x j ) nstead of the sum rate Hj (e)x j + H j (e)x j By the fact that the ndcator functon s symmetrc by defnton, e Hj (e) = H j (e), and by the equa-rate constrant n (3), the rate consumpton becomes max(hj (e)x j,h j (e)x j ) = H j (e)x j +H j (e)x j For each edge e, summng over rates consumed by mutpath routng and by mut-picc network codng eads to the capacty constrant () The non-negatve rate vector x, ncudng a x k and x j, s the subject of rate contro and the edge rate vector r Co(R) s the subject of optma schedung and tme-sharng One advantage of consderng PICC s that unke the exstng butterfy-search approach [], [4], the characterzaton of PICC s path-based rather than structure-based One can thus use any path search agorthm to dentfy possbe consttuent paths and any sx paths P s,d,p s,d,,q s,d can serve as a PICC It s worth pontng out that there s no need to strcty enforce Condton of Theorem durng mpementaton More expcty, snce each edge e knows whether tsef partcpates n a gven path durng the path-search phase, e aso knows ts edgeoverap n the gven sx paths Consder an edge e that s a botteneck, e {e Ps,d } + {e Ps,d } + {e Ps,d } =3 (4) or {e Qs,d } + {e Qs,d } + {e Qs,d } =3 We can st treat the gven sx paths as a PICC n our optmzaton probem () to (3) even though they do not satsfy Condton of Theorem To that end, we smpy need to generaze the ndcator functon Hj (e) and et H j (e) = (rather than ) for any botteneck edge In ths way, we aocate doube the capacty for such e (see ()), whch thus resoves the correspondng botteneck caused by (4) Wth the use of a generazed ndcator Hj (e), searchng for PICCs s equvaent to searchng for paths pus combnng sx paths as a group, whch can be acheved by any path-search agorthms Note that the arger the path coecton P and the PICC coecton PICC j, the hgher achevabe throughput w be Dependng on the avaabe resources, there thus exsts a compexty-performance tradeoff on how exhaustve the pathsearch agorthm shoud be The optma souton of ( 3) can be acheved n a decouped way by sovng ts dua probem va the sub-gradent method Agorthm A: Update For each s, update ts rate vector x [t] = {x k [t],x j [t] : k, j, } for the t-th tme sot by x [t] = arg max x M X U P() x k + k= j:j PICC j = x j P q e [t] H k (e)xk + PICC j Hj (e)x j k= j:j = PICC j qj [t]x j qj [t]x j = P α k= j:j> ( x k y k ) + j:j j:j< PICC j = ( x j yj ), where q e [t] and qj [t] are dua varabes at the t-th tme sot, whose vaues are feedback to s The α are sma constants and y = {y k,y j : j, k, } are auxary varabes of the proxma method n order to emnate oscaton [4] Perodcay, y s set to x [t] and the teraton contnues usng the new y Schedung Update The network seects the optma schedung pocy for the t-th tme sot by r[t] = arg max q e [t]r e (5) r R Queue-ength Update Each nk e updates ts dua varabe q e [t +]accordng to the foowng equaton ( q e [t +]= [q N P e [t]+β e H k (e)x k [t] + (,j): j = k= PICC j = Hj (e)x j [t] +, r e [t])] where [ ] + =max(, Δ ) s the projecton operator and β e s a sma step sze for the sub-gradent method Baance Update Each destnaton d updates the dua varabe qj [t +] for a j> The dua varabe q j accounts Authorzed censed use mted to: Purdue Unversty Downoaded on June, 9 at : from IEEE Xpore Restrctons appy

6 KHREISHAH et a: CROSS-LAYER OPTIMIZATION FOR WIRELESS MULTIHOP NETWORKS WITH PAIRWISE INTERSESSION NETWORK CODING 6 the dfference between packet rates of sources and j that use the same PICC qj[t +]=qj[t]+β ( x j [t] x j[t] ), j : j>,, (6) where β s a sma step sze for the sub-gradent method Proxma Update Perodcay, after every K tme sots, set y x [t] For notatona smpcty, after the proxma update, we reset the tmer vaue t The fve dfferent parts of Agorthm A are couped mpcty va the queue engths q e and the baance nformaton qj at the destnatons One mportant observaton s that wth PINC, ony the rate and the baance updates, performed at the sources s and destnatons d, dffer from ts non-codng counterpart (cf [7]) The schedung and queue-ength updates reman dentca The mpact of PINC on rate-contro and schedung s thus mnma and confned ony n sources and destnatons The compexty of Agorthm A depends many on the number of feedback messages qj [t] that each source receves at each tme sot t, whch s proportona to the number of PICCs n the network Therefore, the number of queueength exchange messages s of the order O(N E (max j: j PICC j )) Ths s of smar compexty to that of tradtona mutpath routng wth schedung and congeston contro [7] typcay proposed for statc mesh networks wthout mobty Dstrbuted methods that reduce the number of need-to-be-consdered PICCs can be found n Secton V and n [9], whch mtgate the compexty of ths agorthm and s executed ony once n the ntazaton phase Another approach for compexty reducton s to ncude the paths that form PICCs one by one n an adaptve way such that the number of contro messages do not exceed a threshod as expaned n [9] For comparson, the number ( of mutcast sessons used n the framework of [9] s N ) V ( V ), and the number of queue-ength exchange messages n the correspondng back-pressure agorthm s thus O(N V 3 nbs), wherenbs s the average number of neghborng nodes for nodes n V A Convergence Anayss of Agorthm A Proposton : Consder a decreasng non-negatve sequence {β τ } such that τ = β τ and τ = (β τ ) < If n the begnnng of each proxma teraton, we reset the step szes β e = β = β τ wth τ = As the nner teraton proceeds, we use β τ, τ =,,K as the step szes n the K nner teratons Then when the update perod K of the proxma varabe y x [t] s suffcenty arge, Agorthm A converges to the optma souton of ( 3), the optma rate assgnment of PINC A sketch of the proof s as foows The boundedness of the rate regon Co(R) and rate-vector x mpes the boundedness of the sub-gradent of the dua probem of ( 3) Proposton 86 n [3] then guarantees the convergence A detaed proof s reegated to [] The convergence wth K bounded away from nfnty and β τ bounded away from zero s emprcay verfed durng our smuatons B Stabty of Agorthm A Defnton : A system oad {w : =,,N} (we sometmes use {w } as shorthand) can be stabzed by Agorthm A f there exsts a non-negatve vector w = {w k,w j :, j, k, } such that P w = w k + k= j:j PICC j = w j,, and wj = w j, (, j) :<j, (7) Moreover, f we repace the rate update n Agorthm A by a fxed rate assgnment x[t] =w, then the dua varabes q e [t] and qj [t] must stay bounded away from nfnty when t tends to nfnty Let Λ denote a set of system oads Λ={{w } } such that for any {w } Λ, there exsts a rate vector r Co(R), a non-negatve vector w = {w k,w j :, j, k, } satsfyng (7), and jonty w and r satsfy P N H k (e)wk + = k= (,j): j PICC j = H j (e)w j r e, e E We then have the foowng stabty resut regardng the system oad regon Λ Proposton : Any system oad {w } that s n the nteror of Λ can be stabzed by the optma rate-contro/schedung n Agorthm A (Proposton can be regarded as a coroary of Proposton 3 that w be ntroduced shorty after) IV PAIRWISE INTERSESSION NETWORK CODING WITH IMPERFECT SCHEDULING In genera, t s computatonay expensve to fnd the optma schedung decson satsfyng (5) n Agorthm A Dependng on dfferent nterference modes, fndng the optma schedung r that maxmzes e q e[t]r e s NP-hard n many cases and generay requres centrazed mpementaton In practce, we woud often have to resort to mperfect schedung schemes that seect the rate vector r[t] that acheves γ fracton of the maxmum vaue Namey, an mperfect schedung pocy choose r[t] satsfyng q e [t]r e [t] γ max r q e [t]r e, (8) where γ s a constant n [, ] Wth mperfect schedung (γ <), the te between Agorthm A and the gradent method for the dua probem s severed and Agorthm A may not converge to any fxed-pont souton The foowng resuts show that even wth mperfect schedung, the proposed PINC scheme wth cross-ayer optmzaton Agorthm A st shows tractabe performance n terms of the stabty regon Proposton 3: Any system oad {w } that s n the nteror of γλ can be stabzed by Agorthm A wth γ-mperfect schedung The sketch of the proof s provded n Appendx A, whch covers Proposton as a speca case A Networks wth Dynamc Arrvas and Departures In addton to networks wth statc arrvas and departures, we aso consder the case of dynamc system oads wth ogarthmc utty functons Consder N casses of users Authorzed censed use mted to: Purdue Unversty Downoaded on June, 9 at : from IEEE Xpore Restrctons appy

7 6 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL 7, NO 5, JUNE 9 For a, users n cass have a common ogarthmc utty functon U (x) = κ og(x) where κ > are predefned system parameters A users n cass w send packets from s to d and w use the same routng paths n P and the same PICCs n PICC j for transmsson We aso assume that users of cass arrve accordng to a Posson process wth rate λ and each user needs to send a fe whose sze s exponentay dstrbuted wth mean μ The system oad of ths network {( ) } wth dynamc arrvas s then defned as λ μ : The dynamc nature of ths settng prompts a sghty dfferent defnton of stabty {( ) } Defnton : A system oad λ μ : can be stabzed by Agorthm A f the dua varabes q e [t] and qj [t] are bounded away from nfnty for each teraton wth probabty one We then have the foowng stabty resut Proposton 4 (Stabty for Dynamc Systems): Consder ogarthmc utty functons U (x) = κ og(x) {( ) Wth } suffcenty sma α, β e,andβ, any system oad λ μ : that s n the nteror of γλ can be stabzed by Agorthm A wth γ-mperfect schedung The proof of Proposton 4 s sketched n Appendx B Proposton 4 mpes that athough the nstantaneous system oad mposed on the network may we exceed the network capacty, as ong as the average system oad s wthn γ tmes the capacty, the queue engths of the network are bounded away from nfnty and the system s stabe Proposton 4 shows that the gracefu stabty degradaton that was prevousy known ony for non-codng transmsson (cf [8]) aso hods for PINC Shftng from non-codng to network-codng soutons enhances the throughput wthout sacrfcng the assocated stabty even wth mperfect schedung B A Generazed Node Excusve Interference Mode Wth the WMA The node excusve mode s a commony used nterference mode for buetooth or for FH-CDMA networks [], [3], [3] that admts effcent and provaby good approxmaton of the optma schedung pocy In the tradtona node excusve mode (wthout takng advantages of the WMA), the data rate of each nk s fxed at c e and each node can ony send to or receve from one other node at any tme The objectve functon of optma schedung s thus equvaent to max r =max M q e [t]r e =max q e [t]c e {e M} M q e [t]c e, (9) e M where M s a matchng of the underyng graph G Fndng the optma schedung of (9) thus becomes a maxmum weghted matchng probem In contrast wth Defnton where the rate update rue s modfed for a statc system oad, for a dynamc system oad, the optma rate update s kept unchanged Ony the schedung update w be changed to ncorporate mperfect schedung as n (8) v 4 u u u 3 v 5 v 6 (a) Consder the WMA of a wreess network u u u u v 4 u 3 v 5 v 6 (b) Three actve auxary nodes correspondng to three uncast transmssons v 4 u 3 v 5 v 6 (c) Two actve auxary nodes correspondng to two broadcast transmsson Fg 7 Iustraton of the node excusve mode when the WMA s taken nto consderaton Nonetheess, when the WMA s taken nto account, e wth the auxary nodes added for the broadcast nature of wreess transmsson as dscussed n Secton II-B, the objectve functon of schedung becomes max r q e [t]r e =max A q e [t]c e {e s adjacent to some node n A}, () where A s a set of actve auxary nodes Snce each network node can ony send to or receve from one auxary node (due to the node-excusveness assumpton), we requre that the node set A satsfes that any node n A does not share any common neghbor wth any other node n A Fg 7(a) depcts a wreess network of sx nodes Nodes u, u,andu 3 woud ke to transmt and the transmsson can be overheard by more than one recevers (see Fg 7(a)) Two possbe schedung poces (two dfferent As) are ustrated In Fg 7(b), the wrene counterpart of Fg 7(a), three auxary nodes are actve and correspond to three uncast transmssons ((u,v 4 ), (u,v 6 ), and (u 3,v 5 )) hghghted by thck edges The other schedung pocy contans two actve auxary nodes as n Fg 7(c) that correspond to two broadcast transmssons ((u, {v 4,u 3 }) and (u, {v 5,v 6 })) It can be shown that maxmzng () s equvaent to sovng a maxmum weghted hypergraph matchng (MWHM) probem In [8] a greedy maxma hypergraph matchng (GMHM) s proposed as an approxmaton of the MWHM More expcty, the network frst seects an auxary node v a that maxmzes e s adjacent to v a q e [t]c e and ncudes v a as part of the schedung pocy A Remove v a and ts neghbors and then restart ths greedy seecton of auxary node unt a maxma schedung pocy A s reached In ths way, -approxmaton of the MWHM, where nbs(v a ) s the set of neghbors around v a We can further sharpen the approxmaton rato as foows GMHM guarantees to fnd a max va nbs(v a) Proposton 5: For any gven network, the GMHM s a 5 - approxmaton agorthm and can thus acheve at east 5 of Authorzed censed use mted to: Purdue Unversty Downoaded on June, 9 at : from IEEE Xpore Restrctons appy

8 KHREISHAH et a: CROSS-LAYER OPTIMIZATION FOR WIRELESS MULTIHOP NETWORKS WITH PAIRWISE INTERSESSION NETWORK CODING 63 the stabty regon Λ when used as an mperfect schedung pocy for the node-excusve mode Proof: Snce each PICC conssts of sx paths, any auxary node partcpatng n a PICC has at most sx outgong branches pus one ncomng edge As a resut, durng the wreess to wrene converson, there s no need to ncude auxary nodes of > (6 + ) neghbors as the neghbors of those nodes w not fuy partcpate n a PICC The above reasonng shows that the GMHM can be made a 7 - approxmaton by emnatng the auxary nodes wth > 7 neghbors By further takng nto account the edge-overap condtons n Theorem, t can be shown that each auxary node needs to have at most four outgong branches (two for the P paths and two for the Q paths) Therefore, the approxmaton rato can be mproved to 5 by emnatng the auxary nodes wth > (4 + ) neghbors The proof s compete It s worth mentonng that the 5-approxmaton s a ower bound of the performance of the GMHM In our numerca study, GMHM has amost dentca performance to the optma MWHM souton Other studes on the performance of greedy maxma matchng for non-codng networks can be found n [6], [44] V DISTRIBUTED CODE DESIGN FOR PINC The rate contro and nk schedung agorthms descrbed thus far aocate optma rates at each nk so that the utty functon can be maxmzed The next queston s what s the network codng scheme that can acheve the optma rate assgnment? For a gven PICC, we proposed a codng scheme based on dentfyng dstrbutedy speca edges n the PICC n [] Specfcay, carefuy chosen decodng operatons are performed on speca decodng edges, whe random network codng s performed on a other edges for the sake of scaabty and dstrbutveness In ths paper we present a new, more effcent approach n whch each edge e decdes the codng operaton based on the subset of the sx paths of the gven PICC that use e Snce each edge naturay knows whether tsef partcpates n a gven route/path or not (a byproduct of the nta path-search phase of Agorthm A), the correspondng codng operaton can be decded ocay wthout knowng the entre topoogy of the network Moreover, ths scheme ony uses bnary XOR operatons, whch has computatona advantages over schemes based on a arge fnte fed GF( 8 ) or GF( 6 ) The new bnary scheme acheves the same optma throughput as that of [9] whe the ater requres the use of a arger fed Practca network codng [7] uses the concept of generatons that synchronze the network operatons as codng s performed ony wthn the same generaton Wth approprate route/path seecton and a carefuy-desgned generatonfushng pocy, packets sedom cyce n the network For the foowng, we thus restrct our attenton to acycc networks In [9], t s observed that some PICCs have neggbe mpact on schedung/rate-contro and can be absorbed by a par of edge-dsjont paths or by other PICCs Fg 8(a) represents one such nsgnfcant PICC The P and Q paths n Fgs 8(b) and 8(c) verfy that Fg 8(a) s ndeed a PICC However, s s d d (a) An Insgnfcant PICC s s d d (c) Paths Q s,d, Q s,d, and Q s,d s s d d (b) Paths P s,d, P s,d,and P s,d s Dsjont path s Dsjont path d d (d) The embedded edge-dsjont paths Fg 8 Iustraton of an nsgnfcant PICC that contans a par of edgedsjont paths as a strct subgraph wthn Fg 8(a), there exsts a par of edge-dsjont paths P s,d and P s,d as ustrated n Fg 8(d) One can send symbos X and Y aong the edge-dsjont paths wthout usng up a avaabe bandwdth n the gven PICC From the throughput/cost perspectve, the par of edge-dsjont paths (Fg 8(d)) domnates the gven PICC (Fg 8(a)), the atter of whch s thus nsgnfcant n the rate-contro/schedung anayss and can be removed from consderaton wthout affectng the optmaty of the souton Let (s,d ) and (s,d ) denote the par of uncast sessons of nterest For acycc networks, the foowng four rues dentfy the nsgnfcant PICCs (each consstng of three P paths and three Q paths) Rue : If P s,d and Q s,d meet at any edge, the PICC s nsgnfcant Dependng on whether P s,d and Q s,d are the same path (and symmetrcay whether P s,d and Q s,d are the same path), we have the foowng three more rues Rue : Suppose P s,d = Q s,d and P s,d Q s,d If there exsts an edge e shared by a three paths P s,d, P s,d,andq s,d, then the PICC s nsgnfcant Rue : Suppose P s,d Q s,d and P s,d = Q s,d If there exsts an edge e shared by a three paths Q s,d, Q s,d,andp s,d, then the PICC s nsgnfcant Rue 3: Suppose P s,d Q s,d and P s,d Q s,d Decare the PICC as nsgnfcant Rues to 3 can be mpemented dstrbutedy n the ntazaton phase by sendng tokens aong the paths to expore whether the paths share a gven edge For exampe, the nsgnfcant PICC n Fg 8(a) can be dentfed by Rue and removed from consderaton Our new XOR-based scheme s then performed on the remanng PICCs that are not removed by the above four rues The detaed descrpton of the code constructon s as foows A detaed proof of the correctness of these four rues and the correspondng dstrbuted mpementaton can be found n [] Authorzed censed use mted to: Purdue Unversty Downoaded on June, 9 at : from IEEE Xpore Restrctons appy

9 64 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL 7, NO 5, JUNE 9 M = M M M node ocaton (,) Cass Source Destnaton M e M = M v (a) Case v M e Y e/ P s,d (b) Case (,) (,) (,) (5,) (35,) M M v M M v Fg The network topoogy, node ocatons, and three casses of uncast traffc used n the smuatons M e Y e/ P s,d e/ P s,d (c) Case 3 M e X + Y e Ps,d e P s,d (d) Case 4 Fg 9 Cases to 4 of the new dstrbuted code constructon usng ony the bnary XOR operaton The thck arrow s the edge e n whch the codng decson w be made M and M are the coded symbos aong the two ncomng edges of v M e s the outgong coded symbo aong edge e The source: Source s sends ts own symbo X aong P s,d, Q s,d,andq s,d,ands sends ts own symbo Y aong P s,d, Q s,d,andp s,d Each ntermedate edge: At each edge e that s the outgong edge of an ntermedate network node v s,s Consder the foowng cases Case : P s,d = Q s,d and P s,d Q s,d Consder four sub-cases Case : If a ncomng edges of v carry the same symbo, then forward that symbo Case : If Case s not satsfed and e / P s,d,thensend Y through e Case 3: If Case s not satsfed, e P s,d,ande/ P s,d,thensendx through e Case 4: If Case s not satsfed, e P s,d,ande P s,d, then send the bnary XORed symbo X + Y through e Case : P s,d Q s,d and P s,d = Q s,d Thss a symmetrc case of Case We perform the symmetrc operatons of Case by swappng the roes of the frst uncast sesson (s,d ), X, andp wth the roes of the second sesson (s,d ), Q and Y Case 3: P s,d = Q s,d and P s,d = Q s,d Perform the same operatons as n Cases to 4 Fg 9 ustrates Cases to 4 for an outgong edge e of an ntermedate node v Proposton 6: For an acycc network, consder a PICC that s not removed by Rues to 3 Then destnaton d (resp d ) s abe to recover the desgnated symbo X (resp Y ) usng the above ocay computed bnary codng scheme The proof s reegated to Appendx C VI NUMERICAL EXPERIMENTS We perform smuatons under the near sgna-to-nose-&- nterference-rato (SINR) mode The objectve s to compare the non-coded and the network codng soutons wth both perfect and mperfect schedung To mpement mperfect schedung, we mantan a sma poo mperfect schedung pocy Θ Θ and choose the mperfect schedung from the smaer pocy poo Θ n a smar way as n [8] accordng to the foowng Every schedung pocy θ Θ s assocated wth a rate vector re θ Further assume that every θ Θ s assocated wth a set of queue engths {qe θ : e} such that the pocy θ s a γ θ -approxmaton pocy satsfyng qθ ere θ γ θ max r qθ er e If the foowng condton hods n the t-th tme sot max θ Θ q e [t]re θ γ mn θ Θ ( [q e [t] qe θ ]+ r max e + ) qθ ere θ, γ θ for some γ, wherere max s the maxmum possbe rate aong edge e, then pocy θa that maxmzes the eft-hand sde s a γ- approxmaton of the optma schedung pocy wth weghts q e [t] on each edge We can use such a schedung pocy θa n the reduced pocy poo Θ wthout the computatonay expensve step of computng the optma schedung pocy θ n the rght-hand sde of (8) If no such θa exsts, we compute drecty one θ[t] satsfyng (8) and store ths new θ[t] and the assocated q e [t] n the sma poo Θ We assume that the tota power assgned to node u at any tme sot s bounded by P u,max To acheve the optma throughput, n each tme sot, each node u shoud ether transmt at fu power P u,max or reman sent For any uncast transmsson from u to v, the data rate r u,v s assumed to be proportona to the SINR eve 3 at the recever v, whchs formay expressed as G(u, v) {(u,v) s actvated} P u,max r uv = W N + w:w u G(w, v), {node w s sendng}p w,max where N s the background nose, W s the bandwdth of the system, and G(u, v) s the path gan between nodes u and v whch s set to (dst(u, v)) 4,wheredst(u, v) s the Eucdean dstance between nodes u and v Wth network codng, the data rate of the broadcast nk wth mutpe recevers s proportona to the mnmum of the SINR eves at those recevers More precsey, f node u s broadcastng to nodes v,,v n, the data rate of ths broadcast nk, r u,{v,,v n}, becomes r u,{v,,v n} = { } G(u, v ) {(u,{v,,v W mn n}) s actvated}p u,max {=,,n} N + w:w u G(w, v ) {w s sendng} P w,max We run the smuatons on the topoogy n Fg The X- and Y-coordnates of the sx networknodes are specfed n the fgure We smuate three casses of users and each cass s aowed to use mut-path or mut-picc communcatons 3 The near SINR mode can be vewed as a frst order approxmaton of the nformaton-theoretc W og( + SINR) mode Authorzed censed use mted to: Purdue Unversty Downoaded on June, 9 at : from IEEE Xpore Restrctons appy

10 KHREISHAH et a: CROSS-LAYER OPTIMIZATION FOR WIRELESS MULTIHOP NETWORKS WITH PAIRWISE INTERSESSION NETWORK CODING 65 The source and destnaton par of each cass s aso shown n the fgure A ogarthmc utty functon U( ) = og( ) s assumed for a casses In our smuaton, we use W =, N =, P u,max =, u, the proxma coeffcent α =, the step szes β e =, e, β =,, and nner teratons wthn each proxma teraton K = Fg represents the resuts for the case of determnstc arrva and departure Usng the non-coded souton wth perfect schedung the rates of casses and converge to about 377 and the rate of cass 3 converges to 35 When mperfect schedung s used wthout network codng and γ = 6 the rates of casses and reman the same as the perfect schedung case and the rate of cass 3 s reduced by ony The number of tme sots n whch new schedues need to be computed s 8 out of totay 5 tme sots (5 proxma teratons) When we further reduce γ to 3, the rate of cass s 4, the rate of cass s 35, and the rate of cass 3 s 3 whch shows a devaton from the optma rates The number of tme sots n whch new schedues need to be computed s 7 out of totay 5 tme sots The PINC souton wth both the perfect and mperfect schedung wth γ = 6, 3 acheves strct farness as the data rates of a casses converge to 4 as shown n Fg Usng mperfect schedung from a reduced poo of schedung poces, the new schedues need to be computed n ony tme sots when γ =6and n ony 3 tme sots when γ = 3 Wth network codng, the computatonay effcent mperfect schedung method outperforms the non-codng souton wth optma schedung from both the throughput and farness perspectves To show that the performance gan of PINC s unversa for other channe modes, we have aso smuated the same topoogy wth a W og( + SINR) mode Smar performance gan s observed n Fg For the dynamc arrva and departure, we smuated the arrva of fes for each cass whose sze s exponentay dstrbuted wth average sze ( μ = ) Each fe arrves accordng to a Posson process wth rate λ Wevarytherate λ and report n Fg 3 the average number of users n the system wth respect to the system oad per user ρ = Δ λ μ As shownnthefgure network codng wth mperfect schedung outperforms the non-coded souton wth perfect schedung by a sgnfcant % VII CONCLUSION For ntersesson network codng, codng across many sessons requres greater transmsson power to broadcast the coded symbo to many recever, whch resuts n hgher nterference n the wreess mut-hop network In both emprca and anaytca studes, t has been shown that for an nterference/energy aware network codng scheme, most of the codng opportuntes nvove ony two sessons, referred heren as parwse ntersesson network codng (PINC) In ths work, we have proposed a jonty optma codng, schedung, and rate-contro scheme for wreess mut-hop networks based on the recent theoretca fndng of PINC The correspondng codng, schedung, and rate-contro components are decouped by the use of queue engths and the ntroducton of rate-baance dua varabes Our resuts have proven that n a wreess mut-hop network, the throughput advantage of PINC can be acheved wthout sacrfcng the stabty condtons Moreover, PINC has mnma mpact on the optma rate-contro/schedung as the ony new component necessary for schedung PINC traffc sthebaance update performed at the recevers Foowng ths new formuaton, we have aso studed the mpact of γ-mperfect schedung on PINC-based rate-contro agorthm and for the correspondng dstrbuted greedy hypergraph matchng agorthm Numerca experments have aso been conducted for the near and the ogarthmc sgna-to-nose/nterference-rato (SINR) modes, whch shows that the achevabe rates usng PINC and effcent mperfect schedung outperforms that of non-codng transmsson wth computatonay expensve optma schedung We provde the foowng future drectons to concude ths paper ) PINC admts a fow-based characterzaton smar to that for non-codng communcatons, whch prompts further extenson of the tradtona wsdoms for noncodng communcatons to PINC We w study the beneft/mpact of PINC for dfferent network objectves, ncudng dstrbuted schedung pocy for genera nterference modes and the jont schedung and energy mnmzaton In [4], the PINC has been extended from two uncast sessons to two mutcast sessons We w deveop the correspondng codng/schedung schemes for mutcast traffc as we ) Another mportant ngredent for practca wreess network codng s the opportunstc way of expotng overhearng and codng opportuntes Opportunstc stenng/codng/routng takes advantages of the randomness of the channe, whch can be vewed as another type of mperfect schedung Specfcay, wth opportunstc routng, the network users can contro ony part of the routng/schedung pocy and the random nature of the wreess channe w decde randomy whch recevers w receve whch packet transmsson Ths opportunstc behavor of wreess mut-hop network provdes a unque chaenge for the anayss of ntersesson network codng as the coded symbos ntended for certan recevers may be receved by a dfferent set of recevers nstead We w bud upon our understandng of mperfect schedung and nvestgate the opportunstc codng opportuntes wth PINC APPENDIX A STABILITY RESULTS WITH DETERMINISTIC ARRIVALS Sketches of the proof of Proposton 3: We frst notce that by (6), qj s a constant and s thus bounded away from nfnty For the foowng, choose the foowng Lyapunov functon V (q) = (q e) e β e for the dua varabes q = {q e : e} Then V (q[t +]) V (q[t]) = ( q e [t] H k (e)xk e k + ) H jx j r e + const, () (,j): j Authorzed censed use mted to: Purdue Unversty Downoaded on June, 9 at : from IEEE Xpore Restrctons appy

11 66 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL 7, NO 5, JUNE 9 5 Non codng wth Perfect Schedung 5 Non codng wth Imperfect Schedung γ=6 5 Non Codng wth Imperfect Schedung γ= Cass Cass Cass Cass Cass Cass Codng wth Perfect Schedung 5 Codng wth mperfect Schedung γ=6 5 Codng wth Imperfect Schedung γ= Cass Cass 3 3 Cass Cass 3 3 Cass Cass Fg The convergence resuts for the case of determnstc arrva wth the near SINR-based nterference mode where const s a constant bounded away from postve and negatve nfnty Snce the fxed rate assgnment x s n the nteror of γλ, the frst term of () can be rewrtten as ɛ q e [t] H k (e)xk + Hj x j e k (,j): j () for some ɛ > Combnng () and (), we have that any component of q tends to nfnty w ead to negatve dfference of the Lyapunov functon V (q) As a resut, a dua varabes q e and qj are bounded away from nfnty Snce the prma varabes have ony bounded domans, the proof s compete transton rate s gven by: n [t] n [t]+ n [t] n [t] wth rate λ wth ( rate μ k xk [t]+ x j [t] ) n [t] A heurstc fudty mode argument s provded as foows By the rate and schedung update rues, we have APPENDIX B STABILITY RESULTS WITH STOCHASTIC ARRIVALS Δ Sketches of the proof of Proposton 4: Let ρ = λ μ Snce {ρ } s n the nteror of γλ, by the defnton of Λ, we can fnd ɛ, ρ k and ρ j satsfyng ρ = k and ( + ɛ) ρ k + k ρ j,, ρ j = ρ j,, j,, H k (e)ρ k + Hj(e)ρ j γre for some r Co(R) (3) Let n denote the number of users n the system The probabty aw of n s determned by a Markov process Its x [t] = arg max U x k + x j x k q e [t] H k (e)x k + H j(e)x j e k qj [t]x j qj [t]x j j:j> α k e j:j< ( x k y k ) + q e [t]r e [t] γ max [r] q e [t]r e e ( x j yj ) (4) Authorzed censed use mted to: Purdue Unversty Downoaded on June, 9 at : from IEEE Xpore Restrctons appy

12 KHREISHAH et a: CROSS-LAYER OPTIMIZATION FOR WIRELESS MULTIHOP NETWORKS WITH PAIRWISE INTERSESSION NETWORK CODING 67 5 Non codng wth Perfect Schedung 5 Non codng wth Imperfect Schedung γ=6 5 Non codng wth Imperfect Schedung γ= Cass Cass Cass Cass Cass Cass Codng wth Perfect Schedung 5 Codng wth Imperfect Schedung γ=6 5 Codng wth Imperfect Schedung γ= Cass Cass 3 3 Cass Cass 3 3 Cass Cass Fg The convergence resuts for the case of determnstc arrva wth the W og( + SINR)-based nterference mode The frst order dervatves of n, q e,andq j become d dt n [t] =λ μ n [t] x k [t]+ k d dt q e[t] ( ( β e n [t] k Hk (e)xk [t] ) ) = H j (e)x j [t] r e [t] + x j [t] f postve or q e [t] > otherwse d ( dt q j[t] =β n [t]x j[t] n j [t]x j[t] ) Consder the foowng V (, ) functon that w be used as the Lyapunov functon of the system where V n (n) = V q (q) = e V (n, q) =V n (n)+v q (q) Kκ n λ (q e ) β e + + α n μ,j:<j, (q j ) β k y k + y j By the fudty mode, we have the foowng expresson for any postve constant K dv n (n[t]) = Kκ n [t] + α y k + dt ρ k ρ n [t] x k [t]+ x j k Therefore dv n dt = ɛ Kκ n + α ρ y k + k + Kκ n + α y k ρ + yj k ( + ɛ)ρ n x k + k Kκ n + α ρ k = ɛ + ( K( + ɛ)κ k xk + x j ( + α y k + )) yj k ( + ɛ)ρ n +(A), k x k + x j y k + x j y j y j y j Authorzed censed use mted to: Purdue Unversty Downoaded on June, 9 at : from IEEE Xpore Restrctons appy

13 68 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL 7, NO 5, JUNE 9 3 technca report [] for each we have expcty construct a postve nteger J such that Number of users Non codng wth Perfect Schedung Non codng wth Imperfect Schedung γ = 8 Non codng wth Imperfect Schedung γ = 5 PINC wth Perfect Schedung PINC wth Imperfect Schedung γ = 8 PINC wth Imperfect Schedung γ = ρ Codng Gan Fg 3 The number of users n the system versus the system oad for the case of dynamc arrva wth the near SINR-based nterference mode where (A) ( ) ( + ɛ) Kκ k xk + n x j ρ ( + ɛ)ρ n x k + x j k In the foowng we w use the foowng notaton y k/ (j) to express that we are teratng over a y k and yj For exampe k,,j,:j yk/ (j) P k= yk + PICCj j:j = yj Smar notaton s used for other varabes such as ρ and q Wedefne F and F such that: F =( + ɛ) α y k/ (j) ρ k / (j ) k, k,j, :(k,j, ) (k,j,) ɛ α ρ y k/ (j) +(A) k,j, F =(+ɛ) α M Σx k, ρk/ (j) HereM Σx s the maxmum rate assgned to any user Snce x [t] soves (4) and U ( ) =κ og( ), thereexst δ k,δ j and δ,m such that κ k,j, :j xk / = q k/ e,(j), where (j ),j,k,: j α (x k/ (j) yk/ (j) )+δk/ (j) δ,m qe, k = q e H k (e) e { qe,j = e q ehj (e)+q j f <j e q ehj (e) q j f >j ( ) We have δ k/ (j) xk/ (j) =and δ,m k, x (j) MP x = due to the compementary sackness condtons In our onne δ,m κ k,j,: j + α yk/ (j), (5) M Σx J Pk,j, yk/ (j) By choosng F =max { κ M Σx +α J bound for a δ,m LetK = +ɛ,wehave dv n ɛ Kκ n dt + q e ( + ɛ) H k (e)ρ k + e k + (,j):<j },F s an upper Hj(e)ρ j n H k (e)x k + n H j(e)x j k ( q j ( + ɛ) ( ρ j ) ( ρ j q j n x j n jx j)) + F + F + F ( + ɛ) k, ρ k/ (j) (6) The detas of how (6) s obtaned n our onne technca report [] Snce dv q (q) dt = ( ( q e n H k (e)x k + e k ) ) H j(e)x j r e + qj ( n x j n jx ) j, (,j):<j and ρ j = ρ dv j, the overa drft dt = dvn dt + dvq dt becomes dv dt ɛ Kκ n ɛ ( q e H k (e)ρk e k ) + Hj (e)ρ j + F + F + F ( + ɛ) ρ k/ (j) k, Here, we used (3) The Lyapunov functon w have a negatve drft and the system s stabe A fu proof that takes nto account the second-order varaton can be obtaned accordngy APPENDIX C THE VALIDITY OF THE DISTRIBUTED CODE CONSTRUCTION Proof of Proposton 6: We prove ths theorem by nducton We perform codng operatons sequentay from the most upstream edges to the most downstream edges Let M e represent the symbo transmtted aong edge e Wehave Authorzed censed use mted to: Purdue Unversty Downoaded on June, 9 at : from IEEE Xpore Restrctons appy

14 KHREISHAH et a: CROSS-LAYER OPTIMIZATION FOR WIRELESS MULTIHOP NETWORKS WITH PAIRWISE INTERSESSION NETWORK CODING 69 the foowng nducton hypothess: If Case s satsfed, then for any edge e we have: X or X + Y f e P s,d M e = Y or X + Y f e P s,d (7) X or Y f e P s,d or e Q s,d Case s a symmetrc verson of Case and the dscusson s thus omtted If Case 3 s satsfed, for any edge e we have: X or X + Y f e P s,d M e = Y or X + Y f e P s,d (8) X or Y f e P s,d or e Q s,d Both hypotheses (7), (8) are satsfed on the mmedate outgong edges of sources s and s whch carry X and Y respectvey To show that the nducton hods for a edges, we need to consder 3 scenaros when Case s satsfed as n Tabe I and 5 scenaros when Case 3 s satsfed as n Tabe II The entres n the second coumn n both Tabe I and II represent the paths that share edge e and the entres n the thrd coumn represent the correspondng codng operaton that w be performed on edge e Snce Case 3 s smper than Case, we dsccuss Case 3 frst and then move on to Case For Case 3 we have P s,d = Q s,d,p s,d = Q s,d Therefore, we have four dstnct paths P s,d,p s,d,p s,d,andq s,d n the PICC If an edge s shared by three paths, t s ether the case that the PICC s nsgnfcant by Rue or condton of Theorem s not satsfed There are ( 4 ) = 6 scenaros n whch two paths meet at a snge edge Snce by Rue, P s,d and Q s,d do not meet, we are eft wth 5 scenaros to consder as n Tabe II In the foowng, we prove that the nducton hypothess foows for the 5 scenaros n Case 3 Scenaro as n Tabe II The paths that meet at edge e are P s,d and P s,d The symbos carred by the paths on the respectve prevous edges can be the same or not If they are the same, the symbos must be X + Y accordng to the hypothess, whch s the ntersecton of the frst two cases of (8) The coded symbo X + Y w be forwarded and the nvarant hods for the target edge e If the symbos that enter edge v, the ta of e are dfferent, then node v can decode both X and Y and compute the coded symbo X + Y accordng to Case 34 (or equvaenty Case 4) n Secton V and send X + Y aong e The hypothess hods n ths scenaro that e P s,d P s,d Scenaro and 3 as n Tabe II The paths that meet at edge e are P s,d and Q s,d (P s,d and P s,d ) The symbos carred by the paths on the respectve prevous edges can be the same or not If they are the same, the symbos must be X accordng to the hypothess, whch s the ntersecton of the frst and thrd cases of (8) The symbo X w be forwarded and the nvarant hods for the target edge e If the symbos that enter edge v, the ta of e are dfferent, then node v can compute X accordng to Case 33 (or equvaenty Case 3) n Secton V and send X aong e The hypothess hods n ths scenaro that e P s,d Q s,d (e P s,d P s,d ) Scenaro4and5asnTabeIIThe paths that meet at edge e are P s,d and Q s,d (P s,d and P s,d ) The symbos carred by the paths on the respectve prevous edges can be the same or not If they are the same, the symbos must be Y accordng to the hypothess, whch s the ntersecton of the ast two cases of (8) The symbo Y w be forwarded and the nvarant hods for the target edge e If the symbos that enter edge v, the ta of e are dfferent, then node v can compute the symbo Y accordng to Case 3 (or equvaenty Case ) n Secton V and send Y aong e The hypothess hods n ths scenaro that e P s,d Q s,d (e P s,d P s,d ) For Case we have P s,d = Q s,d Therefore, we have fve dstnct paths P s,d,p s,d,p s,d,q s,d,andq s,d n the PICC By Rue P s,d and Q s,d w not meet at a snge edge Aso P s,d,p s,d,q s,d w not meet at a snge edge due to Rue Therefore, any scenaro n whch an edge s used by fve or four of the paths s mpossbe We have ( 5 3) = dfferent scenaros n whch edge e s used by three dstnct paths The scenaros n whch an edge s used by P s,d,p s,d,p s,d or P s,d,q s,d,q s,d voate condton of Theorem The scenaro n whch an edge s used by P s,d,p s,d,q s,d, w be removed because t satsfes Rue The scenaros n whch an edge s used by (P s,d,p s,d,q s,d ), (P s,d,q s,d,q s,d ), or (P s,d,q s,d,q s,d ) w be removed because t satsfes Rue As a resut, we need to ony consder 4 scenaros n whch e s used by three dstnct paths (Scenaros 4, 5,, n Tabe I) Snce by Rue P s,d and Q s,d do not use the same edge we have ony 9 scenaros n whch e s used by two paths The tota s 3 scenaros as n Tabe I In the foowng, we prove that the nducton hypothess hods for the 3 scenaros of Case Scenaros,3,4,7andasnTabeIThe paths that meet at edge e are (P s,d and P s,d ), (P s,d and Q s,d ), (P s,d, P s,d,andq s,d ), (P s,d and Q s,d ), or (Q s,d, P s,d,andq s,d ) The symbos carred by the paths on the respectve prevous edges can be the same or not If they are the same, the symbos must be Y accordng to the hypothess, whch s the ntersecton of the ast two cases of (7) The symbo Y w be forwarded and the nvarant hods for the target edge e If the symbos that enter edge v, the ta of e are dfferent, then node v can compute the symbo Y accordng to Case n Secton V and send Y aong e The hypothess hods n these scenaros Scenaros, 5 and 3 as n Tabe I The paths that meet at edge e are (P s,d and P s,d ), (P s,d, P s,d,andq s,d ), or (P s,d and Q s,d ) The symbos carred by the paths on the respectve prevous edges can be the same or not If they are the same, the symbos must be X accordng to the hypothess, whch s the ntersecton of the frst and thrd cases of (7) The symbo X w be forwarded and the nvarant hods for the target edge e If the symbos that enter edge v, the ta of e are dfferent, then node v can compute X accordng to Case 3 n Secton V and send X aong e The hypothess hods n these scenaros Scenaros 6 and as n Tabe I The paths that meet at edge e are (P s,d and P s,d )or(q s,d, P s,d,andp s,d ) The symbos carred by the paths on the respectve prevous edges can be the same or not If they are the same, the symbos Authorzed censed use mted to: Purdue Unversty Downoaded on June, 9 at : from IEEE Xpore Restrctons appy

15 6 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL 7, NO 5, JUNE 9 TABLE I THE LIST OF POSSIBLE CODING OPERATIONS A NODE HAS TO PERFORM IF THE PICC SATISFIES CASE PICC satsfes Case Paths sharng edge e Symbos transmtted on edge e Paths sharng edge e Symbos transmtted on edge e Scenaro P s,d P s,d Y Scenaro 8 Q s,d P s,d X or X + Y Scenaro P s,d P s,d X Scenaro 9 Q s,d P s,d Y or X + Y Scenaro 3 P s,d Q s,d Y Scenaro Q s,d Q s,d Y or X + Y Scenaro 4 P s,d P s,d Q s,d Y Scenaro Q s,d P s,d P s,d X + Y Scenaro 5 P s,d P s,d Q s,d X Scenaro Q s,d P s,d Q s,d Y Scenaro 6 P s,d P s,d X + Y Scenaro 3 P s,d Q s,d X Scenaro 7 P s,d Q s,d Y TABLE II THE LIST OF POSSIBLE CODING OPERATIONS A NODE HAS TO PERFORM IF THE PICC SATISFIES CASE 3 PICC satsfes Case 3 Paths sharng edge e Symbos transmtted on edge e Scenaro P s,d P s,d X + Y Scenaro P s,d Q s,d X Scenaro 3 P s,d P s,d X Scenaro 4 P s,d Q s,d Y Scenaro 5 P s,d P s,d Y must be X + Y accordng to the hypothess, whch s the ntersecton of the frst two cases of (7) The coded symbo X + Y w be forwarded and the nvarant hods for the target edge e If the symbos that enter edge v, the ta of e are dfferent, then node v can decode both X and Y and compute the coded symbo X + Y accordng to Case 4 n Secton V and send X + Y aong e The hypothess hods n these scenaros Scenaros 9 and as n Tabe I The paths that meet at edge e are (Q s,d and P s,d ), or (Q s,d and Q s,d ) The symbos carred by the paths on the respectve prevous edges can be the same or not If they are the same, the symbos must be ether Y or X + Y accordng to the hypothess The symbo Y or X + Y w be forwarded and the nvarant hods for the target edge e If the symbos that enter edge v, the ta of e are dfferent, then node v can compute the symbo Y accordng to Case n Secton V and send Y aong e The hypothess hods n these scenaros Scenaro 8 as n Tabe I The paths that meet at edge e are (Q s,d and P s,d ) The symbos carred by the paths on the respectve prevous edges can be the same or not If they are the same, the symbos must be ether X or X + Y accordng to the hypothess The symbo X or X + Y w be forwarded and the nvarant hods for the target edge e If the symbos that enter edge v, the ta of e are dfferent, then node v can compute X accordng to Case 3 n Secton V and send X aong e The hypothess hods n ths scenaro REFERENCES [] R Ahswede, N Ca, S-Y R L, and R Yeung, Network nformaton fow, IEEE Trans Inform Theory, vo 46, no 4, pp 4 6, Juy [] D Baker, J Wesether, and A Ephremdes, Dstrbuted agorthm for schedung the actvaton of nks n a sef-organzng mobe rado network, n Proc IEEE Int Conf Commun, 98 [3] D Bertsekas, A Nedć, and A Ozdagar, Convex Anayss and optmzaton Athena Scentfc, 3 [4] D Bertsekas and J Tstsks, Parae and dstrbuted computaton: Numerca methods Athena Scentfc, 997 [5] S Chachusk, M Jennngs, S Katt, and D Katab, Tradng structure for randomness n wreess opportunstc routng, n Proc ACM Speca Interest Group on Data Commun (SIGCOMM) Kyoto, Japan, August 7 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16 KHREISHAH et a: CROSS-LAYER OPTIMIZATION FOR WIRELESS MULTIHOP NETWORKS WITH PAIRWISE INTERSESSION NETWORK CODING 6 [5] S-Y L, R Yeung, and N Ca, Lnear network codng, IEEE Trans Inform Theory, vo 49, no, pp 37 38, February 3 [6] Z L and B L, Network codng: the case of mutpe uncast sessons, n Proc 4nd Annua Aerton Conf on Comm, Contr, and Computng Montceo, Inos, USA, September 4 [7] X Ln and N Shroff, Jont rate contro and schedung n muthop wreess networks, n Proc IEEE Conf Decson & Contr Paradse Isand, Bahamas, December 4 [8], The mpact of mperfect schedung on cross-ayer congeston contro n wreess networks, IEEE Trans Networkng, vo 4, no, pp 3 35, Apr 6 [9], Utty maxmzaton for communcaton networks wth mutpath routng, IEEE Trans Automat Contr, vo 5, no 5, pp , May 6 [3] J Lu, D Goecke, and D Towsey, Bounds on the gan of network codng and bradcastng n wreess networks, n Proc 6th IEEE Conference on Computer Communcatons (INFOCOM) Anchorage, Aaska, USA, May 7 [3] D Lun, N Ratnakar, M 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Commun, vo 3, no, pp 36 5, January 5 [44] G Zussman, A Brzeznsk, and E Modano, Muthop Loca Poong for Dstrbuted Throughput Maxmzaton n Wreess Networks, n Proc 7th IEEE Conference on Computer Communcatons (INFO- COM) Phoenx, USA, Apr 8 Abdaah Khreshah receved the BS degree from Jordan Unversty of Scence and Technoogy (JUST), Irbd, Jordan n 4, and the MS degree from the Schoo of Eectrca and Computer Engneerng, Purdue Unversty, West Lafayette, IN, n 6 He s currenty workng toward the PhD degree at Purdue Unversty Hs research nterests ncude network codng, congeston contro, and cross ayer desgn n wreess networks Chh-Chun Wang joned the Schoo of Eectrca and Computer Engneerng n January 6 as an Assstant Professor He receved the BE degree n EE from Natona Tawan Unversty, Tape, Tawan n 999, the MS degree n EE, the PhD degree n EE from Prnceton Unversty n and 5, respectvey He worked n Comtrend Corporaton, Tape, Tawan, as a desgn engneer durng n and spent the summer of 4 wth Faron Technooges, New Jersey In 5, he hed a post-doc poston n the Eectrca Engneerng Department of Prnceton Unversty He receved the Natona Scence Foundaton Facuty Eary Career Deveopment (CAREER) Award n 9 Hs current research nterests are n the graph-theoretc and agorthmc anayss of teratve decodng and of network codng Other research nterests of hs fa n the genera areas of optma contro, nformaton theory, detecton theory, codng theory, teratve decodng agorthms, and network codng Ness B Shroff receved hs PhD degree from Coumba Unversty, NY, n 994 and joned Purdue unversty as an Assstant Professor At Purdue, he became Professor of the Schoo of Eectrca and Computer Engneerng n 3 and drector of CWSA n 4, a unversty-wde center on wreess systems and appcatons In Juy 7, he joned The Oho State Unversty as the Oho Emnent Schoar of Networkng and Communcatons, a chared Professor of ECE and CSE Hs research nterests span the areas of wreess and wrene communcaton networks He s especay nterested n fundamenta probems n the desgn, performance, prcng, and securty of these networks Dr Shroff s a past edtor for IEEE/ACM Trans on Networkng and the IEEE Communcatons Letters and current edtor of the Computer Networks Journa He has served as the technca program co-char and genera co-char of severa major conferences and workshops, such as the IEEE INFOCOM 3, ACM Mobhoc 8, IEEE CCW 999, and WICON 8 He was aso a co-organzer of the NSF workshop on Fundamenta Research n Networkng, hed n Are House Vrgna, n 3 Dr Shroff s a feow of the IEEE He receved the IEEE INFOCOM 8 best paper award, the IEEE INFOCOM 6 best paper award, the IEEE IWQoS 6 best student paper award, the 5 best paper of the year award for the Journa of Communcatons and Networkng, the 3 best paper of the year award for Computer Networks, and the NSF CAREER award n 996 (hs INFOCOM 5 paper was aso seected as one of two runner-up papers for the best paper award) Authorzed censed use mted to: Purdue Unversty Downoaded on June, 9 at : from IEEE Xpore Restrctons appy