Backpressure with Adaptive Redundancy (BWAR)

Backpressure with Adapti v e Redundanc y (BW AR) Majed Alresaini alresain A T usc . edu Maheswaran Sathiamoorthy msathiam A T usc . edu Bhaska r Krishnamac hari bkrishna A T usc . edu Michael J. Neely mjneely A T usc . edu Abstract —Backpressure scheduling and routing, in which packets are prefer entially transmitted over links with high queue differentials, offers the promise of throughput-optimal operation fo r a wide range of communication networks. Howe ver , when th e traffic load is l ow , due to the corres pondin g low queue occupancy , backpressure scheduling/routing experiences long delays. Th is i s particularly of concern in intermittent encounter-based mobile networks which are alr eady delay-limited due to th e sparse and highly dynamic network connectivity . While state of the art mechanisms f or such networks ha ve pr oposed the use o f redundant tra nsmissions to impro ve delay , they do not work well wh en the traffic load is h igh. W e propose in th is paper a nov el hybrid appro ach that we r efer to as b ackpressure with adaptive redundancy (BW AR) , whi ch provides th e best of both worlds. This approach is highly robust and d istributed and does not require any prior k nowledge of network load conditions. W e ev aluate BW AR th rough both mathematical analysis and simula- tions b ased on cell-partitioned model. W e p ro ve th eoretically that BW AR does not p erf orm worse than tradit ional backpressur e in term s of the max imum throughput, while yielding a better delay b ound. The simulations confirm that BW AR outperform s traditional backpressure at low load, whil e outperfo rming a state of the art encounter -routing scheme (S pray and W ait) at high load. I . I N T R O D U C T I O N Queue-d ifferential backpressure schedu ling and r outing was shown b y T assiulas and Ephremid es to be thr ough put optimal in terms o f being able to stabilize the n etwork under any fea- sible traffic rate vector [1]. Additio nal research has extended the or iginal result to sho w th at back pressure techniq ues can be c ombined with utility optim ization, resulting in simple, throug hput-o ptimal, cross-layer network protocols f or all kinds of network s [2]–[5], [ 29]. Recently , some of th ese tech niques have be en translated to practically implem ented ro uting and rate-contr ol protoco ls for wireless networks [6]–[1 0]. The basic idea of b ackpressure mechan isms is to p rioritize transmissions over link s that have the h ighest qu eue differen- tials. Backp ressure effectively makes packets flow throug h th e network as tho ugh p ulled by gr avity tow ards the destination, which has the sm allest q ueue size of 0 . Un der hig h traffic con- ditions, this works very well, and back pressure is able to fully utilize the a vailable network resour ces in a high ly dynam ic fashion. Under low traffi c con ditions, howe ver , because many other n odes may also h av e a small or 0 qu eue size, th ere is This materia l is supported in part by the followi ng: NSF grant 1049541, the Networ k Science Collabo rati ve T echnolo gy Alliance sponsored by the U.S. Army Research L aborator y W 911NF-09-2-0053 , and King Saud Uni versity . inefficiency in term s o f an in crease in delay , as p ackets m ay loop or take a long time to make their way to the destination . In this paper , we focu s primarily on intermittently connected networks, such a s enc ounter-based mob ile networks (some - times also referre d to as delay or disru ption to lerant networks (DTN)). In such networks, conventional path-d iscovery-based MANET r outing tech niques like A ODV [11] an d DSR [ 12] are not feasible becau se the n etwork m ay not form a single connected partition at any time, and thus a full path may never exist between the source and th e destination . Instead, it is n ecessary to u se store- and-fo rward type proto cols that can handle the un derlying mobility . A back pressure based routing scheme ca n be easily implem ented in such a network, with the d ecision of what information to exchange being made between each pair of nodes based on th eir queu e differentials whenever they en counter each other . Howe ver the above- mentioned delay inefficiency o f the backp ressure mec hanism at low traffic loads is fur ther exacerbated in such networks, because they a re already de lay-limited d ue to spa rse network connectivity . In the literature on intermitten tly con nected networks, ther e are se veral p roposed schemes for store-and-f orward based routing , such as [13]–[ 18]. Some of these, such as Spr ay and W ait, advocate the use o f r edund ant transmissions, to make add itional copies of the commu nicated inform ation in the network. Th e r eplication of th e content makes it faster for the destin ation to access a co py . Howe ver, as the addition al replication always inc reases th e network loa d, these protoc ols, which are not thr ough put-op timal to b egin with, suffer addi- tional congestion . In this paper, we propose a novel hybrid approach , an adaptive redundancy techniq ue for ba ckpressure routing , that yields the benefits of replicatio n to re duce delay un der low load conditions, while at the sam e time preserving the perfor- mance and ben efits o f traditional back pressure ro uting under high tra ffic co nditions. This techniqu e, which we ref er to as backpr essure with a daptive r edund ancy (BW AR), essentially creates copies o f packets in a new duplicate buffer upon an en- counter, when the transmitter ’ s queue o ccupan cy is low . T hese duplicate p ackets are tr ansmitted on ly when th e or iginal qu eue is empty . This me chanism can dramatically improve delay of backpr essure d uring low load conditio ns due to two reaso ns: (1) d ue to the existence o f m ultiple co pies o f th e sam e p ackets at m ultiple n odes, th e destina tion is more likely to encoun ter a massage inten ded fo r it. (2) this way , th e alg orithm builds u p 2 gradients towards the destination s faster and red uces pa cket looping . The add itional tran smissions incurred by BW AR d ue to th e dup licates utilize a vailable slots which would other wise go idle, in order to redu ce the delay . Particularly fo r networks that are not en ergy-limited, this offers a more efficient way to utilize the available b andwidth durin g lo w load con ditions. In order to min imize th e sto rage re source u tilization of d uplicate packets, id eally , these duplicate packets should be removed from the network when ever a copy is delivered to the d es- tination. Since this may be difficult to implemen t (except in some kind s of networks with a separa te contro l p lane), we also propo se and e valuate a practical timeou t mech anism for auto matic duplicate removal. Und er high lo ad c ondition s, because q ueues are rarely empty , duplica tes a re r arely created, and BW AR effectively reverts to traditiona l backp ressure and inherits its throu ghput optimality property . By design , BW AR is highly robust and distributed and doe s no t re quire prior knowledge of locations, mobility patterns, and load conditions. The following are the key co ntributions of this work: • W e propose BW AR, a new adaptive redunda ncy technique for backp ressure scheduling/ro uting in intermittently con- nected n etworks. And we present a timeout mec hanism for du plicate removal, which a llows BW AR to be easily implemented in practice. • W e develop an an alytical model of BW AR, an d prove theoretically tha t it yield s a smaller upper bound o n the av erage queu e size (and hence the av erage delay ) than trad itional ba ckpressure, wh ile re taining th rough put optimality . • Th roug h simulations using an idealized cell-p artition mo- bility model, we quantify th e be nefits f rom u sing BW AR. Specifically , w e show th at it outperfor ms b oth traditional backpr essure and Spray & W ait [1 5], a state of the a rt DTN/ICN routin g mechanism. The rest of th e pape r is o rganized a s follows. In section II, we in troduc e and describ e BW AR. In section III-A, we revie w the theo ry behin d tradition al backpressur e sch eduling and routing . W e show in sectio n III-B the qu eue d ynamics for BW AR and how it can improve the delay the oretically . In section IV we pre sent our mod el-based simulation results. In section V, we d escribe related work in th is subject to plac e our co ntributions in con text. W e con clude in section VI and discuss future work. I I . B AC K P R E S S U R E W I T H A D A P T I V E R E D U N DA N C Y In this section, we first describe traditional b ackpres- sure sche duling an d routin g and then our n ew proposal for b ackpressur e sch eduling/r outing with ad aptive redun dancy (BW AR). In b oth cases, we a ssume that the re are N nod es in th e network, and time is discretized. W e assume a multi- commod ity flow system in which ev ery no de could be a po- tential d estination ( correspo nding to a p articular co mmod ity). A. T raditional Backpr essure Scheduling and Rou ting W e assume that each n ode maintain s N − 1 queues, one f or each com modity , with the j th queue at each n ode con taining packets that are destined f or n ode j . Let Q c i ( t ) ind icate the number of pa ckets destined to n ode c q ueued at no de i at time t . N aturally , Q i i ( t ) = 0 ∀ t . Let µ c ij ( t ) be the schedulin g and r outing variable that in dicates the number o f packets of commodity c to be sched uled o n link ( i, j ) . T r aditional backpr essure sche duling/r outing [1], [ 2] selects th e µ c ij ( t ) that solve the follo wing problem ( a form of maximum weight indepen dent set selection): max X i,j,c ∆ c ij ( t ) · µ c ij ( t ) sub ject to , X c µ c ij ( t ) ≤ θ ij ( t ) , ∀ i, ∀ j µ c ij ( t ) · µ d km ( t ) = 0 , (( i, j ) , ( k , m )) ∈ Ω( t ) , ∀ c, ∀ d (1 ) Where ∆ c ij ( t ) = Q c i ( t ) − Q c j ( t ) is the lin k we ight, which denotes the qu eue differential fo r co mmod ity c on link ( i, j ) at slot t an d the feasibility constrain ts on µ c ij ( t ) pertain to the av ailab le network cap acity , taking into acco unt the interference between nodes. θ ij ( t ) is the channel state in term s of number of pa ckets that ca n be tr ansmitted over link ( i, j ) d uring slot t . Ω( t ) is the link interfer ence set a t slot t such that if link ( i, j ) interferes with link ( i ′ , j ′ ) at slot t then (( i, j ) , ( i ′ , j ′ )) ∈ Ω( t ) and hen ce, th ose two links can not be bo th sch eduled at slot t . The maximiza tion prob lem in (1) can b e solved by find ing the maxim um commo dity c ∗ ij ( t ) for each link ( i, j ) at slot t that maximizes ∆ c ij ( t ) and a ssign µ c ij ( t ) = 0 for all c 6 = c ∗ ij ( t ) and then solve, max X i,j ∆ c ∗ ij ( t ) ij ( t ) · µ c ∗ ij ( t ) ij ( t ) sub ject to , µ c ∗ ij ( t ) ij ( t ) ≤ θ ij ( t ) , ∀ i, ∀ j µ c ∗ ij ( t ) ij ( t ) · µ c ∗ km ( t ) km ( t ) = 0 , (( i, j ) , ( k , m )) ∈ Ω( t ) (2) B. BW AR Scheduling a nd Routing Our pro posed en hancemen t of bac kpressure with adap tiv e redund ancy work s as follows. W e have an addition al set of N − 1 dup licate b uffers o f size D max at each node. Besides the original queue occup ancy Q c i ( t ) which has the same meaning as in tradition al b ackpressure , the duplicate qu eue occ upancy is de noted by D c i ( t ) , that in dicates the num ber of dup licate packets at node i that are destined to nod e c at time t . Again , Q i i ( t ) = D i i ( t ) = 0 ∀ t since d estinations need not buf fer any pac kets inten ded fo r themselves. Th e d uplicate q ueues are maintained and utilized as follows: • Or iginal packets when transmitted are removed f rom the main queue; howe ver, if th e queu e size is lower than a certain thresh old q th , th en the transmitted packet is duplicated and kept in the duplicate buffer associated with its d estination if it is not fu ll oth erwise no duplicate is created. W e fou nd that setting bo th q th and D max to the 3 value of the m aximum link service rate is enou gh and giv es sup erior delay results. • Du plicate packets a re not removed fr om the duplicate buf fer when transmitted. T hey a re only rem oved when they are no tified to be r eceived by the d estination, or a pre-defin ed timeout has occurr ed. • When a certain link is sched uled for tr ansmission, th e original packets in the ma in queu e are tran smitted first. If no more orig inal packets are lef t, only then d uplicates are transmitted. T hus the duplicate queu e has a strictly lower priority . Similar to original backpressure scheduling/ro uting, the BW AR scheduling/r outing also re quires th e so lution of a similar maximu m weight ind ependen t set problem : max X i,j,c ∆ c BW AR ,ij ( t ) · µ c ij ( t ) sub ject to , X c µ c ij ( t ) ≤ θ ij ( t ) , ∀ i, ∀ j µ c ij ( t ) · µ d km ( t ) = 0 , (( i, j ) , ( k , m )) ∈ Ω( t ) , ∀ c, ∀ d (3) W e define an enhan ced link weight for BW AR, ∆ c BW AR ,ij ( t ) as follows, to take into accoun t the occ upancy of the dup licate buf fer . ∆ c BW AR ,ij ( t ) =  Q c i ( t ) − Q c j ( t )  + 1 2  1 j = c And Q c i ( t )+ D c i ( t ) > 0  + 1 4 1 D max  D c i ( t ) − D c j ( t )  (4) Here t he indicator f unction 1 j = c And Q c i ( t )+ D c i ( t ) > 0 denotes that node j is the final destinatio n f or the considered comm odity c . This giv es higher weight to commodities that encounter their destinatio ns. W e show later how th is effecti vely results in dr amatic delay improvement. Similarly , the maximizatio n problem in (3) can be solved first by findin g the m aximum commod ity c ∗ BW AR ,ij ( t ) f or each link ( i, j ) at slot t that m ax- imizes ∆ c BW AR ,ij ( t ) f ollowed by the same app roach discussed earlier in II-A. It is im portant to notice that a solution to (3) is indeed a solution to (1) assuming that Q c i ( t ) and µ c ij ( t ) are integers. The small weight added in (4) giv es advantage fir st to link s/commod ities which encounter the destination and then to higher d uplicate buffer d eferential to in crease the c hance of serving dup licates. The small fr actions in (4) assures this priority when there are ties in (1) to boost delay p erform ance. C. Backpr essure r outing in in termittently c onnected networks In gener al b ackpressur e schedu ling is NP-hard, owing to the M WIS problem that needs to be solved at each tim e. Howe ver, in this paper, we fo cus on in termittently co nnected networks, that consist of sp arse en counter s between p airs o f nodes. Ther efore, at any g iv en time, the size of any connected compon ent of the network is very small. In this case, the scheduling prob lem is dram atically simplified. D. Practical Dup licate Removal As can be seen fr om the above descriptio n, BW AR cr eates duplicate packets whenever the transmitter’ s queue occup ancy is low . In an ideal setting , for efficiency , the duplicated packets in the network shou ld be d eleted instantaneo usly wh en any copy is deli vered to the inten ded destinatio n. Th is could only be imp lemented pr actically in interm ittently conn ected networks whe re a centralized contr ol plan e is av ailable tha t can provide such an instantaneous acknowledgement to all nodes in the network. In other cases, som e o ther m echanism is sou ght, so we propo se the fo llowing time out mecha nism. Whenever a packet arri ves into the netw ork, it is time-stamped. After a tim eout p eriod P fro m that arr iv al time, any dup licate copies of that packet at any node in the network will be deleted. T o obtain higher d elay performanc e improvement, when an original packet is duplicated, it is placed in the duplicated buffer giving it lo wer ser vice priority , howe ver , it is flagged an d not deleted wh en a timeout occurred. It is on ly re moved when it g ets a cknowledged d irectly b y the destination. In the next section we un dertake an analy sis o f the perfo r- mance of BW AR a nd com pare it with the k nown results for traditional bac kpressure ro uting. Specifically , we prove that any feasible rate vector is also stabilized by BW AR, and the bound that we can give on the expected qu eue occup ancy f or BW AR is b etter th an that for regular backpr essure. I I I . M AT H E M AT I C A L A N A LY S I S A. Review of the Analysis of Basic Backpr essur e W e consider a timeslotted network with N nodes that commun icate with each other . Packets ar riv e to each node, and each packet must be delivered to a specific d estination, possibly via a multi- hop path. Each node main tains several queues, one per destination, to store packets. Eac h q ueue h as the f ollowing dyna mics: Q ( t + 1) = max[ Q ( t ) − µ ( t ) , 0] + A ( t ) (5) Where Q ( t ) is the queue size at time t , µ ( t ) is the transmission rate out of the queu e at time t , and A ( t ) is the total p acket arr iv als to the qu eue at time t . Each time slot, we observe the qu eue states an d the chann el states and make sch eduling an d ro uting decisions b ased on this inf ormation . T o clear this out, let Q c n ( t ) be the qu eue backlog (n umber of p ackets) in n ode n ∈ { 1 , 2 , ..., N } tha t are destined fo r nod e c ∈ { 1 , ..., N }\{ n } at slot t . Let A c n ( t ) be the exogenous pa cket arriv als that come to node n and destined to nod e c at time t with rate λ c n . Exog enous ar riv a ls are the packets tha t just enter ed the network. Endogeno us arrivals, howe ver , are arr iv als f rom other n odes and we re already inside the network. P ackets may be forwarded to se veral nodes before reaching the destination. Let us define the capacity region Λ to be the set of all po ssible arriv al rate vectors ( λ c n ) n,c that are stabilizing by some schedu ling and rou ting strategy . Let θ ab ( t ) be the channel state from n ode a to node b at tim e t in terms of how many packets can be transmitted. Let µ ab ( t ) b e 4 the scheduled service rate fro m nod e a to n ode b at slot t . Let µ c ab ( t ) be the service r ate for comm odity c routed f rom node a to nod e b at time t an d m ust satisfy: X c µ c ab ( t ) ≤ µ ab ( t ) ≤ θ ab ( t ) (6) The qu eue dynam ics for each time slot and for each qu eue is the following: Q c n ( t + 1) = max[ Q c n ( t ) − X b µ c nb ( t ) , 0] + A c n ( t ) + X a ˜ µ c an ( t ) (7) Where ˜ µ is the actual transfer rate due to insuf ficient packets in the que ue. For example, o n some slots we may a ble to send 5 p ackets, but we o nly send 3, because only 3 were available in the q ueue. I n equation ( 7), A c n ( t ) are the exogeno us arriv als and P a ˜ µ c an ( t ) a re the endo genous arr i vals to nod e n . Define the vector Q ( t ) = ( Q c n ( t )) n,c to be the vector of all queu es in th e network at time t . Th e L yapu nov function L ( Q ( t )) can be define d as f ollowing: L ( Q ( t )) = X n,c Q c n ( t ) 2 (8) The L yapu nov drift ∆( Q ( t )) is d efined as fo llowing: ∆( Q ( t )) = E { L ( Q ( t + 1 )) − L ( Q ( t )) | Q ( t ) } (9) It has been already pr oven by [1], [2] that: ∆( Q ( t )) ≤ X n,c E { β c n ( t ) } − 2 X n,c Q c n ( t ) E { ψ c n ( t ) | Q ( t ) } (10) Such that: β c n ( t ) = X b µ c nb ( t ) ! 2 + A c n ( t ) + X a µ c an ( t ) ! 2 (11) and, ψ c n ( t ) = X b µ c nb ( t ) − X a µ c an ( t ) − A c n ( t ) (12) Maximizing P n,c Q c n ( t ) E { ψ c n ( t ) | Q ( t ) } in (1 0) which is equiv alent to the max imization p roblem defin ed in (1) yields the backpre ssure algorith m for sched uling an d r outing and it has been p roven b y [1], [2] that it supports the max imum capacity Λ . The average qu eue occup ancy bo und for back - pressure schedulin g and routin g is: ¯ Q ≤ ¯ β 2 ǫ (13) such that, ¯ Q = lim T → ∞ 1 T T X τ =0 X n,c E { Q c n ( τ ) } (14) ¯ β = lim T → ∞ 1 T T X τ =0 X n,c E { β c n ( τ ) } (15) ǫ = arg max x ≥ 0 ( λ c n + x ) n,c ∈ Λ (16) Where, ¯ Q is the average of total queue backlo g occ upancy . ¯ β is the sum o f the second moment of the scheduled tr ansmission rate out of each que ue plus the second moment of the sum of the ar riv als and scheduled tr ansmission rate in to ea ch q ueue and sum med over all queues. ǫ is the max imum positive number such that adding ǫ to each arrival r ate still makes them inside the capacity region Λ . B. Analysis of BW AR Here is a formal mathematical descrip tion of backp ressure with adaptive redundan cy . As before , let Q c n ( t ) to be qu eue backlog in node n of commod ity c at time slot t . W e define D c n ( t ) to be nu mber o f redund ant packets in nod e n of co m- modity c at time t . Redun dant packets are stored separa tely in redund ant buffers. Redund ant packets h ave lower p riority in such a way that no redun dant packet i s served unless the qu eue of orig inal packets is empty . For all time slots t , A c n ( t ) , θ ab ( t ) , µ ab ( t ) , µ c ab ( t ) an d ˜ µ c ab ( t ) are defined exactly as before. Arri val rates λ c n are also d efined as b efore. The qu eue dynamics in equation (7) is upda ted for ada ptiv e redu ndancy to be: Q c n ( t + 1) = max[ Q c n ( t ) − γ c n ( t ) − X b µ c nb ( t ) , 0] + A c n ( t ) + X a ˜ µ c an ( t ) (17) Where γ c n ( t ) is the number of origin al packets inside node n of commod ity c at time slot t that are known to be deliv ered by some dup licates to th e de stination using our BW AR strategy . One ideal model is th at we find out which packets are delivered immediately , ano ther is tha t we find ou t after some delay . Ou r analysis allows fo r any such k nowledge of d eliv ered p ackets. W e show later a practical timeout-based strategy for duplicate removals. Those γ c n ( t ) pac kets are needed to be r emoved from the queu e since they are alrea dy kn own to be delivered. W e assume that the d eletion happ ens during the time slot t hence at the beginning of time slot t non e of th ose p ackets are deleted yet but are kn own to be d eleted. The queu e dynamics in (1 7) consider on ly orig inal packets an d d oes n ot take into account the duplicate packets. W e define the re dunda nt buffer dynamics that are isolated f rom the original queue dynamics as following: D c n ( t + 1) = D c n ( t ) − ˜ γ c n ( t ) + δ c n ( t ) + X a ω c an ( t ) (18) Where ˜ γ c n ( t ) denotes the numb er of duplicates in node n o f com modity c at time t that are kn own to be a lready delivered to the destination and hence they must be removed. δ c n ( t ) is n umber of dup licates created at n ode n durin g slot t acc ording to the adaptive re dundan cy criteria. ω c ab ( t ) is the ac tual d uplicate tran smissions f rom node a to n ode b o f commod ity c at time t . BW AR algo rithm choo ses δ c n ( t ) an d ω c ab ( t ) in such away to assure that D c n ( t + 1) ≤ D max ∀ t . As befor e, Q ( t ) = ( Q c n ( t )) n,c is the vector of all queue backlog s at time t . Let U c n ( t ) to be the undelivered qu eue 5 backlog in nod e n of com modity c at tim e t . Hence, U c n ( t ) = Q c n ( t ) − γ c n ( t ) (19) Let U ( t ) = ( U c n ( t )) n,c be the vector of all queue backlogs of undelivered p ackets at time t . Let Γ ( t ) = ( γ c n ( t )) n,c be the vector of all remov ed dup licates at time t . Define the L y apunov function L ( X ) = P ( X i ) 2 . Assume th at ¯ Q , ¯ β and ǫ are defined as before in (14), (15) and (16) r espectively . Let also define, ¯ U = lim T → ∞ 1 T T X τ =0 X n,c E { U c n ( τ ) } (20) Γ 2 = lim T → ∞ 1 T T X τ =0 X n,c E n ( γ c n ( τ )) 2 o (21) Q. Γ = lim T → ∞ 1 T T X τ =0 X n,c E { Q c n ( τ ) .γ c n ( τ ) } (22) U. Γ = lim T → ∞ 1 T T X τ =0 X n,c E { U c n ( τ ) .γ c n ( τ ) } (23) Where, ¯ U is the average of total q ueue backlog occ upancy for un delivered p ackets in the main queu es. Γ 2 is the second moment of nu mber of removed pac kets in each original queue becau se those packets are known that are deliv ered b y duplicates to the destination a nd summ ed over all queues. Q. Γ is the jo int second moment o f n umber o f removed packets and the queu e backlog summ ed over all queu es. U. Γ is the joint second moment of numb er of removed packets and the queu e backlog of undelivered packets summ ed over all queues. For simplicity of exposition, we prove the r esult in the simple case whe n ar riv al r ates A c n ( t ) and the ch annel states θ ab ( t ) are i.i.d. over slots. This can be extend ed to g eneral ergodic (p ossibly non -i.i.d.) p rocesses using a T -slot drift argument as in [1 9]. Theorem 1 . If the chan nel states θ ab ( t ) ar e i.i.d . an d the arrival pr ocesses A c n ( t ) ar e i.i.d. with rates λ c n that are in side the capacity r egion Λ such tha t ( λ c n + ǫ ) n,c ∈ Λ for some ǫ > 0 , then BW A R stab ilizes a ll q ueues with the follo wing bou nd on the average of tota l queue occupan cy of un deliver ed p ackets ¯ U , ¯ U ≤ ¯ β − Γ 2 − 2 U. Γ 2 ǫ (24) Pr oo f: Sq uaring both sides of (17), Q c n ( t + 1) 2 ≤ ( Q c n ( t ) − γ c n ( t )) 2 + β c n ( t ) − 2 ( Q c n ( t ) − γ c n ( t )) ψ c n ( t ) (25) where β c n ( t ) and ψ c n ( t ) are defined as before in (11) an d (12) respectively . Summing over all n and c , X n,c Q c n ( t + 1) 2 ≤ X n,c ( Q c n ( t ) − γ c n ( t )) 2 + X n,c β c n ( t ) − 2 X n,c ( Q c n ( t ) − γ c n ( t )) ψ c n ( t ) (26) T aking the con ditional exp ectation E { . | Q ( t ) − Γ ( t ) } , E { L ( Q ( t + 1)) − L ( Q ( t ) − Γ ( t )) | Q ( t ) − Γ ( t ) } ≤ E ( X n,c β c n ( t ) − 2 X n,c ( Q c n ( t ) − γ c n ( t )) ψ c n ( t )      Q ( t ) − Γ ( t ) ) (27) Since our BW AR po licy maximizes (3) and hen ce (1) taking into acc ount the u ndelivered packets U ( t ) only , it will also maximize: E ( X n,c ( Q c n ( t ) − γ c n ( t )) ψ c n ( t )      Q ( t ) − Γ ( t ) ) (28) Howe ver, because ( λ c n + ǫ ) n,c are inside the capacity region Λ , we kn ow fro m [19] that the re exists a stationar y an d ran- domized algo rithm al g ∗ , which makes d ecisions indep enden t of Q ( t ) − Γ ( t ) , yielding ψ ∗ c n ( t ) that satisfy: E { ψ ∗ c n ( t ) } ≤ − ǫ ∀ n, c Because BW AR m aximizes (2 8), it follows that: E ( X n,c ( Q c n ( t ) − γ c n ( t )) ψ c n ( t )      Q ( t ) − Γ ( t ) ) ≤ E ( X n,c ( Q c n ( t ) − γ c n ( t )) ψ ∗ c n ( t ) ) = − X n,c ( Q c n ( t ) − γ c n ( t )) ǫ (29) Using this in (27) yields, E { L ( Q ( t + 1)) − L ( Q ( t ) − Γ ( t )) | Q ( t ) − Γ ( t ) } ≤ X n,c E { β c n ( t ) | Q ( t ) − Γ ( t ) } − 2 ǫ X n,c ( Q c n ( t ) − γ c n ( t )) (30) T aking iterative expectation, E { L ( Q ( t + 1)) } − E { L ( Q ( t ) − Γ ( t )) } ≤ X n,c E { β c n ( t ) } − 2 ǫ X n,c E { ( Q c n ( t ) − γ c n ( t )) } (31) Notice that: E { L ( Q ( t ) − Γ ( t )) } = E { L ( Q ( t ) } + E { L ( Γ ( t )) } − 2 E { Q ( t ) . Γ ( t ) } ( 32) Hence by summ ing over tim e slots τ ∈ { 0 , ..., T } and by telescoping, E { L ( Q ( T )) } − E { L ( Q (0)) } − T X τ =0 E { L ( Γ ( τ )) } + 2 T X τ =0 E { Q ( τ ) . Γ ( τ ) } ≤ T X τ =0 X n,c E { β c n ( τ ) } − 2 ǫ T X τ =0 X n,c E { ( Q c n ( τ ) − γ c n ( τ )) } (33) Dividing by T and tak ing the lim fo r T → ∞ im plies: 6 ¯ Q − ¯ Γ ≤ ¯ β + Γ 2 − 2 Q. Γ 2 ǫ (34) Now for und eliv ered p ackets ¯ U , we have by (19) and (3 4), ¯ U ≤ ¯ β − Γ 2 − 2 U. Γ 2 ǫ Remark: No te th at the computa tion of Γ 2 and U . Γ is determined b y the du plicate removal strategies. Dep ending o n these terms, the queue bound in this ab ove th eorem could be much lower than the queu e oc cupancy bou nd for regular backpr essure in (13). Thus we h av e a form al g uarantee that BW AR is no worse in terms of throug hput than backp ressure, and poten tially mu ch better in te rms of delay , since by Little’ s theorem a verage de lay is proportiona l to the average number of undelivered p ackets. W e will validate this find ing with mo del in the next sectio n. I V . M O D E L - B A S E D S I M U L AT I O N S A. The Cell-P artitioned Model The model in this pap er simp lifies th e control variables to be the whole transmission rates µ ab ( t ) for schedu ling and th e commod ity transmission rates µ c ab ( t ) for rou ting. W e simulate BW AR in the con text of en counter-based scheduling and ro uting for a simple model (cell-p artitioned network), which yields useful insights on its perfo rmance. In this idealized mo del the network deploym ent a rea is separ ated into disjoint cells and nodes have i.i. d. mobility m odel [2 0] as fo llows. W e have N no des an d C cells. At each slot t , node n can be inside any cell with equal prob abilities of 1 C . For collision an d interference simplicity , only on e transmission (one packet) is allowed in each cell in each time slot. Because of this we set q th = D max = 1 . An other simplifyin g assumption is that the nodes in the network are organized into pairs, acting as destinations to each oth er . Each nod e h as Bernoulli exog enous arrivals intended fo r its pa ir . Depen ding on the nu mber of cells C in the network we ca n choose the right number of th e nodes N ≈ 1 . 79 · C in order to maximize throug hput as shown in [ 20]. Our simulation resu lts show that by optimizing number o f nodes based o n the number o f cells to maximize th rough put, the d elay also is improved. W e co nsider in our simulations, ne tworks of sizes 9 , 12 , 16, 20, and 25 cells in the network. And for optim ality , nu mber of nodes are chosen to be 16, 20 , 28, 34 , a nd 44 respectively . For timeout duplicate removals we set the time out value P = C . Here we show how BW AR works in the cell-partitioned network with the simplifyin g assump tion that only on e trans- mission is allo wed per c ell per tim e slot. Each tim e slot t an d for ea ch c ell l we choose two n odes a ∗ and b ∗ and co mmod ity c ∗ such that: • a ∗ and b ∗ are in cell l . • Q c ∗ a ∗ ( t ) − Q c ∗ b ∗ ( t ) ≥ Q c a ( t ) − Q c b ( t ) ; f or all c , for all a an d b in cell l at time slot t . T his captures the maximiz ation of queue differentials of the main qu eues. • I f th ere exists a, b in cell l such tha t, Q b a ( t ) − Q b b ( t ) = Q c ∗ a ∗ ( t ) − Q c ∗ b ∗ ( t ) then c ∗ = b ∗ . This captures th e destination advantage. • I f th ere exists a, b in cell l and c such that Q c a ( t ) − Q c b ( t ) = Q c ∗ a ∗ ( t ) − Q c ∗ b ∗ ( t ) a nd { c ∗ 6 = b ∗ or [( c = b ) and ( c ∗ = b ∗ )] } then ( Q c a ( t ) + D c a ( t )) − ( Q c b ( t ) + D c b ( t )) ≤ ( Q c ∗ a ∗ ( t ) + D c ∗ a ∗ ( t )) − ( Q c ∗ b ∗ ( t ) + D c ∗ b ∗ ( t )) . This captures th e maximization of duplicate b uffer dif f erentials if there are some ties in main queue differentials. The alg orithm simply assigns µ c ∗ a ∗ b ∗ ( t ) a value of 1 , and assigns all other µ c ab ( t ) a value of 0 such that a, b in c ell l . When a transmission is made fro m node a to nod e b of commod ity c a t time slot t an d that transm ission will make Q c a ( t + 1) + D c a ( t + 1) = 0 then th is tran smitted p acket is duplicated an d store d in the dup licate buffer of no de a ma king D c n ( t ) = 1 instead o f 0 . Duplicate pac kets are served o nly if there are n o orig inal packets to tran smit. There is strict lower priority of duplicate packets compar ed to origin al packets. B. Pr oto col V ariants In the simulation s, we impleme nt and compar e fi ve different routing proto col variants. They ar e described as follows: • Reg ular Backpressur e (RB) : This is the basic back pres- sure scheduling a nd routing mechanism, wh ere decisions are made purely based on queu e d ifferentials. • Reg ular Backpressur e with Destination Advantage (RB-D A) : This is a sligh t m odification in which packets correspo nding to the destinatio n are prioritized when the destination is encounte red. As we show , this alread y yields significan t delay improvements over regular back - pressure. • BW AR with Ideal pack et removal and original packets retained in t he Main queue (BW AR -IM) : This is o ur novel bac kpressure with ad aptive redun dancy in wh ich the destination advantage is also holds. Her e, when an original packet is duplicated th e or iginal packet remains in the m ain queue wh ile the d uplicate is sto red in the du- plicate buf fer . W e assume he re whenever a packet reaches the destination, all of its duplicates are deleted inclu ding the o riginal o ne in the ma in q ueue in stantaneou sly . • BW AR with Ideal pack et removal and original packets moved to Duplicate bu ffer upon copy (BW AR-ID) : This is very similar to BW AR-IM. Th e only difference is that when ev er an original packet is d uplicated b oth the original pa cket an d the du plicate are stored in the duplicate buf fer (of course in two different no des o ne in the r eceiver and the other in th e sen der respectively). • BW AR with T ime-out based packet remov al a nd original packets moved to Dupl icate buffer up on copy (BW AR-TD) : This is a practical impleme ntation of BW AR in which du plicates are de leted from the d uplicate buf fer after a pred efined timeo ut v alue P ha s passed since th e first time the original packet is admitted to the network . H owe ver, th e orig inal packet that is kept in dup licate buf fer is flagg ed and will n ot be deleted 7 15 20 25 30 35 40 45 0 10 20 30 40 50 60 70 80 Number of Nodes (N) Averae Delay RB RB−DA BWAR−IM BWAR−ID BWAR−TD S&W (a) Delay as we v ary N for low λ = 0 . 001 10 −3 10 −2 10 −1 10 0 10 1 10 2 10 3 Load( λ ) Average Delay RB RB−DA BWAR−IM BWAR−ID BWAR−TD (b) Delay as we va ry λ for N = 44 of backpressure vari ants Fig. 1. Comparing delay performance of protocol varian ts: RB, RB-D A, BW AR-IM, BW AR-ID, BW AR-TD and S&W under the cell-pa rtition ed model. when a timeout occu rred. It is only deleted if it gets acknowledged directly by the destination if its alrea dy received or otherwise it moved back to the main que ue when it encou nters the destination . • Spra y and W ait (S&W) : This is not a backpressure based mec hanism. Sp ray and W ait is pr esented by T . Spyropou los et a l. [14] which is a state of the art routing scheme in intermitten tly co nnected mobile n et- works. S&W crea tes a pred efined fixed n umber of co pies (spraying ) of the pa cket when admitted to the network. Those c opies ar e d istributed to d istinct nodes and then each co py waits until it enco unters the destination . W e implemented S&W for co mparison with BW AR. Our results show that BW AR outperf orms S& W especially in h igh load scenarios. The ev aluations are conducted using a custom simulator written in C++ (fo r repeatability , we make our code av ailable online at http ://anrg.usc.edu /downloads/ ). E ach s imulation runs for one million time slots. In figure 1(a), we sh ow av erage delay of all ab ove p rotoco l variants as nu mber o f n odes N vary f or low lo ad λ = 0 . 00 1 out of the per no de cap acity region Λ node = [0 , 0 . 1 4] . Delay is reduced significan tly when BW AR is used . For this low load scenario all BW AR variants ha ve almo st the same average delay and they perf orm sligh tly better than Spray and W ait. Figure 1 (a) also shows the great dramatic d elay imp rovement of destination advantage with out any redund ancy in RB-D A compare d to regular b ackpressur e RB. Figure 1(b) compa res the average delay o f all variants of b ackpressure -based pro tocols as we vary the load. As expected, as the loa d increases the d elay improvement of BW AR declines com pared to RB-D A. Figur e 1(b) also shows how BW AR-ID performs m uch b etter co mpared to BW AR-IM beyond so me thr eshold of load( λ ). This shows how moving the du plicated origin al p acket to the dup licate buffer has g reat delay enhan cement for high load scen arios. In Figure 2, results show h ow BW AR mechan ism outper- forms Spray and W ait (S&W) delay p erform ance for high load. It shows also how BW AR suppo rts almo st twice th e capacity region of S&W . 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 20 40 60 80 100 120 140 Load( λ ) Average Delay S&W BWAR−TD BWAR−ID Fig. 2. Compari ng S&W delay with BW AR-ID and BW AR-TD as we vary λ for N = 44 10 −3 10 −2 10 −1 0 2 4 6 8 10 12 14 x 10 6 Load( λ ) Number of Transmissions RB RB−DA BWAR−IM BWAR−ID BWAR−TD S&W Fig. 3. Comparin g energy consumption as we vary λ for N = 44 under the cell-p artitio ned m odel. Surprisingly in figure 3, BW AR-I M has a better total number of transmissions comp ared to regula r backpressur e RB-D A for low load d espite the flooding duplicates natu re of BW AR at low load. Spray and W ait ha s sup erior energy consump tion perfor mance co mpared to all backpressure- based protoco l variants con sidered. For futur e work, we intend to 8 study the po ssibility of having both power o ptimization and adaptive redundan cy featu res to be en abled o n backpre ssure. 10 0 10 1 10 2 10 3 10 4 10 0 10 1 10 2 10 3 Timeout Average Delay λ = 0.001 (BWAR−TD) λ = 0.016 (BWAR−TD) λ = 0.064 (BWAR−TD) λ = 0.128 (BWAR−TD) λ = 0.001 (BWAR−ID) λ = 0.016 (BWAR−ID) λ = 0.064 (BWAR−ID) λ = 0.128 (BWAR−ID) Fig. 4. Time out effe ct of BW AR-TD and comparing it with BW AR-ID for dif ferent λ ∈ { 0 . 001 , 0 . 016 , 0 . 064 , 0 . 128 } for N = 44 under the cell- partit ioned model. Figure 4 studies the effect of timeo ut value P of BW AR- TD f or r emoving dup licates un der d ifferent load scen arios a nd compare s its delay p erform ance with ideal duplicate r emovals in BW AR-ID. V . R E L AT E D W O R K The first theoretical work on backpressure schedulin g is the classic result by T assiulas and Eph remides in 1 992, proving that this queue-d ifferential based scheduling mechanism is throug hput op timal (i.e., it can stabilize any feasible rate vector in a n etwork) [1]. Sin ce then, r esearchers have combined the basic back pressure mech anism with utility op timization to provide a c ompreh ensive app roach to stochastic n etwork optimization [2], [21], [22]. Of most relev ance to this work are paper s on delay enhance- ments to backpressure. A num ber o f p apers [23]–[25] address the u tility-delay trad eoff in o ptimization- oriented b ackpres- sure, to ob tain a tra deoff based on a V parame ter such that the utility is improved by a factor of O (1 /V ) while the delay is made to be p olylogar ithmic in V . Su ch a tradeoff has been shown to be practically achievable using LIFO queu eing in [26], at the cost of a small pr obability of d roppin g p ackets. The first-e ver im plementation of dynamic ba ckpressure routing aimed f or wireless sensor networks (BCP) [9] uses such a LIFO mechan ism. As our focus in this work is no t on utility optimization , the techn iques p resented in these works are somewhat orth ogona l to the red undan cy ap proach we develop here. Anoth er set of paper s [3], [27], [2 8] consider the u se of shortest path rou ting in con junction with backpre ssure to im- prove the delay perfor mance. These techn iques are well su ited for static networks in which such p aths can be comp uted; howe ver , since our fo cus is on en counter based networks with limited connectivity , such an app roach is not ap plicable. In [29 ], the autho rs present a mechanism whe reby o nly one real qu eue is maintained for ea ch neighb or, along with virtual co unters/shadow queues for all destination s, and sh ow that this y ields delay improvements. And in [ 5], a novel variant of backpressure sched uling mech anism is prop osed which u ses head o f line packet delay instead of queue lengths as the basis o f th e bac kpressure weight calculation for each link/comm odity , also yielding enhan ced delay per forman ce. Howe ver, these w orks both assume the existence of static fixed routes. It would be interesting to explo re in futur e work whether their tech niques c an be app lied to intermittently connected enc ounter-based mobile network s, and if so , how these approach can be fu rther en hanced by the use of the adaptive redundan cy that we pr opose in this work. Ryu et al. present two works on back pressure rou ting aimed sp ecifically fo r cluster-based intermittently co nnected networks [10], [30]. I n [30], the authors develop a two- phase r outing sch eme, co mbining backpr essure r outing with source rou ting for clu ster-based networks, separating intra- cluster routin g from inter-cluster rou ting. They show that this ap proach results in large queues at only a subset o f the nod es, y ielding smaller delay s than conventional back- pressure. In [1 0], the authors impleme nt th e above-mentio ned algorithm in a real experimental network and show the delay improvements empirically . The key difference of these works from our s is that we do no t make any assumption ab out the intermittently co nnected network being o rganized in a cluster- based hierarch y . Dvir and V asilakos [31] also consider back pressure rout- ing for interm ittently con nected n etworks, with lin k weights similar to that used in BCP [9]. They ev aluate W eighted Fair Queueing in add ition to LI FO and show th rough simula tions that it offers energy imp rovements. Th eir work do es n ot explicitly address additional delay improvemen ts needed f or these k inds of networks. There is a rich literature on routing in delay toleran t / intermittently co nnected encou nter based mobile networks (see [32] for a com prehen si ve sur vey). Altho ugh there ex- ist single- copy routin g me chanisms f or such networks [1 3], it has be en well-recog nized that rep lication is helpf ul in reducing delay . While basic epid emic routing [33] creates multiple message replicas for reliab le, fast d eliv ery , it incurs too h igh o f a transmission cost. Sma rter mu lti-copy rou ting mechanisms have therefor e bee n developed such as Spra y and W ait [14], and SARP [34]. These work s introduce redund ant packet tran smissions to impr ove d elay . Howe ver, all of these approa ches ar e no t adaptive to the traffic and th erefor e will hurt the thro ughpu t p erform ance of the network. This h as been noted b efore, by the au thors of [10], who write that “replication- based algorith ms such as epidemic r outing for DTNs ... result in lower thro ughp ut since m ultiple copies of a piece o f data need to be fo rwarded an d stored ( and therefor e not thro ughpu t optimal). ” I n fact, in [20], it has been theoretically proved th at capacity of such sch emes that use fixed redun dancy is necessarily lower . In this work , we presen t the first back pressure algorithm th at uses replica tion in an adaptive manner so as to maintain thr oughp ut optimality while reducing delay . W e explicitly comp are ou r BW AR scheme with Spray a nd W ait, an d sho w th roug h our ev alu ation that not 9 only d oes it provide similar , even better , delay p erform ance, it does so witho ut hur ting thr ough put optimality; specifically , we show that BW AR can hand le m uch high er tra ffic loa d than Spray and W ait. T o summarize, this paper on BW AR is the fir st work that explicitly combines the b est of both worlds: multi-copy r outing for intermitten tly connected networks and thr ough put-op timal backpr essure sche duling. This combination yields better d elay perfor mance than trad itional backp ressure, particular ly at low loads, and better ability to han dle hig h tr affic than traditional DTN/ICN routin g schemes. V I . C O N C L U S I O N A N D F U T U R E W O R K W e have presented in this p aper BW AR, an enh anced backpr essure algorithm tha t intro duces ada ptiv e redu ndancy to improve d elay per forman ce. W e h av e proved analytically that this algor ithm is also thro ughpu t o ptimal while p roviding a better dela y bou nd, particu larly at low load settings. Thr ough simulation results we ha ve shown that BW AR ou tperfor ms both trad itional bac kpressure (at low load s) an d conventional DTN-routin g m echanisms (a t high lo ads) in encou nter-based mobile networks. There a re a few open avenues for futur e work suggested b y our study . First, we would like to und ertake a more carefu l analysis of the delay improvements ob tained, relating them more explicitly , fo r instance, to ar riv al pro cess parameters and th e u nderly ing mobility m odel. Second, th e im provements obtained by BW AR in terms of delay are ob tained at the ex- pense of greater number of transmissions due to the introd uced redund ancy . While th is may be accep table in som e n etworks, for e nergy-con strained networks this could be a co ncern. W e therefor e plan to explore the desig n of energy-efficient variants o f BW AR in the future, in which the redu ndancy can be con trolled to provide a tu nable tradeo ff b etween en ergy and delay . W e would also like to investigate automated self- configur ation of the timeo ut parameter for duplicate removal throug h a distributed mecha nism, as th is is curren tly statically configur ed in BW AR. R E F E R E N C E S [1] L. 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