Backpressure with Adaptive Redundancy (BWAR)

Transcription

1 Bakpressure with Adaptive Redundany (BWAR) Majed Alresaini alresain AT us. edu Maheswaran Sathiamoorthy msathiam AT us. edu Bhaskar Krishnamahari bkrishna AT us. edu Mihael J. Neely mjneely AT us. edu arxiv: v [s.ni] 9 Aug 20 Abstrat Bakpressure sheduling and routing, in whih pakets are preferentially transmitted over links with high queue differentials, offers the promise of throughput-optimal operation for a wide range of ommuniation networks. However, when the traffi load is low, due to the orresponding low queue oupany, bakpressure sheduling/routing experienes long delays. This is partiularly of onern in intermittent enounter-based mobile networks whih are already delay-ited due to the sparse and highly dynami network onnetivity. While state of the art mehanisms for suh networks have proposed the use of redundant transmissions to improve delay, they do not work well when the traffi load is high. We propose in this paper a novel hybrid approah that we refer to as bakpressure with adaptive redundany (BWAR), whih provides the best of both worlds. This approah is highly robust and distributed and does not require any prior knowledge of network load onditions. We evaluate BWAR through both mathematial analysis and simulations based on ell-partitioned model. We prove theoretially that BWAR does not perform worse than traditional bakpressure in terms of the maximum throughput, while yielding a better delay bound. The simulations onfirm that BWAR outperforms traditional bakpressure at low load, while outperforming a state of the art enounter-routing sheme (Spray and Wait) at high load. I. INTRODUCTION Queue-differential bakpressure sheduling and routing was shown by Tassiulas and Ephremides to be throughput optimal in terms of being able to stabilize the network under any feasible traffi rate vetor []. Additional researh has extended the original result to show that bakpressure tehniques an be ombined with utility optimization, resulting in simple, throughput-optimal, ross-layer network protools for all kinds of networks [2] [5], [29]. Reently, some of these tehniques have been translated to pratially implemented routing and rate-ontrol protools for wireless networks [6] [0]. The basi idea of bakpressure mehanisms is to prioritize transmissions over links that have the highest queue differentials. Bakpressure effetively makes pakets flow through the network as though pulled by gravity towards the destination, whih has the smallest queue size of 0. Under high traffi onditions, this works very well, and bakpressure is able to fully utilize the available network resoures in a highly dynami fashion. Under low traffi onditions, however, beause many other nodes may also have a small or 0 queue size, there is This material is supported in part by the following: NSF grant 04954, the Network Siene Collaborative Tehnology Alliane sponsored by the U.S. Army Researh Laboratory W9NF , and King Saud University. ineffiieny in terms of an inrease in delay, as pakets may loop or take a long time to make their way to the destination. In this paper, we fous primarily on intermittently onneted networks, suh as enounter-based mobile networks (sometimes also referred to as delay or disruption tolerant networks (DTN)). In suh networks, onventional path-disovery-based MANET routing tehniques like AODV [] and DSR [2] are not feasible beause the network may not form a single onneted partition at any time, and thus a full path may never exist between the soure and the destination. Instead, it is neessary to use store-and-forward type protools that an handle the underlying mobility. A bakpressure based routing sheme an be easily implemented in suh a network, with the deision of what information to exhange being made between eah pair of nodes based on their queue differentials whenever they enounter eah other. However the abovementioned delay ineffiieny of the bakpressure mehanism at low traffi loads is further exaerbated in suh networks, beause they are already delay-ited due to sparse network onnetivity. In the literature on intermittently onneted networks, there are several proposed shemes for store-and-forward based routing, suh as [3] [8]. Some of these, suh as Spray and Wait, advoate the use of redundant transmissions, to make additional opies of the ommuniated information in the network. The repliation of the ontent makes it faster for the destination to aess a opy. However, as the additional repliation always inreases the network load, these protools, whih are not throughput-optimal to begin with, suffer additional ongestion. In this paper, we propose a novel hybrid approah, an adaptive redundany tehnique for bakpressure routing, that yields the benefits of repliation to redue delay under low load onditions, while at the same time preserving the performane and benefits of traditional bakpressure routing under high traffi onditions. This tehnique, whih we refer to as bakpressure with adaptive redundany (BWAR), essentially reates opies of pakets in a new dupliate buffer upon an enounter, when the transmitter s queue oupany is low. These dupliate pakets are transmitted only when the original queue is empty. This mehanism an dramatially improve delay of bakpressure during low load onditions due to two reasons: () due to the existene of multiple opies of the same pakets at multiple nodes, the destination is more likely to enounter a massage intended for it. (2) this way, the algorithm builds up

2 2 gradients towards the destinations faster and redues paket looping. The additional transmissions inurred by BWAR due to the dupliates utilize available slots whih would otherwise go idle, in order to redue the delay. Partiularly for networks that are not energy-ited, this offers a more effiient way to utilize the available bandwidth during low load onditions. In order to minimize the storage resoure utilization of dupliate pakets, ideally, these dupliate pakets should be removed from the network whenever a opy is delivered to the destination. Sine this may be diffiult to implement (exept in some kinds of networks with a separate ontrol plane), we also propose and evaluate a pratial timeout mehanism for automati dupliate removal. Under high load onditions, beause queues are rarely empty, dupliates are rarely reated, and BWAR effetively reverts to traditional bakpressure and inherits its throughput optimality property. By design, BWAR is highly robust and distributed and does not require prior knowledge of loations, mobility patterns, and load onditions. The following are the key ontributions of this work: We propose BWAR, a new adaptive redundany tehnique for bakpressure sheduling/routing in intermittently onneted networks. And we present a timeout mehanism for dupliate removal, whih allows BWAR to be easily implemented in pratie. We develop an analytial model of BWAR, and prove theoretially that it yields a smaller upper bound on the average queue size (and hene the average delay) than traditional bakpressure, while retaining throughput optimality. Through simulations using an idealized ell-partition mobility model, we quantify the benefits from using BWAR. Speifially, we show that it outperforms both traditional bakpressure and Spray & Wait [5], a state of the art DTN/ICN routing mehanism. The rest of the paper is organized as follows. In setion II, we introdue and desribe BWAR. In setion III-A, we review the theory behind traditional bakpressure sheduling and routing. We show in setion III-B the queue dynamis for BWAR and how it an improve the delay theoretially. In setion IV we present our model-based simulation results. In setion V, we desribe related work in this subjet to plae our ontributions in ontext. We onlude in setion VI and disuss future work. II. BACKPRESSURE WITH ADAPTIVE REDUNDANCY In this setion, we first desribe traditional bakpressure sheduling and routing and then our new proposal for bakpressure sheduling/routing with adaptive redundany (BWAR). In both ases, we assume that there are N nodes in the network, and time is disretized. We assume a multiommodity flow system in whih every node ould be a potential destination (orresponding to a partiular ommodity). A. Traditional Bakpressure Sheduling and Routing We assume that eah node maintains N queues, one for eah ommodity, with the j th queue at eah node ontaining pakets that are destined for node j. Let Q i (t) indiate the number of pakets destined to node queued at node i at time t. Naturally, Q i i (t) = 0 t. Let µ ij (t) be the sheduling and routing variable that indiates the number of pakets of ommodity to be sheduled on link (i, j). Traditional bakpressure sheduling/routing [], [2] selets the µ ij (t) that solve the following problem (a form of maximum weight independent set seletion): max i,j, ij (t) µ ij (t) subjet to, µ ij(t) θ ij (t), i, j µ ij (t) µd km (t) = 0, ((i,j),(k,m)) Ω(t),, d () Where ij (t) = Q i (t) Q j (t) is the link weight, whih denotes the queue differential for ommodity on link (i, j) at slot t and the feasibility onstraints on µ ij (t) pertain to the available network apaity, taking into aount the interferene between nodes. θ ij (t) is the hannel state in terms of number of pakets that an be transmitted over link (i, j) during slot t. Ω(t) is the link interferene set at slot t suh that if link (i,j) interferes with link (i,j ) at slot t then ((i,j),(i,j )) Ω(t) and hene, those two links an not be both sheduled at slot t. The maximization problem in () an be solved by finding the maximum ommodity ij (t) for eah link (i,j) at slot t that maximizes ij (t) and assign µ ij (t) = 0 for all ij (t) and then solve, subjet to, max i,j ij (t) ij (t) µ ij (t) (t) µ ij (t) ij (t) θ ij (t), i, j µ ij (t) ij (t) µ km (t) km (t) = 0, ((i,j),(k,m)) Ω(t) (2) B. BWAR Sheduling and Routing Our proposed enhanement of bakpressure with adaptive redundany works as follows. We have an additional set of N dupliate buffers of size D max at eah node. Besides the original queue oupany Q i (t) whih has the same meaning as in traditional bakpressure, the dupliate queue oupany is denoted by Di (t), that indiates the number of dupliate pakets at node i that are destined to node at time t. Again, Q i i (t) = Di i (t) = 0 t sine destinations need not buffer any pakets intended for themselves. The dupliate queues are maintained and utilized as follows: Original pakets when transmitted are removed from the main queue; however, if the queue size is lower than a ertain threshold q th, then the transmitted paket is dupliated and kept in the dupliate buffer assoiated with its destination if it is not full otherwise no dupliate is reated. We found that setting both q th and D max to the ij

3 3 value of the maximum link servie rate is enough and gives superior delay results. Dupliate pakets are not removed from the dupliate buffer when transmitted. They are only removed when they are notified to be reeived by the destination, or a pre-defined timeout has ourred. When a ertain link is sheduled for transmission, the original pakets in the main queue are transmitted first. If no more original pakets are left, only then dupliates are transmitted. Thus the dupliate queue has a stritly lower priority. Similar to original bakpressure sheduling/routing, the BWAR sheduling/routing also requires the solution of a similar maximum weight independent set problem: max i,j, BWAR,ij(t) µ ij(t) subjet to, µ ij(t) θ ij (t), i, j µ ij (t) µd km (t) = 0, ((i,j),(k,m)) Ω(t),, d (3) We define an enhaned link weight for BWAR, BWAR,ij (t) as follows, to take into aount the oupany of the dupliate buffer. ( ) j= And Q i (t)+d i (t)>0 BWAR,ij (t) = ( Q i (t) Q j (t)) ( D 4D i (t) Dj(t) ) max (4) Here the indiator funtion j= And Q i (t)+di (t)>0 denotes that node j is the final destination for the onsidered ommodity. This gives higher weight to ommodities that enounter their destinations. We show later how this effetively results in dramati delay improvement. Similarly, the maximization problem in (3) an be solved first by finding the maximum ommodity BWAR,ij (t) for eah link (i,j) at slot t that maximizes BWAR,ij (t) followed by the same approah disussed earlier in II-A. It is important to notie that a solution to (3) is indeed a solution to () assuming that Q i (t) and µ ij (t) are integers. The small weight added in (4) gives advantage first to links/ommodities whih enounter the destination and then to higher dupliate buffer deferential to inrease the hane of serving dupliates. The small frations in (4) assures this priority when there are ties in () to boost delay performane. C. Bakpressure routing in intermittently onneted networks In general bakpressure sheduling is NP-hard, owing to the MWIS problem that needs to be solved at eah time. However, in this paper, we fous on intermittently onneted networks, that onsist of sparse enounters between pairs of nodes. Therefore, at any given time, the size of any onneted omponent of the network is very small. In this ase, the sheduling problem is dramatially simplified. D. Pratial Dupliate Removal As an be seen from the above desription, BWAR reates dupliate pakets whenever the transmitter s queue oupany is low. In an ideal setting, for effiieny, the dupliated pakets in the network should be deleted instantaneously when any opy is delivered to the intended destination. This ould only be implemented pratially in intermittently onneted networks where a entralized ontrol plane is available that an provide suh an instantaneous aknowledgement to all nodes in the network. In other ases, some other mehanism is sought, so we propose the following timeout mehanism. Whenever a paket arrives into the network, it is time-stamped. After a timeout period P from that arrival time, any dupliate opies of that paket at any node in the network will be deleted. To obtain higher delay performane improvement, when an original paket is dupliated, it is plaed in the dupliated buffer giving it lower servie priority, however, it is flagged and not deleted when a timeout ourred. It is only removed when it gets aknowledged diretly by the destination. In the next setion we undertake an analysis of the performane of BWAR and ompare it with the known results for traditional bakpressure routing. Speifially, we prove that any feasible rate vetor is also stabilized by BWAR, and the bound that we an give on the expeted queue oupany for BWAR is better than that for regular bakpressure. III. MATHEMATICAL ANALYSIS A. Review of the Analysis of Basi Bakpressure We onsider a timeslotted network with N nodes that ommuniate with eah other. Pakets arrive to eah node, and eah paket must be delivered to a speifi destination, possibly via a multi-hop path. Eah node maintains several queues, one per destination, to store pakets. Eah queue has the following dynamis: Q(t+) = 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 queue at time t, and A(t) is the total paket arrivals to the queue at time t. Eah time slot, we observe the queue states and the hannel states and make sheduling and routing deisions based on this information. To lear this out, let Q n(t) be the queue baklog (number of pakets) in noden {,2,...,N} that are destined for node {,...,N}\{n} at slot t. Let A n(t) be the exogenous paket arrivals that ome to node n and destined to node at time t with rate λ n. Exogenous arrivals are the pakets that just entered the network. Endogenous arrivals, however, are arrivals from other nodes and were already inside the network. Pakets may be forwarded to several nodes before reahing the destination. Let us define the apaity region Λ to be the set of all possible arrival rate vetors (λ n ) that are stabilizing by some sheduling and routing strategy. Let θ ab (t) be the hannel state from node a to node b at time t in terms of how many pakets an be transmitted. Let µ ab (t) be

4 4 the sheduled servie rate from node a to node b at slot t. Let µ ab (t) be the servie rate for ommodity routed from node a to node b at time t and must satisfy: µ ab(t) µ ab (t) θ ab (t) (6) The queue dynamis for eah time slot and for eah queue is the following: Q n (t+) = max[q n (t) b µ nb (t),0] +A n (t)+ µ an (t) (7) a Where µ is the atual transfer rate due to insuffiient pakets in the queue. For example, on some slots we may able to send 5 pakets, but we only send 3, beause only 3 were available in the queue. In equation (7), A n (t) are the exogenous arrivals and a µ an(t) are the endogenous arrivals to node n. Define the vetor Q(t) = (Q n(t)) to be the vetor of all queues in the network at time t. The Lyapunov funtion L(Q(t)) an be defined as following: L(Q(t)) = Q n(t) 2 (8) The Lyapunov drift (Q(t)) is defined as following: (Q(t)) = E{L(Q(t+)) L(Q(t)) Q(t)} (9) It has been already proven by [], [2] that: (Q(t)) E{β n(t)} 2 Q n(t)e{ψ n(t) Q(t)} Suh that: ( 2 ( βn(t) = µ nb(t)) + A n(t)+ a and, b (0) µ an(t)) 2 () ψn(t) = µ nb(t) µ an(t) A n(t) (2) b a Maximizing Q n(t)e{ψn(t) Q(t)} in (0) whih is equivalent to the maximization problem defined in () yields the bakpressure algorithm for sheduling and routing and it has been proven by [], [2] that it supports the maximum apaity Λ. The average queue oupany bound for bakpressure sheduling and routing is: suh that, Q = β = ǫ = Q β 2ǫ (3) T E{Q n(τ)} (4) T E{βn (τ)} (5) argmax(λ n +x) Λ (6) x 0 Where, Q is the average of total queue baklog oupany. β is the sum of the seond moment of the sheduled transmission rate out of eah queue plus the seond moment of the sum of the arrivals and sheduled transmission rate into eah queue and summed over all queues. ǫ is the maximum positive number suh that adding ǫ to eah arrival rate still makes them inside the apaity region Λ. B. Analysis of BWAR Here is a formal mathematial desription of bakpressure with adaptive redundany. As before, let Q n(t) to be queue baklog in node n of ommodity at time slot t. We define Dn(t) to be number of redundant pakets in node n of ommodity at time t. Redundant pakets are stored separately in redundant buffers. Redundant pakets have lower priority in suh a way that no redundant paket is served unless the queue of original pakets is empty. For all time slots t, A n (t), θ ab(t), µ ab (t), µ ab (t) and µ ab (t) are defined exatly as before. Arrival rates λ n are also defined as before. The queue dynamis in equation (7) is updated for adaptive redundany to be: Q n (t+) = max[q n (t) γ n (t) b +A n(t)+ a µ nb (t),0] µ an(t) (7) Whereγn(t) is the number of original pakets inside noden of ommodityat time slot t that are known to be delivered by some dupliates to the destination using our BWAR strategy. One ideal model is that we find out whih pakets are delivered immediately, another is that we find out after some delay. Our analysis allows for any suh knowledge of delivered pakets. We show later a pratial timeout-based strategy for dupliate removals. Those γn (t) pakets are needed to be removed from the queue sine they are already known to be delivered. We assume that the deletion happens during the time slot t hene at the beginning of time slot t none of those pakets are deleted yet but are known to be deleted. The queue dynamis in (7) onsider only original pakets and does not take into aount the dupliate pakets. We define the redundant buffer dynamis that are isolated from the original queue dynamis as following: D n (t+) = D n (t) γ n (t)+δ n (t)+ a ωan (t) (8) Where γ n (t) denotes the number of dupliates in node n of ommodity at time t that are known to be already delivered to the destination and hene they must be removed. δn (t) is number of dupliates reated at node n during slot t aording to the adaptive redundany riteria. ωab (t) is the atual dupliate transmissions from node a to node b of ommodity at time t. BWAR algorithm hooses δn (t) and ωab (t) in suh away to assure that D n(t+) D max t. As before, Q(t) = (Q n (t)) is the vetor of all queue baklogs at time t. Let Un (t) to be the undelivered queue

5 5 baklog in node n of ommodity at time t. Hene, U n(t) = Q n(t) γ n(t) (9) Let U(t) = (Un (t)) be the vetor of all queue baklogs of undelivered pakets at time t. Let Γ(t) = (γn(t)) be the vetor of all removed dupliates at time t. Define the Lyapunov funtionl(x) = (X i ) 2. Assume that Q, β andǫare defined as before in (4), (5) and (6) respetively. Let also define, T Ū = E{U n(τ)} (20) Γ 2 = Q.Γ = U.Γ = T E T { (γ n (τ))2} (2) E{Q n (τ).γ n (τ)} (22) T E{Un(τ).γ n(τ)} (23) Where, Ū is the average of total queue baklog oupany for undelivered pakets in the main queues. Γ 2 is the seond moment of number of removed pakets in eah original queue beause those pakets are known that are delivered by dupliates to the destination and summed over all queues. Q.Γ is the joint seond moment of number of removed pakets and the queue baklog summed over all queues. U.Γ is the joint seond moment of number of removed pakets and the queue baklog of undelivered pakets summed over all queues. For simpliity of exposition, we prove the result in the simple ase when arrival rates A n(t) and the hannel states θ ab (t) are i.i.d. over slots. This an be extended to general ergodi (possibly non-i.i.d.) proesses using a T-slot drift argument as in [9]. Theorem. If the hannel states θ ab (t) are i.i.d. and the arrival proesses A n (t) are i.i.d. with rates λ n that are inside the apaity regionλsuh that(λ n+ǫ) Λ for someǫ > 0, then BWAR stabilizes all queues with the following bound on the average of total queue oupany of undelivered pakets Ū, Ū β Γ 2 2U.Γ 2ǫ Proof: Squaring both sides of (7), Q n(t+) 2 (Q n(t) γ n(t)) 2 +β n(t) (24) 2(Q n (t) γ n (t))ψ n (t) (25) whereβn(t) andψn(t) are defined as before in () and (2) respetively. Summing over all n and, Q n(t+) 2 (Q n(t) γn(t)) 2 + βn(t) 2 (Q n (t) γ n (t))ψ n (t) (26) Taking the onditional expetation E{. Q(t) Γ(t)}, E{L(Q(t+)) L(Q(t) Γ(t)) Q(t) Γ(t)} { } E βn (t) 2 (Q n (t) γ n (t))ψ n (t) Q(t) Γ(t) (27) Sine our BWAR poliy maximizes (3) and hene () taking into aount the undelivered pakets U(t) only, it will also maximize: { } E (Q n(t) γn(t))ψ n(t) Q(t) Γ(t) (28) However, beause (λ n +ǫ) are inside the apaity region Λ, we know from [9] that there exists a stationary and randomized algorithm alg, whih makes deisions independent of Q(t) Γ(t), yielding ψ n (t) that satisfy: E{ψ n (t)} ǫ Beause BWAR maximizes (28), it follows that: { } E (Q n(t) γn(t))ψ n(t) Q(t) Γ(t) { } E (Q n (t) γ n (t))ψ n (t) = (Q n (t) γ n (t))ǫ (29) Using this in (27) yields, E{L(Q(t+)) L(Q(t) Γ(t)) Q(t) Γ(t)} E{βn (t) Q(t) Γ(t)} 2ǫ (Q n (t) γ n (t)) (30) Taking iterative expetation, E{L(Q(t+))} E{L(Q(t) Γ(t))} E{βn (t)} 2ǫ E{(Q n (t) γ n (t))} (3) Notie that: E{L(Q(t) Γ(t))} = E{L(Q(t)}+E{L(Γ(t))} 2E{Q(t).Γ(t)} (32) Hene by summing over time slots τ {0,...,T} and by telesoping, T E{L(Q(T))} E{L(Q(0))} E{L(Γ(τ))} +2 T E{Q(τ).Γ(τ)} τ=0 τ=0 T T E{βn (τ)} 2ǫ E{(Q n (τ) γ n (τ))} (33) Dividing by T and taking the for T implies:

6 6 Q Γ β +Γ 2 2Q.Γ (34) 2ǫ Now for undelivered pakets Ū, we have by (9) and (34), Ū β Γ 2 2U.Γ 2ǫ Remark: Note that the omputation of Γ 2 and U.Γ is determined by the dupliate removal strategies. Depending on these terms, the queue bound in this above theorem ould be muh lower than the queue oupany bound for regular bakpressure in (3). Thus we have a formal guarantee that BWAR is no worse in terms of throughput than bakpressure, and potentially muh better in terms of delay, sine by Little s theorem average delay is proportional to the average number of undelivered pakets. We will validate this finding with model in the next setion. IV. MODEL-BASED SIMULATIONS A. The Cell-Partitioned Model The model in this paper simplifies the ontrol variables to be the whole transmission rates µ ab (t) for sheduling and the ommodity transmission rates µ ab (t) for routing. We simulate BWAR in the ontext of enounter-based sheduling and routing for a simple model (ell-partitioned network), whih yields useful insights on its performane. In this idealized model the network deployment area is separated into disjoint ells and nodes have i.i.d. mobility model [20] as follows. We have N nodes and C ells. At eah slot t, node n an be inside any ell with equal probabilities of C. For ollision and interferene simpliity, only one transmission (one paket) is allowed in eah ell in eah time slot. Beause of this we set q th = D max =. Another simplifying assumption is that the nodes in the network are organized into pairs, ating as destinations to eah other. Eah node has Bernoulli exogenous arrivals intended for its pair. Depending on the number of ells C in the network we an hoose the right number of the nodes N.79 C in order to maximize throughput as shown in [20]. Our simulation results show that by optimizing number of nodes based on the number of ells to maximize throughput, the delay also is improved. We onsider in our simulations, networks of sizes 9, 2, 6, 20, and 25 ells in the network. And for optimality, number of nodes are hosen to be 6, 20, 28, 34, and 44 respetively. For timeout dupliate removals we set the timeout value P = C. Here we show how BWAR works in the ell-partitioned network with the simplifying assumption that only one transmission is allowed per ell per time slot. Eah time slot t and for eah ell l we hoose two nodes a and b and ommodity suh that: a and b are in ell l. Q a (t) Q b (t) Q a(t) Q b (t); for all, for all a and b in ell l at time slot t. This aptures the maximization of queue differentials of the main queues. If there exists a,b in ell l suh that, Q b a (t) Qb b (t) = Q a (t) Q b (t) then = b. This aptures the destination advantage. If there exists a,b in ell l and suh that Q a (t) Q b (t) = Q a (t) Q b (t) and { b or [( = b) and ( = b )]} then (Q a (t)+d a (t)) (Q b (t)+d b (t)) (Q a (t)+d a (t)) (Q b (t) + D b (t)). This aptures the maximization of dupliate buffer differentials if there are some ties in main queue differentials. The algorithm simply assigns µ a b (t) a value of, and assigns all other µ ab (t) a value of 0 suh that a,b in ell l. When a transmission is made from node a to node b of ommodity at time slot t and that transmission will make Q a (t + ) + D a (t + ) = 0 then this transmitted paket is dupliated and stored in the dupliate buffer of node a making Dn(t) = instead of 0. Dupliate pakets are served only if there are no original pakets to transmit. There is strit lower priority of dupliate pakets ompared to original pakets. B. Protool Variants In the simulations, we implement and ompare five different routing protool variants. They are desribed as follows: Regular Bakpressure (RB): This is the basi bakpressure sheduling and routing mehanism, where deisions are made purely based on queue differentials. Regular Bakpressure with Destination Advantage (RB-DA): This is a slight modifiation in whih pakets orresponding to the destination are prioritized when the destination is enountered. As we show, this already yields signifiant delay improvements over regular bakpressure. BWAR with Ideal paket removal and original pakets retained in the Main queue (BWAR-IM): This is our novel bakpressure with adaptive redundany in whih the destination advantage is also holds. Here, when an original paket is dupliated the original paket remains in the main queue while the dupliate is stored in the dupliate buffer. We assume here whenever a paket reahes the destination, all of its dupliates are deleted inluding the original one in the main queue instantaneously. BWAR with Ideal paket removal and original pakets moved to Dupliate buffer upon opy (BWAR-ID): This is very similar to BWAR-IM. The only differene is that whenever an original paket is dupliated both the original paket and the dupliate are stored in the dupliate buffer (of ourse in two different nodes one in the reeiver and the other in the sender respetively). BWAR with Time-out based paket removal and original pakets moved to Dupliate buffer upon opy (BWAR-TD): This is a pratial implementation of BWAR in whih dupliates are deleted from the dupliate buffer after a predefined timeout value P has passed sine the first time the original paket is admitted to the network. However, the original paket that is kept in dupliate buffer is flagged and will not be deleted

7 Averae Delay RB RB DA BWAR IM BWAR ID BWAR TD S&W Number of Nodes (N) (a) Delay as we vary N for low λ = 0.00 Average Delay BWAR TD Load(λ) RB RB DA BWAR IM BWAR ID (b) Delay as we vary λ for N = 44 of bakpressure variants Fig.. Comparing delay performane of protool variants: RB, RB-DA, BWAR-IM, BWAR-ID, BWAR-TD and S&W under the ell-partitioned model. when a timeout ourred. It is only deleted if it gets aknowledged diretly by the destination if its already reeived or otherwise it moved bak to the main queue when it enounters the destination. Spray and Wait (S&W): This is not a bakpressure based mehanism. Spray and Wait is presented by T. Spyropoulos et al. [4] whih is a state of the art routing sheme in intermittently onneted mobile networks. S&W reates a predefined fixed number of opies (spraying) of the paket when admitted to the network. Those opies are distributed to distint nodes and then eah opy waits until it enounters the destination. We implemented S&W for omparison with BWAR. Our results show that BWAR outperforms S&W espeially in high load senarios. The evaluations are onduted using a ustom simulator written in C++ (for repeatability, we make our ode available online at Eah simulation runs for one million time slots. In figure (a), we show average delay of all above protool variants as number of nodes N vary for low load λ = 0.00 out of the per node apaity region Λ node = [0,0.4]. Delay is redued signifiantly when BWAR is used. For this low load senario all BWAR variants have almost the same average delay and they perform slightly better than Spray and Wait. Figure (a) also shows the great dramati delay improvement of destination advantage without any redundany in RB-DA ompared to regular bakpressure RB. Figure (b) ompares the average delay of all variants of bakpressure-based protools as we vary the load. As expeted, as the load inreases the delay improvement of BWAR delines ompared to RB-DA. Figure (b) also shows how BWAR-ID performs muh better ompared to BWAR-IM beyond some threshold of load(λ). This shows how moving the dupliated original paket to the dupliate buffer has great delay enhanement for high load senarios. In Figure 2, results show how BWAR mehanism outperforms Spray and Wait (S&W) delay performane for high load. It shows also how BWAR supports almost twie the apaity region of S&W. Average Delay S&W BWAR TD BWAR ID Load(λ) Fig. 2. Comparing S&W delay with BWAR-ID and BWAR-TD as we vary λ for N = 44 Number of Transmissions 4 x Load(λ) RB RB DA BWAR IM BWAR ID BWAR TD Fig. 3. Comparing energy onsumption as we vary λ for N = 44 under the ell-partitioned model. Surprisingly in figure 3, BWAR-IM has a better total number of transmissions ompared to regular bakpressure RB-DA for low load despite the flooding dupliates nature of BWAR at low load. Spray and Wait has superior energy onsumption performane ompared to all bakpressure-based protool variants onsidered. For future work, we intend to S&W

8 8 study the possibility of having both power optimization and adaptive redundany features to be enabled on bakpressure. Average Delay Timeout λ = 0.00 (BWAR TD) λ = 0.06 (BWAR TD) λ = (BWAR TD) λ = 0.28 (BWAR TD) λ = 0.00 (BWAR ID) λ = 0.06 (BWAR ID) λ = (BWAR ID) λ = 0.28 (BWAR ID) Fig. 4. Timeout effet of BWAR-TD and omparing it with BWAR-ID for different λ {0.00,0.06,0.064,0.28} for N = 44 under the ellpartitioned model. Figure 4 studies the effet of timeout value P of BWAR- TD for removing dupliates under different load senarios and ompares its delay performane with ideal dupliate removals in BWAR-ID. V. RELATED WORK The first theoretial work on bakpressure sheduling is the lassi result by Tassiulas and Ephremides in 992, proving that this queue-differential based sheduling mehanism is throughput optimal (i.e., it an stabilize any feasible rate vetor in a network) []. Sine then, researhers have ombined the basi bakpressure mehanism with utility optimization to provide a omprehensive approah to stohasti network optimization [2], [2], [22]. Of most relevane to this work are papers on delay enhanements to bakpressure. A number of papers [23] [25] address the utility-delay tradeoff in optimization-oriented bakpressure, to obtain a tradeoff based on a V parameter suh that the utility is improved by a fator of O(/V) while the delay is made to be polylogarithmi in V. Suh a tradeoff has been shown to be pratially ahievable using LIFO queueing in [26], at the ost of a small probability of dropping pakets. The first-ever implementation of dynami bakpressure routing aimed for wireless sensor networks (BCP) [9] uses suh a LIFO mehanism. As our fous in this work is not on utility optimization, the tehniques presented in these works are somewhat orthogonal to the redundany approah we develop here. Another set of papers [3], [27], [28] onsider the use of shortest path routing in onjuntion with bakpressure to improve the delay performane. These tehniques are well suited for stati networks in whih suh paths an be omputed; however, sine our fous is on enounter based networks with ited onnetivity, suh an approah is not appliable. In [29], the authors present a mehanism whereby only one real queue is maintained for eah neighbor, along with virtual ounters/shadow queues for all destinations, and show that this yields delay improvements. And in [5], a novel variant of bakpressure sheduling mehanism is proposed whih uses head of line paket delay instead of queue lengths as the basis of the bakpressure weight alulation for eah link/ommodity, also yielding enhaned delay performane. However, these works both assume the existene of stati fixed routes. It would be interesting to explore in future work whether their tehniques an be applied to intermittently onneted enounter-based mobile networks, and if so, how these approah an be further enhaned by the use of the adaptive redundany that we propose in this work. Ryu et al. present two works on bakpressure routing aimed speifially for luster-based intermittently onneted networks [0], [30]. In [30], the authors develop a twophase routing sheme, ombining bakpressure routing with soure routing for luster-based networks, separating intraluster routing from inter-luster routing. They show that this approah results in large queues at only a subset of the nodes, yielding smaller delays than onventional bakpressure. In [0], the authors implement the above-mentioned algorithm in a real experimental network and show the delay improvements empirially. The key differene of these works from ours is that we do not make any assumption about the intermittently onneted network being organized in a lusterbased hierarhy. Dvir and Vasilakos [3] also onsider bakpressure routing for intermittently onneted networks, with link weights similar to that used in BCP [9]. They evaluate Weighted Fair Queueing in addition to LIFO and show through simulations that it offers energy improvements. Their work does not expliitly address additional delay improvements needed for these kinds of networks. There is a rih literature on routing in delay tolerant / intermittently onneted enounter based mobile networks (see [32] for a omprehensive survey). Although there exist single-opy routing mehanisms for suh networks [3], it has been well-reognized that repliation is helpful in reduing delay. While basi epidemi routing [33] reates multiple message replias for reliable, fast delivery, it inurs too high of a transmission ost. Smarter multi-opy routing mehanisms have therefore been developed suh as Spray and Wait [4], and SARP [34]. These works introdue redundant paket transmissions to improve delay. However, all of these approahes are not adaptive to the traffi and therefore will hurt the throughput performane of the network. This has been noted before, by the authors of [0], who write that repliation-based algorithms suh as epidemi routing for DTNs... result in lower throughput sine multiple opies of a piee of data need to be forwarded and stored (and therefore not throughput optimal). In fat, in [20], it has been theoretially proved that apaity of suh shemes that use fixed redundany is neessarily lower. In this work, we present the first bakpressure algorithm that uses repliation in an adaptive manner so as to maintain throughput optimality while reduing delay. We expliitly ompare our BWAR sheme with Spray and Wait, and show through our evaluation that not

9 9 only does it provide similar, even better, delay performane, it does so without hurting throughput optimality; speifially, we show that BWAR an handle muh higher traffi load than Spray and Wait. To summarize, this paper on BWAR is the first work that expliitly ombines the best of both worlds: multi-opy routing for intermittently onneted networks and throughput-optimal bakpressure sheduling. This ombination yields better delay performane than traditional bakpressure, partiularly at low loads, and better ability to handle high traffi than traditional DTN/ICN routing shemes. VI. CONCLUSION AND FUTURE WORK We have presented in this paper BWAR, an enhaned bakpressure algorithm that introdues adaptive redundany to improve delay performane. We have proved analytially that this algorithm is also throughput optimal while providing a better delay bound, partiularly at low load settings. Through simulation results we have shown that BWAR outperforms both traditional bakpressure (at low loads) and onventional DTN-routing mehanisms (at high loads) in enounter-based mobile networks. There are a few open avenues for future work suggested by our study. First, we would like to undertake a more areful analysis of the delay improvements obtained, relating them more expliitly, for instane, to arrival proess parameters and the underlying mobility model. Seond, the improvements obtained by BWAR in terms of delay are obtained at the expense of greater number of transmissions due to the introdued redundany. While this may be aeptable in some networks, for energy-onstrained networks this ould be a onern. We therefore plan to explore the design of energy-effiient variants of BWAR in the future, in whih the redundany an be ontrolled to provide a tunable tradeoff between energy and delay. 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