IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 19 ...inlab.lab.asu.edu/Publications/YinShaRed_11.pdf842 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 19, NO. 3, JUNE 2011 where is the rate

IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 19, NO. 3, JUNE 2011 841

On Combining Shortest-Path and Back-PressureRouting Over Multihop Wireless Networks

Lei Ying, Member, IEEE, Sanjay Shakkottai, Member, IEEE, Aneesh Reddy, and Shihuan Liu, Student Member, IEEE

Abstract—Back-pressure-type algorithms based on the algo-rithm by Tassiulas and Ephremides have recently received muchattention for jointly routing and scheduling over multihop wirelessnetworks. However, this approach has a significant weakness inrouting because the traditional back-pressure algorithm exploresand exploits all feasible paths between each source and destination.While this extensive exploration is essential in order to maintainstability when the network is heavily loaded, under light or mod-erate loads, packets may be sent over unnecessarily long routes,and the algorithm could be very inefficient in terms of end-to-enddelay and routing convergence times. This paper proposes a newrouting/scheduling back-pressure algorithm that not only guaran-tees network stability (throughput optimality), but also adaptivelyselects a set of optimal routes based on shortest-path informationin order to minimize average path lengths between each sourceand destination pair. Our results indicate that under the tradi-tional back-pressure algorithm, the end-to-end packet delay firstdecreases and then increases as a function of the network load(arrival rate). This surprising low-load behavior is explained dueto the fact that the traditional back-pressure algorithm exploitsall paths (including very long ones) even when the traffic loadis light. On the other-hand, the proposed algorithm adaptivelyselects a set of routes according to the traffic load so that longpaths are used only when necessary, thus resulting in muchsmaller end-to-end packet delays as compared to the traditionalback-pressure algorithm.

Index Terms—Back-pressure routing, delay reduction, shortest-path routing, throughput-optimal.

I. INTRODUCTION

D UE TO the scarcity of wireless bandwidth resources, it isimportant to efficiently utilize resources to support high-

throughput, high-quality communications over multihop wire-less networks. In this context, good routing and scheduling al-gorithms are needed to dynamically allocate wireless resourcesto maximize the network throughput region. To address this,

Manuscript received August 18, 2009; revised April 04, 2010 and September09, 2010; accepted October 20, 2010; approved by IEEE/ACM TRANSACTIONS

ON NETWORKING Editor C. Westphal. Date of publication December 23, 2010;date of current version June 15, 2011. This work was supported in part byNSF Grants CNS-0347400, CNS-0519535, CNS-0721380, and CNS-0953165;the DARPA ITMANET Program; and DTRA Grants HDTRA1-08-1-0016 andHDTRA1-09-1-0055. An earlier version of this paper appeared in the Proceed-ings of the IEEE International Conference on Computer Communications (IN-FOCOM), Rio de Janeiro, Brazil, April 19–25, 2009.

L. Ying and S. Liu are with the Department of Electrical and Com-puter Engineering, Iowa State University, Ames, IA 50011 USA (e-mail:[email protected]; [email protected]).

S. Shakkottai and A. Reddy are with the Department of Electrical and Com-puter Engineering, The University of Texas at Austin, Austin, TX 78712 USA(e-mail: [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TNET.2010.2094204

throughput-optimal1 routing and scheduling, first developed inthe seminal work of [2], has been extensively studied [3]–[14].We refer to [15] and [16] for a comprehensive survey. Whilethese algorithms maximize the network throughput region, ad-ditional issues need to be considered for practical deployment.

With the significant increase of real-time traffic, end-to-enddelay becomes very important in network algorithm design. Thetraditional back-pressure algorithm stabilizes the network byexploiting all possible paths between source–destination pairs(thus load balancing over the entire network). While this mightbe needed in a heavily loaded network, this seems unneces-sary in a light or moderate load regime. Exploring all paths isin fact detrimental—it leads to packets traversing excessivelylong paths between sources and destinations, leading to largeend-to-end packet delays.

This paper proposes a new routing/scheduling back-pres-sure algorithm that minimizes the path lengths betweensources and destinations while simultaneously being overallthroughput-optimal. The proposed algorithm results in muchsmaller end-to-end packet delay as compared to the traditionalback-pressure algorithm. The main contributions of this paperare summarized next.

A. Main Contributions

We define a flow using its source and destination. Let de-note a flow in network, denote the set of all flows in thenetwork, and denote the number of packets generated byflow at time . We first consider the case where each flow as-sociates with a hop constraint . The routing and schedulingalgorithm needs to guarantee that the packets from flow aredelivered in no more than hops. Note that this hop constraintis closely related to the end-to-end propagation delay. For thisproblem, we propose a shortest-path-aided back-pressure algo-rithm that exploits the shortest-path information to guarantee thehop constraint and is throughput-optimal; i.e., if there exists arouting/scheduling algorithm that can support the traffic with thegiven hop constraints, then the shortest-path-aided back-pres-sure can support the traffic as well.

We then consider a case where no per-flow hop constraint isimposed. The objective is to minimize the average number ofhops per packet delivery (or the average path lengths betweensources and destinations). Mathematically, given a traffic load

, the objective is

1A routing/scheduling algorithm is throughput-optimal if it can stabilize anytraffic that can be stabilized by any other routing/scheduling algorithm.

1063-6692/$26.00 © 2010 IEEE

842 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 19, NO. 3, JUNE 2011

where is the rate that flow delivers packets using paths

with hops, and . This objectivehas two interpretations.

• First, can be thought of as the number oftransmissions needed to support traffic (transmittinga packet over an -hop path requires transmissions).Thus, minimizing can be regarded as min-imizing the network resource used to support the trafficdemand.

• Second, note that the number of hops is closely related tothe end-to-end delay, so is related to the averageend-to-end delay of flow . Thus, minimizingcan potentially be used as a surrogate for minimizing theaverage end-to-end delay over all flows in the network (thedifference being that the MAC delays are ignored in thehop-count metric).

To solve this problem, we propose a joint traffic-control andshortest-path-aided back-pressure algorithm that not onlyguarantees the network stability (throughput-optimal), butalso adaptively selects the optimal routes according to thetraffic demand. When the traffic is light, the algorithm onlyuses shortest paths; when the traffic increases, more paths areexploited to support the traffic. Our simulations show thatthe joint traffic-control and shortest-path-aided back-pressurealgorithm leads to a much smaller end-to-end delay comparedto the traditional back-pressure algorithm: 5 time slots/packetversus 1000 time slots/packet when the traffic load is light and2000 time slots/packet versus 3000 time slots/packet when thetraffic load is high as illustrated in the example in Section II.

B. Related Work

Throughput-optimal routing/scheduling was first proposedin [2], and has then been studied for varied networks includingcellular networks [17], cooperative relay networks [11], [12],and multihop wireless networks [6], [7], [9]. Low-complexityimplementations have been proposed in [13] and [18]–[28].Joint scheduling/routing/power control has been developedin [6] and [10]. Throughput-optimal routing/scheduling formulticast flows has been considered in [29]. The idea of usingthe shortest path information to enhance the performance ofthe back-pressure algorithm has been studied in [30]. The maindifference is that the proposed algorithm provably minimizesthe average path lengths, whereas the enhanced algorithm in[30] uses the shortest path information in a heuristic manner.An alternate algorithm that deals with minimizing the numberof hops has been recently independently obtained in [31]. Theobjective function in [31] is the same as in this paper, howeverthe proposed algorithms are different. In [32] and [33], the au-thors have proposed throughput-optimal routing policies basedon new Lyapunov functions that improve the delay performancecompared to the original back-pressure algorithm.

II. ILLUSTRATIVE EXAMPLE

As was discussed in the Introduction, the back-pressure al-gorithm exploits all feasible paths, which is critical to maintain

Fig. 1. Back pressure via our joint traffic splitting and shortest-path-aided backpressure.

stability when the network is heavily loaded. However, when thetraffic load is light, packets may be sent over unnecessary longpaths and the algorithm could be very inefficient.

In this section, we present a simulation result to demonstratethe weakness of the back-pressure algorithm and the significantend-to-end delay reduction that results under the proposed al-gorithm (the algorithm will be described in Section V).

Define the end-to-end delay of a packet to be the time in-terval from when the packet enters the source to when the packetreaches the destination (this includes the MAC delay at inter-mediate nodes). Fig. 1 illustrates the average end-to-end delaysunder the back-pressure algorithm and the proposed algorithmunder different traffic loads. The network used in the simulationis a grid-like network with 64 nodes and 8 data flows. A detaileddescription of the network and simulation settings will be pre-sented in Section VI. From Fig. 1, we have two observations.

1) Under the back-pressure algorithm, surprisingly, the delayfirst decreases and then increases as the traffic load in-creases. The second part is easy to understand: The queuesbuild up when the traffic load increases, which increasesthe queuing delays. The first part is because the back-pres-sure algorithm uses all paths even when the traffic load islight. For example, in a light traffic regime, using shortestpaths is sufficient to support the traffic flows. However,under the back-pressure algorithm, long paths and pathswith loops are also used. Furthermore, the lighter the trafficload, the more loops are involved in the route. Hence, theend-to-end delay is large.

2) In the proposed algorithm, the set of routes used is intel-ligently selected according to the traffic load so that longpaths are used only when necessary. We can see that underthe proposed algorithm, not only is the delay significantlyreduced, but also the delay monotonically increases withthe traffic load.

We would like to emphasize that under the proposed al-gorithm, the delay improvement is achieved without losingthe throughput-optimality. The proposed algorithm is stillthroughput-optimal, but yields much smaller end-to-end delaysas compared to the traditional back-pressure algorithm.

YING et al.: ON COMBINING SHORTEST-PATH AND BACK-PRESSURE ROUTING OVER MULTIHOP WIRELESS NETWORKS 843

III. BASIC MODEL

Network Model: Consider a network represented by a graph, where is the set of nodes and is the set of

directed links. We assume that and . Denoteby the link from node to node . Furthermore, let

denote a link-rate vector such that is thetransmission rate over link . A link-rate vector is saidto be admissible if the link-rates specified by can be achievedsimultaneously. Define to be the set of all admissible link-ratevectors. It is easy to see that depends on the choice of inter-ference model and might not be a convex set. Furthermore, istime-varying if link-rates are time-varying. To simplify our no-tations, we assume time-invariant link-rates in this paper. How-ever, our results can be extended to time-varying link-rates ina straightforward manner. Furthermore, we assume that thereexist and such that for all

and all admissible .Next, we define a link vector to be obtainable if

, where denotes the convex hull of .Note that an admissible rate-vector is a set of rates at whichthe links can transmit simultaneously, while an obtainablerate-vector is a set of rates that can be achieved including usingtime sharing. As a simple example, consider a network withtwo nodes {1, 2} and two links {(1, 2), (2, 1)}. Assume thelink capacity is one packet per time slot for both links, andhalf-duplex constraint so that only one link can transmit at onetime. Then, is not an admissible rate-vectorsince two links cannot transmit at the same time. However, it isobtainable by time sharing.

Traffic Model: For network traffic, we let denote a flow,denote the source of the flow, and the destination of

the flow. We use to denote the set of all flows in the net-work. Assume that time is discretized, and letdenote the number of packets injected by flow at time . In thispaper, we assume is random and independent and identi-cally distributed (i.i.d.) across time slots, for all if

, for all and , and .

IV. THROUGHPUT-OPTIMAL ROUTING/SCHEDULING

WITH HOP CONSTRAINTS

In this section, we consider the case where each flow isassociated with a hop constraint . Packets of flow need tobe delivered within hops. We propose a shortest-path-aidedback-pressure algorithm, which is throughput-optimal underhop-constraints. The algorithm is also a building block forthe algorithm to be proposed in Section V, which seamlesslyintegrates the back-pressure and the shortest-path routing.

Next, we characterize the network throughput region underhop constraints.

A. Network Throughput Region Under Hop Constraints

We denote by the indicator function with condition ,i.e., if condition holds, and otherwise. Giventraffic and hop constraint ,we define by saying that if there exists

such that the following conditions hold.

(i) For any three-tuple such that and, we have

(1)(ii) If , then

(2)

where is the minimum number of hops from nodeto node .

(iii)

(3)

where

and is the set of all destinations.We can regard as the average transmission rate

over link for transmitting those packets that are requiredto be delivered to node within hops. Note that when apacket is sent to node from node , the hop constraint associ-ated with the packet reduces by one. Then, the conditions abovecan be explained as follows.

a) Condition (i) is the flow-conservation constraint, whichstates that the number of incoming packets to nodewith hop constraint is equal to the number of outgoingpackets from node with hop constraint . Note thatthe hop constraint reduces by one after a packet is sent outby node because it takes one hop to transmit the packetfrom node to one of its neighbors. We only considerhop constraints up to because the longest loop-freepath has no more than hops, and considering onlyloop-free routes does not change the network throughputregion.

b) Condition (ii) states that a packet should not be trans-mitted from node to node if node cannot deliverthe packet within the required number of hops.

c) Condition (iii) is the capacity constraint, which states thatthe rate-vector should be obtainable.

We say traffic can be stabilized if there exists somerouting/scheduling algorithm under which the mean of thenumber of packets queued in the network is bounded. From dis-cussions a)–c), it is easy to see that if can be stabilized,then there must exist satisfying conditions (i)–(iii). Thus,is named as the the throughput region of network .

B. Queue Management

We introduce our queue management scheme. Recallis the minimum number of hops from node to node (orthe length of the shortest path from node to node ). Notethat can be computed in a distributed fashion using al-gorithms such as the Bellman–Ford algorithm. Thus, we assumethat node knows for all destinations , andfor such that .


Fig. 2. Illustration of queue management and computation of back pressure.

We assume node maintains a separate queue, named queue, for those packets required to be delivered to node

within hops. For destination , node maintains queues for, where is a universal upper bound

on the number of hops along loop-free paths.As an example, consider the directed network shown in Fig. 2,

and assume that (i.e., there is only one destination).Each nondestination node maintains up to three queues (becausefor this topology, there are no loop-free paths longer than threehops). Node 1 has queues corresponding to , 2, 3, re-spectively. Node 2 does not have a direct path to node 4 (i.e.,

), hence it maintains only two queues correspondingto , 3 (and implicitly, we set to ensurethat no packets enter ). Node 3 maintains three separatequeues corresponding to , 2, 3, in spite of the observa-tion that there is only one feasible route from node 3 to node 4.We maintain these additional queues because the global networktopology is not known by individual nodes. Finally, all queuesat the destination for packets meant to itself are set to zero (e.g.,

). In Fig. 2, queues into which packets potentiallyarrive are marked in solid lines, and the “virtual” queues that arefixed at are in dotted lines.

C. Queue Dynamics

Let denote the queue length at time slot , and

denote the service rate allocated to transmit packetsfrom queue to queue over link attime . Since the packets in queue need to be deliveredwithin hops, they can be only deposited to queuesfor . For example, packets from queue {2, 4, 3}can be transferred to queue {3, 4, 2} or queue {3, 4, 1}. Thus,we impose the following constraint on routing: The packets inqueue can be only transferred to queues for

, i.e., for all .

The dynamics of queue is as follows:

where is the actual number of packets transferredfrom queue to queue and is smaller than

when there are not enough packets in queue

. Define to be the unused service. We have

We also define for all , i.e., packets delivered areremoved from the network immediately.

In Section IV-D, we propose a shortest-path-aided back-pres-sure algorithm that stabilizes the network given any

.

D. Shortest-Path-Aided Back-Pressure Algorithm

Recall that we have per-hop queues for each destination,which is different from the back-pressure algorithm in [2].Thus, we first define the back pressure of link under ourqueue management scheme. We define , the backpressure between queue and queue overlink , as follows:

• if and;

• otherwise (note that queuedoes not exist if ).

The back pressure of link is defined to be

Considering the example shown in Fig. 2, it can be verifiedthat , ,

, , and.

Shortest-Path-Aided Back-Pressure Algorithm2

Consider time slot .Step 0: The packets injected by flow are deposited intoqueue maintained at node .Step 1: The network first computes that solves thefollowing optimization problem:

(4)

2In this algorithm, we allow the packets in queue �� to be transferredto queues �� for any � such that � � �� , which is more general thanthe algorithm proposed in [1], where the packets in queue �� can betransmitted only to queue �� .


where is an admissible link-rate vector and is therate over link .Step 2: Consider link . If and

, node selects a pair of queues, sayand , such that

and transfers packets from queue toqueue at rate .

We again consider the example in Fig. 2. Assume thenode-exclusive interference model where adjacent linkscannot be active at the same time. Furthermore, assumethat link capacity is equal to one packet/time slot for alllinks. Then, given the queue states shown in the figure,we can easily verify that and

. Therefore, node 1transmits one packet from queue {1, 4, 1} to its destination(node 4), and node 2 transmits one packet from queue {2, 4, 2}to queue {3, 4, 1} at node 3.

Note that the optimization problem defined by (4) is a central-ized problem. There has been a lot of recent work on distributedsolutions, e.g., [18], [19], [21]–[23], and [26]–[28]. These dis-tributed algorithms can be used in step 2 of the proposed algo-rithm. Distributed implementation, however, is not the focus ofthis paper.

The next theorem shows that the shortest-path-aided back-pressure algorithm is throughput-optimal under per-flow hopconstraints, and the proof is presented in Appendix A.

Theorem 1: Given traffic and hop constraint such thatfor some , the network can be sta-

bilized under the shortest-path-aided back-pressure algorithm,and packets delivered are routed over paths that satisfy corre-sponding hop constraints.

V. THROUGHPUT-OPTIMAL AND HOP-OPTIMAL

ROUTING/SCHEDULING

In Section IV, we proposed the shortest-path-aidedback-pressure algorithm that is throughput-optimal and sup-ports per-flow hop constraint.

In this section, we consider the scenario where no hop con-straint is imposed. Recall that is an upper bound on thenumber of hops of loop-free paths. Define such that

for all . Then, we can assume that a flow is al-ways associated with hop constraint , i.e., all loop-free pathsare allowed. Note that considering only loop-free paths does notchange the network throughput region. Thus, we say is withinthe network throughput region if , which is alsowritten as .

In this section, we propose an algorithm that is boththroughput-optimal and hop-count optimal, i.e., minimizingthe average path lengths. Recall that the motivation to developa hop-optimal algorithm is that such an algorithm will not onlyminimize the number of transmissions required to support thetraffic, but also reduce the average end-to-end transmissiondelay. (As we will later see from simulations, minimizing hopcount does seem to result in smaller end-to-end delays.)

A. Hop Minimization

Given traffic , we let denote the set of routing/scheduling policies that stabilize the network. We further define

to be the rate at which flow delivers packets overpaths with exactly hops under policy , which is well definedwhen the network can be stabilized. Our objective is to find apolicy such that

(5)

Note that each stabilizing policy yields an obtainable ratevector . Recall that is the averagerate over link used to transmit packets destined to nodeand delivered with exactly more hops. Thus, problem (5) isequivalent to the following optimization problem:

(6)

such that (7)

if (8)

(9)

(10)

(11)

(12)

To understand problem (6), we can think that we split flowinto flows , allocate fraction offlow to flow , and impose hop constraint to flow . Then,the average number of hops per packet delivery of flow is

Thus, problem (6) is to find a splitting that is supportable andalso minimizes the number of hops used to support the traffic.

B. Dual Decomposition

To solve optimization problem (6), we define to be theLagrange multiplier associated with (7). Then, we can obtain apartial Lagrange dual function as follows:

subject to: (8)–(12)


where

and

According to the Slater’s condition [34], the strong dualityholds. Thus, there exist such that is theoptimal solution to problem (6), and

From the equality above, we can thus conclude that there existsuch that the following equations hold:

(13)


(14)


(15)

where equality (15) holds according to the definition ofLagrange multipliers.

C. Joint Traffic-Splitting and Shortest-Path-AidedBack-Pressure Algorithm

Now motivated by (13) and (14), we propose a joint traffic-splitting and shortest-path-aided back-pressure algorithm.

First, note that

is linear in terms of . Thus, we have

Note that the Lagrange multiplier is related to queuelength , and (7)–(10) are the same as conditions (i)-(iii)defined in Section IV-A, so equality (14) motivates us to use theshortest-path-aided back pressure defined by (4).

Furthermore, equality (13) motivates us to propose a traffic-splitting scheme such that, at time slot , the arrivals offlow are deposited in queue that minimizes

The parameter in the traffic splitting is a tuningparameter, which plays an important role when the proposedalgorithm is used in a stochastic network (stochastic arrivalsand fading channels). In theory, the value of controls thetradeoff between the overall backlog in the network and the op-timality of the steady-state resource allocation solution. The al-gorithm asymptotically solves the hop minimization problem as

, but pays a price of increasingly large backlogs in thenetwork. In previous works on stochastic control of wireless net-works [5], [7], similar tuning parameters have also been intro-duced and studied.

Joint Traffic-Splitting and Shortest-Path-AidedBack-Pressure Algorithm

Traffic Splitting: At time , external arrivals of flow aredeposited into queue , where is thesmallest integer of the following set:

(16)

Routing/Scheduling: The shortest-path-aided back-pressurealgorithm without step 0.

We first show that the above algorithm is throughput-optimal.We denote by the number of packets that are injectedby flow at time , and assigned a hop constraint under thejoint traffic control and shortest-path-aided back-pressure algo-rithm with parameter .

Theorem 2: Given such that for some, the network is stochastically stable under joint traffic-

splitting and shortest-path-aided back-pressure algorithm.Proof: It can be easily verified that is a Markov

chain. We define a Lyapunov function

and prove that there exists such that iffor some , then

(17)

which implies the positive recurrence of the Markov chain. Thedetails are presented in Appendix B.

Now given such that , we further define

Note that is well defined because the network isstable according to Theorem 2.


The next theorem states that the algorithm asymptoticallysolves the optimization problem (6) as .

Theorem 3: Given such that for some, under the joint traffic-allocation and shortest-path-aided

back pressure, we have

(18)where is the optimal solution to problem (6).

Proof: Based on Theorem 2, we can first show that thereexists such that

(19)Furthermore, it is easy to see that

holds for any and . Thus, the theorem holds. The details aregiven in Appendix C.

According to Theorem 3, we should choose a large to min-imize the average-number of hops per packet delivery. How-ever, we notice that with a large , packets are assigned toqueue only when queue hasa large backlog, which could lead to a large queueing delay (i.e.,large MAC delay). Thus, there is a tradeoff choosing the valueof (to trade off between reducing hop count and queueingdelay). A similar tradeoff resulted from a “drift-plus-penalty”technique of Lyapunov optimization for wireless networks hasbeen observed and analyzed in [7], [35], where the orderwisetradeoff is quantified. In this paper, we will study the impact of

on network performances using simulations in Section VI.

VI. SIMULATIONS

In this section, we use simulations to study the performanceof the proposed joint traffic-splitting and shortest-path-aidedback-pressure algorithm. We use the term the joint algorithm torefer to the joint traffic-splitting and shortest-path-aided back-pressure algorithm. The simulations were implemented usingOMNeT++.

A. Simulation Setup

We consider a network with 64 nodes as shown in Fig. 3. Thenetwork consists of four clusters, and each cluster is a 4 4 reg-ular grid with two randomly added links. Two neighboring clus-ters are connected by two links. Here, only two links are used toconnect two clusters instead of four or more. This is to “force”intercluster flows to be routed over long paths when the trafficload is high so that the traffic-splitting behavior of the joint al-gorithm can be easily observed. All links are bidirectional linkswith capacity one packet/time slot for both directions. All linksare assumed to be orthogonalized so they can transmit simulta-neously. The propagation delay of a link is assumed to be zero.

Eight traffic flows were created in the network, as listed inTable I. Flows 1–5 are intercluster flows, and the rest are intr-acluster flows. The packet arrivals of all flows follow Poissonprocesses. We fixed the arrival rates of intracluster flows to be

Fig. 3. Topology of the network used in the simulations.

TABLE IFLOWS IN THE NETWORK

0.2 packets/time slot. All intercluster flows have the same ar-rival rate, denoted by (packets/time slot).

In the simulations, we varied to observe the performanceof the back-pressure algorithm and the joint algorithms underdifferent traffic loads. For each , the simulation is executed for100 000 iterations. When ties occurred in deciding the trafficsplit or computing the back pressure of a link, we selected thefirst obtained solution.

B. Average Number of Hops per Packet Delivery

We first study the average number of hops per packet delivery,called average hop count, which is averaged over all success-fully delivered packets. We implemented the back-pressure al-gorithm and the joint algorithm with and .We note that when , in the traffic splitting, a queue withhop constraint is chosen over a queue with hopconstraint as long as the first queue is smaller. Thus, a smallresults in a small penalty on long paths.

From Fig. 4, we have the following observations.• When is small, the joint algorithm has significantly

smaller average hop counts than that of the back-pressurealgorithm (4 hops/packet delivery versus 180 hops/packetdelivery). This is because the back pressure exploits allfeasible paths, while the joint algorithm only utilizes shortpaths.

• When is large (the network is critically loaded), the av-erage hop counts of the joint algorithm became closer tothat of the back-pressure algorithm. This is because in a


Fig. 4. Hop counts of the back-pressure algorithm and the joint algorithm withdifferent �’s.

Fig. 5. Hop counts of the joint algorithm with different�’s.

heavy traffic regime, the joint algorithm also exploits longpaths to maintain stability.

Fig. 5 is the “zoomed-in” picture of Fig. 4, which shows thehop counts of the joint algorithm with different values of . Weobserve that the average hop count increases as decreases.In Theorem 3, we have proved that the average path lengthsare asymptotically minimized when . Our simulationresults are consistent with the theorem.

C. End-to-End Packet Delays

We also computed the average end-to-end packet delay, av-eraging over all successfully delivered packets. Similar to thehop count, in Fig. 6, we observe that the back pressure performsvery poorly when is small. This can be attributed to the ex-cessive looping in the route of each packet and can roughly beinterpreted as a random walk on the two-dimensional network.

When is large, we also observe some improvement of thejoint algorithm, with and , over the back-pressurealgorithm. The improvement decreases because the joint algo-rithm has to exploit long paths in a heavy traffic regime. Wefurther note that the joint algorithm with performsvery poorly in terms of end-to-end packet delay while it has thesmallest average hop count. As we have seen in the analysis ofTheorem 3, minimizes the average hop count, but re-sults in large queues, hence large end-to-end packet delays.

Fig. 6. Average end-to-end packet delays under the back-pressure algorithmand the joint algorithm with different �’s.

Fig. 7. CDF of end-to-end packet delays.

Fig. 7 illustrates the cumulative distribution function (cdf) ofend-to-end packet delays for and . We observethat the joint algorithm with has a much steeper slopecompared to the back-pressure algorithm, which again indicatesthat the joint algorithm has a much better delay performancecompared to the back-pressure algorithm.

D. Queue Lengths

Here, we study the total queue length at each node. The av-erage queue length was obtained by averaging over the 100 000iterations and over all nodes in the network. Fig. 8 illustrates thecomparison between the back-pressure and the joint algorithmwith in light and medium traffic regimes. Fig. 9 illus-trates the average queue lengths in medium and heavy trafficregimes. We observe that in a light traffic regime, the averagequeue length of the joint algorithm is close to 0, while the oneunder the back-pressure algorithm is more than 20. The two al-gorithms, however, perform similarly in a heavy traffic regime.We note that the joint algorithm still has smaller end-to-endpacket delays, as shown in Fig. 6, because the average hop countis smaller, as shown in Fig. 4.

Fig. 10 illustrates the average queue lengths under the jointalgorithm with different ’s. We observe that the average queuelength increases as increases.


Fig. 8. Back-pressure versus the joint algorithm in light and medium trafficregimes.

Fig. 9. Back-pressure versus the joint algorithm in medium and heavy trafficregimes.

Fig. 10. Performance of the joint algorithm with different values of� .

E. File Transfer Delay

We also investigated file transfer delays (the duration from thetime a file enters the network until it is received at the destina-tion). We compared the back-pressure algorithm with the jointalgorithm with . In this simulation, files belonging to thesame flow are injected into the source of the flow one by one,

Fig. 11. Back-pressure versus the joint algorithm with� � �.

and the second file arrives after all packets of the first file aresent out from the source. After a file arrives, the packets of thefile are injected into the source node with a constant rate untilthe complete file is injected. The file size follows a Poison distri-bution. We considered two file-size distributions: 1) a Poissondistribution with mean 50; and 2) a Poisson distribution withmean 1000. Similar to previous simulations, we fixed the ofintracluster flows and varied the of intercluster flows.

Under back-pressure algorithm and the joint algorithm, somepackets may be queued in the network for a very long time.We therefore assume the packets of a file are coded using rate-less codes so that a file can be completely recovered when 90%of the coded packets are received. Fig. 11 illustrates the filetransfer delays of the joint algorithm with and theback-pressure algorithm. As we can see, when the mean file sizeis 50, the joint algorithm performs significantly better than theback-pressure algorithm in both light or medium traffic regimes,but performs similarly to the back-pressure algorithm in theheavy traffic regime. This is because in the heavy traffic regime,the end-to-end packet delays of the two algorithms are similar.When file sizes are large, the two algorithms perform similarlyregardless of the traffic load. This is because, for a large-size file,the dominant component of the file transfer delay is the trans-mission delay, the number of time slots required to inject all thepackets of a file into the network, which is independent of therouting algorithm.

VII. DISCUSSION

A. Minimum-Weight-Aided Back Pressure

In Sections IV and V, the scheduling/routing algorithms wedeveloped use the shortest-path information in finding the nexthop. The length of a path is defined to be the number of hopsalong the path. Instead of counting the number of hops, we canassign different weights to different links. The weight can bethe propagation time of the link, the geographic distance be-tween two nodes, etc. Then, letting denote the minimumaggregated weight from node to node , we can use this infor-mation to replace to have algorithms that support otherquality-of-service constraints.


B. Elastic Flows and Utility Maximization

In this paper, we primarily focused on inelastic flows. Thealgorithms can be easily extended to elastic flows by consideringthe following utility maximization problem:

where is the utility function associated with flow , andthe constant controls the tradeoff between the network utilityand the average path length. Exploiting the dual decomposition,a rate control algorithm can be obtained following a similar ap-proach used in [5]–[7] and [9]. Combining the rate control al-gorithm and the short-path-aided back-pressure algorithm de-veloped in this paper, the algorithm can maximize the networkutility while minimizing the average path length required forsupporting the maximum utility (by choosing a sufficientlylarger than ).

C. Virtual Queues

In the joint traffic-splitting and shortest-path-aided back-pres-sure algorithm, we impose artificial hop constraints in order tominimize the average path length. Note that for the packets atnode with destination , interchanging their hop constraintswill not change the routing/scheduling decisions, hence that willnot change the average number of transmissions per time slot,which is the same as the average path length. This suggests thatper-hop queues do not need to be real queues. We can maintainper-destination real queues as in the traditional back-pressurealgorithm, but have per-hop counters (virtual queues). This ideaof using virtual queues (or shadow queues) to reduce the queuecomplexity has been proposed in [31].

Let denote the value of the corresponding virtualqueue at time , and the length of the real queue main-tained for destination at node at time . The joint algorithmwith virtual queues works as follows: 1) the virtual queues areupdated as defined in the joint algorithm; and 2) at each timeslot, we transfer packets from the real queueto queue for destination such that there exist andsatisfying

By utilizing virtual queues, the number of real queues re-quired in the system will be the same as that in the originalback-pressure algorithm.

VIII. CONCLUSION

In this paper, we have proposed new routing/scheduling algo-rithms that integrate the back-pressure algorithm and shortest-path routing. Using simulations, we have demonstrated a sig-nificant end-to-end delay performance improvement using theproposed algorithm.

APPENDIX APROOF OF THEOREM 1

Some steps of the following proofs are similar to previousanalysis of back-pressure-based algorithms. They are includedfor exhaustiveness.

First, it is easy to verify that is Markovian since theshortest-path-aided back-pressure algorithm makes routing andscheduling decisions based on the queue lengths and link statesat time . Defining a Lyapunov function

the drift of the Lyapunov function is as follows:

where

Recall that ,

, and . The following in-equalities can be verified easily:

• ;

• ;

• only if since,otherwise, there are enough packets in queue tobe transmitted.

Based on these inequalities and following the argument in [36],we can obtain the following inequality:


where

and

Note that implies that there existand that satisfy the conditions (1)–(3), where

if . We then obtain that under theshortest-path-aided back pressure, for any

(20)

where equality ( ) yields from the definitionof , equality ( ) holds because

is linear

over , and inequality ( ) holds results from the definition ofthe back pressure.

By adding and substituting terms

and using equality (1), we conclude that

(21)

where inequality is a result of inequality (20).We note that for all . Thus, given

, there exists some linkfor which we have ,which implies that

where the last inequality holds because

Taking expectation (over ) at both sides of the inequalityyields the following inequality:

Then, by summing up both sides of the inequality above fromto , we obtain

which implies that

Hence, the network is stable.Next, we will show that no packet will violate the hop

constraint under the proposed algorithm. From the definitionof the back pressure and the optimization (4), we can seethat the packets in queue are transmitted only toqueues for and . This guar-antees there exists at least one feasible path from node todestination with no more than hops.

Also, the packets of flow are first queued atqueue . Based on the facts above, it can be


easily verified that if a packet is received by its destination ,then

where is the number of hops the packet has been transmittedover. We therefore conclude that every delivered packet is de-livered within the required number of hops.

APPENDIX BPROOF OF THEOREM 2

First, it can be easily verified that is a Markov chain.Define the Lyapunov function to be

Note that implies that there exist andsuch that , and conditions (i)–(iii)defined in Section IV-A hold. Similar to the proof of Theorem 1,we can first show that

(22)

(23)

Given and , the traffic-splitting algorithmguarantees

so

Furthermore, since , we have

which implies that (22) (23) 0.According to inequality (20), the following inequality also

holds:

Therefore, we obtain

where .The rest of the proof is identical to the proof of Theorem 1.

APPENDIX CPROOF OF THEOREM 3

Recall that and are the optimal solutions tooptimization problem (6). Thus, ,and and satisfy conditions (i)–(iii) definedin Section IV-A. Similar to the proof of Theorem 1, we can showthat


(24)

Similar to the analysis in Appendix B, we can further obtain

and

Substituting these two inequalities into (24), we conclude that

which further implies that

holds for any , and

We therefore have that

(25)

According to the definition of , the following inequalityholds for all and :

so

Letting go to infinity, we can obtain that

and the theorem holds.

ACKNOWLEDGMENT

The authors gratefully acknowledge the useful discussionswith Prof. R. Srikant, University of Illinois at Urbana–Cham-paign, and the insightful comments from the reviewers andassociate editor.

REFERENCES

[1] L. Ying, S. Shakkottai, and A. Reddy, “On combining shortest-pathand back-pressure routing over multihop wireless networks,” in Proc.IEEE INFOCOM, Rio de Janeiro, Brazil, 2009, pp. 1674–1682.

[2] L. Tassiulas and A. Ephremides, “Stability properties of constrainedqueueing systems and scheduling policies for maximum throughput inmultihop radio networks,” IEEE Trans. Autom. Control, vol. 37, no. 12,pp. 1936–1948, Dec. 1992.

[3] L. Tassiulas and A. Ephremides, “Dynamic server allocation to parallelqueues with randomly varying connectivity,” IEEE Trans. Inf. Theory,vol. 39, no. 2, pp. 466–478, Mar. 1993.

[4] X. Lin and N. Shroff, “Joint rate control and scheduling in multihopwireless networks,” in Proc. IEEE CDC, Paradise Island, Bahamas,Dec. 2004, vol. 2, pp. 1484–1489.

[5] A. Eryilmaz and R. Srikant, “Fair resource allocation in wireless net-works using queue-length-based scheduling and congestion control,”in Proc. IEEE INFOCOM, 2005, vol. 3, pp. 1794–1803.

[6] A. Stolyar, “Maximizing queueing network utility subject to stability:Greedy primal-dual algorithm,” Queue. Syst., vol. 50, no. 4, pp.401–457, Aug. 2005.

[7] M. Neely, E. Modiano, and C. Li, “Fairness and optimal stochastic con-trol for heterogeneous networks,” in Proc. IEEE INFOCOM, Miami,FL, Mar. 2005, vol. 3, pp. 1723–1734.

[8] M. J. Neely, “Optimal backpressure routing for wireless networks withmulti-receiver diversity,” in Proc. CISS, 2006, pp. 18–25.

[9] A. Eryilmaz and R. Srikant, “Joint congestion control, routing andMAC for stability and fairness in wireless networks,” IEEE J. Sel. AreasCommun., vol. 24, no. 8, pp. 1514–1524, Aug. 2006.

[10] M. Neely, “Energy optimal control for time-varying wireless net-works,” IEEE Trans. Inf. Theory, vol. 52, no. 7, pp. 2915–2934, Jul.2006.

[11] E. Yeh and R. Berry, “Throughput optimal control of wireless networkswith two-hop cooperative relaying,” in Proc. IEEE ISIT, Jun. 2007, pp.351–355.


[12] E. Yeh and R. Berry, “Throughput optimal control of cooperative relaynetworks,” IEEE Trans. Inf. Theory, vol. 53, no. 10, pp. 3827–3833,Oct. 2007.

[13] K. Jung and D. Shah, “Low delay scheduling in wireless network,” inProc. IEEE ISIT, 2007, pp. 1396–1400.

[14] L. Ying, R. Srikant, and D. Towsley, “Cluster-based back-pressurerouting algorithm,” in Proc. IEEE INFOCOM, 2008, pp. 484–492.

[15] X. Lin, N. Shroff, and R. Srikant, “A tutorial on cross-layer optimiza-tion in wireless networks,” IEEE J. Sel. Areas Commun., vol. 24, no. 8,pp. 1452–1463, Aug. 2006.

[16] L. Georgiadis, M. J. Neely, and L. Tassiulas, Resource Allocation andCross-Layer Control in Wireless Networks. Hanover, MA: NOW,2006, Foundations and Trends in Networking.

[17] M. Andrews, K. Kumaran, K. Ramanan, A. Stolyar, R. Vijayakumar,and P. Whiting, “CDMA data QoS scheduling on the forward link withvariable channel conditions,” Bell Labs, Tech. Memo, Apr. 2000.

[18] X. Lin and S. Rasool, “Constant-time distributed scheduling poli-cies for ad hoc wireless networks,” in Proc. IEEE CDC, 2006, pp.1258–1263.

[19] X. Wu and R. Srikant, “Scheduling efficiency of distributed greedyscheduling algorithms in wireless networks,” in Proc. IEEE IN-FOCOM, 2006, pp. 1–12.

[20] A. Dimakis and J. Walrand, “Sufficient conditions for stability oflongest queue first scheduling,” Adv. Appl. Prob., pp. 505–521, 2006.

[21] A. Eryilmaz, A. Ozdaglar, and E. Modiano, “Polynomial complexityalgorithms for full utilization of multi-hop wireless networks,” in Proc.IEEE INFOCOM, 2007, pp. 499–507.

[22] A. Gupta, X. Lin, and R. Srikant, “Low-complexity distributed sched-uling algorithms for wireless networks,” in Proc. IEEE INFOCOM,2007, pp. 1631–1639.

[23] S. Sanghavi, L. Bui, and R. Srikant, “Distributed link scheduling withconstant overhead,” in Proc. ACM SIGMETRICS, San Diego, CA, Jun.2007, pp. 313–324.

[24] L. Lin, X. Lin, and N. Shroff, “Low-complexity and distributed en-ergy minimization in multi-hop wireless networks,” in Proc. IEEE IN-FOCOM, 2007, pp. 1685–1693.

[25] C. Joo, X. Lin, and N. B. Shroff, “Understanding the capacity regionof the greedy maximal scheduling algorithm in multi-hop wirelessnetworks,” in Proc. IEEE INFOCOM, Phoenix, AZ, Apr. 2008, pp.1103–1111.

[26] L. Jiang and J. Walrand, “A distributed CSMA algorithm forthroughput and utility maximization in wireless networks,” in Proc.46th Annu. Allerton Conf. Commun., Control, Comput., 2008, pp.1511–1519.

[27] J. Liu, Y. Yi, A. Proutiere, M. Chiang, and H. V. Poor, “Maximizingutility via random access without message passing,” Microsoft Re-search, Tech. Rep., Sep. 2008.

[28] J. Ni and R. Srikant, “Distributed CSMA/CA algorithms for achievingmaximum throughput in wireless networks,” Tech. Rep., 2009.

[29] L. Bui, R. Srikant, and A. L. Stolyar, “Optimal resource allocationfor multicast flows in multihop wireless networks,” Philos. Trans.Royal Soc. London A, Math. Phys. Eng. Sci., vol. 366, no. 1872, pp.2059–2074, 2008.

[30] M. J. Neely, “Dynamic power allocation and routing for satellite andwireless networks with time varying channels,” Ph.D. dissertation,Dept. Elect. Eng. Comput. Sci., Massachusetts Inst. Technol., Cam-bridge, Nov. 2003.

[31] L. Bui, R. Srikant, and A. L. Stolyar, “Novel architectures and algo-rithms for delay reduction in back-pressure scheduling and routing,” inProc. IEEE INFOCOM, 2009, pp. 2936–2940.

[32] P. Gupta and T. Javidi, “Towards throughput and delay-optimal routingfor wireless ad-hoc networks,” in Proc. Asilomar Conf. Signals, Syst.Comput., Nov. 2007, pp. 249–254.

[33] M. Naghshvar, H. Zhuang, and T. Javidi, “A general class ofthroughput optimal routing policies in multi-hop wireless networks,”in Proc. 47th Annu. Allerton Conf. Commun., Control, Comput., 2009,pp. 1395–1402.

[34] S. Boyd and L. Vandenberghe, Convex Optimization. New York:Cambridge Univ. Press, 2004.

[35] M. Neely and R. Urgaonkar, “Cross-layer adaptive control for wirelessmesh networks,” Ad Hoc Netw., vol. 5, no. 6, pp. 719–743, 2007.

[36] A. Eryilmaz, R. Srikant, and J. R. Perkins, “Stable scheduling policiesfor fading wireless channels,” IEEE/ACM Trans. Netw., vol. 13, no. 2,pp. 411–424, Apr. 2005.

Lei Ying (M’08) received the B.E. degree fromTsinghua University, Beijing, China, in 2001, andthe M.S. and Ph.D. degrees in electrical engineeringfrom the University of Illinois at Urbana–Champaignin 2003 and 2007, respectively.

During Fall 2007, he was a Post-Doctoral Fellowwith the University of Texas at Austin. He is currentlyan Assistant Professor with the Department of Elec-trical and Computer Engineering, Iowa State Univer-sity, Ames. He has been named the Litton AssistantProfessor in the department for 2010–2011. His re-

search interest is broadly in the area of information networks, including wirelessnetworks, mobile ad hoc networks, P2P networks, and social networks.

Dr. Ying received a Young Investigator Award from the Defense Threat Re-duction Agency (DTRA) in 2009 and a National Science Foundation (NSF)CAREER Award in 2010.

Sanjay Shakkottai (M’02) received the Ph.D. degreein electrical and computer engineering from the Uni-versity of Illinois at Urbana–Champaign in 2002.

He is with the University of Texas at Austin,where he is currently an Associate Professor andthe Engineering Foundation Centennial TeachingFellow in the Department of Electrical and ComputerEngineering. His current research interests includenetwork architectures, algorithms, and performanceanalysis for wireless and sensor networks.

Dr. Shakkottai received the National ScienceFoundation (NSF) CAREER Award in 2004.

Aneesh Reddy is currently pursuing the Ph.D. de-gree in electrical and computer engineering under theguidance of Dr. Sanjay Shakkottai at the Universityof Texas at Austin.

His research interests include distributed sched-uling algorithms in wireless ad hoc networks.

Shihuan Liu (S’10) received the B.E. degree inelectronics engineering from Tsinghua University,Beijing, China, in 2008, and is currently pursuing thePh.D. degree in electrical and computer engineeringat Iowa State University, Ames.

He currently performs research on resource alloca-tion in wireless networks.

Documents

IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 19 ...inlab.lab.asu.edu/Publications/YinShaRed_11.pdf842 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 19, NO. 3, JUNE 2011 where is the rate