2. 130 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 19, NO. 1, FEBRUARY 2011be formed in the network. We demonstrated the use of exclu- single-queue systems; we derive sample path bounds on asive sets for the purpose of deriving lower bounds on delay for a group of queues upstream of a bottleneck;wireless network with single-hop trafﬁc in [11]. In this paper, we • derivation of a fundamental lower bound on the system-further generalize the typical notion of a bottleneck. In our ter- wide average queuing delay of a packet in multihop wire-minology, we deﬁne a -bottleneck to be a set of links less network, regardless of the scheduling policy used, bysuch that no more than of them can simultaneously transmit. analyzing the single-queue systems obtained;Fig. 1 shows bottlenecks for the network under the 1-hop • extensive numerical studies and discussion on the usefulinterference model. In this paper, we develop new analytical insights into the design of optimal or nearly optimal sched-techniques that focus on the queueing due to the -bot- uling policies gained by the lower bound analysis.tlenecks and thereby avoid the complex interactions in the net- We begin with the description of the system model. We thenwork. One of the techniques, which we call the “reduction tech- present our methodology for obtaining reductions and usingnique,” simpliﬁes the analysis of the queueing upstream of a them to lower-bound the system-wide average delay of packets. -bottleneck to the study of a single queue system with We then address the question of designing delay-efﬁcient servers as indicated in the ﬁgure. schedulers. We then provide concrete examples illustrating the It is also possible to derive stochastic upper bounds on the methodology and comparison of the back-pressure policy toaverage delay of the network using the techniques of Lyapunov the lower bound. We also describe how the proposed approachdrifts [8]. We were able to obtain sharper upper bounds in differs fundamentally from the existing techniques and can be[11] by using a different Lyapunov function, but we do not used to gain deeper understanding of the scheduling policiespursue them here because they focus on a speciﬁc scheduling for wireless networks.scheme. Our focus, on the other hand, is to derive a funda-mental bound on the performance of any policy. Moreover,our lower bound techniques capture the effect of statistical II. SYSTEM MODELmultiplexing of packets due to several ﬂows passing through acommon -bottleneck, which cannot be analyzed using We consider a wireless network , where is thethe method of Lyapunov drifts. As a result, the upper bounds set of nodes and is the set of links. Each link has unit capacity.computed using these techniques [8] tend to be quite loose in There are ﬂows, each distinguished by its source–destinationmost practical scenarios. pair . There is a ﬁxed route (set of links) between the We consider the lower bound analysis as an important ﬁrst source and corresponding destination . Each ﬂow has itsstep toward a complete delay analysis of multihop wireless sys- own exogenous arrival stream .tems. For a tandem queue network, the average delay of a delay- Each packet has a deterministic service time equal to one unit.optimal policy proposed by [24] numerically coincides with the The exogenous arrivals at each source are assumed to be inde-lower bound provided in this paper. A clique network is a spe- pendent. Let represent the vectorcial graph where at most one link can be scheduled at any given of exogenous arrivals, where is the number of packets in-time. Using existing results on work-conserving queues, we de- jected into the system by the source during time slot (forsign a delay-optimal policy for a clique network and compare ). Let represent the corre-it to the lower bound. For a network with node-exclusive in- sponding arrival rate vector.terference, our lower bound is tight in the sense that it goes to The path on which ﬂow is routed is speciﬁed asinﬁnity whenever the delay of any throughput-optimal policy is , where is a node at a -hop dis-unbounded. tance from the source node . The source node is denoted by We will see that although delay-optimal policies can be de-rived for some simple networks like the clique and the tandem, and the destination node by , where is the pathderiving such policies in general is extremely complex. Instead, length. The packets arriving at each node are queued. Each nodewe reengineer a well-known throughput-optimal scheduling maintains a separate queue for each ﬂow that passes throughpolicy known as back-pressure policy and demonstrate that, for the node. Let denote the queue length at node corre-certain representative topologies, its delay performance is close sponding to ﬂow . After reaching the destination node, eachto the fundamental lower bound. Finally, we also present a packet leaves the system, i.e., . The queue lengthcase where neither back-pressure policy nor the shadow queue vector is denoted by . Mul-approach (proposed in [2]) are close to the lower bound. For tiple ﬂows can share a link . A link can be activated in a timethis case, we design a new handcrafted policy whose delay slot only if the corresponding queue is nonempty. We use theperformance is actually close to the lower bound. Thus, the term activation (scheduling) of a link or a queue interchange-lower bound analysis provides useful insights into the design ably. At most, one packet is served at a queue in a given timeof optimal or nearly optimal scheduling policies. slot. The service structure is slotted. We now summarize our main contributions in this paper: The set of links that do not cause mutual interference and • development of a new queue grouping technique to handle hence can be scheduled simultaneously are called activation the complex correlations of the service process resulting vectors (matchings). Let be the collection of all activation from the multihop nature of the ﬂows; we also introduce a vectors . We allow the activation vectors to be arbitrary, i.e., novel concept of -bottlenecks in the network; they can characterize any interference model. At each time slot, • development of a new technique to reduce the analysis an activation vector is scheduled depending on the sched- of queueing upstream of a bottleneck to studying simple uling policy and the underlying interference model. The indi-
3. GUPTA AND SHROFF: DELAY ANALYSIS AND OPTIMALITY OF SCHEDULING POLICIES FOR MULTIHOP WIRELESS NETWORKS 131cator function indicates whether or not ﬂow received ser- simultaneously. For example, [11] and [14] identify cliques invice at the th hop from source at time slot . Note that the conﬂict graph as the bottlenecks. This corresponds to a set of links, among which only one link can be scheduled at any given time. We call these sets of links exclusive sets. We also discuss another type of bottleneck in the case of a cycle graph, where no if and link is scheduled (1) more than two links can be scheduled simultaneously. Some of otherwise the important exclusive sets for the wireless grid example under The evolution of the queues in the system is as follows: the 2-hop interference model are highlighted in Fig. 9. We use the indicator function to indicate whether the ﬂow passes through the -bottleneck. The total ﬂow if rate crossing the bottleneck is given by (2) otherwise We use the 2-hop interference model in most of our simula- (3)tion studies since it has often been used to model the behavior ofa large class of MAC protocols based on virtual carrier sensingusing RTS/CTS messages, which includes the IEEE 802.11 pro- Let the ﬂow enter the -bottleneck at the node andtocol [1]. Under an -hop interference model, any two active leave it at the node . Hence, equals the number oflinks in are always separated by or more hops in the un- links in the -bottleneck that are used by ﬂow .derlying network graph. We deﬁne and as follows: III. DERIVING LOWER BOUNDS ON AVERAGE DELAY (4) In this section, we present our methodology to derive lowerbounds on the system-wide average packet delay for a given (5)multihop wireless network. At a high level, we partitionthe ﬂows into several groups. Each group passes through a -bottleneck, and the queueing for each group is ana-lyzed individually. The grouping is done so as to maximizethe lower bound on the system wide expected delay. For ﬂows B. Reduction Techniquepassing through a given bottleneck (a group), we lower boundthe sum of queues upstream and downstream of the bottleneck In this section, we demonstrate our methodology to deriveseparately. We reduce the analysis of queueing upstream of lower bounds on the average size of the queues correspondinga -bottleneck to studying single-queue systems fed to the ﬂows that pass through a -bottleneck.by appropriate arrival processes. These arrival processes are By deﬁnition, the number of links/packets scheduled in thesimple functions of the exogenous arrival processes of the bottleneck, , is no more than , i.e.,original network. For example, Fig. 1 shows the reduction oftwo -bottlenecks in the network. A separate lower boundcan be established for the queues downstream of the network. (6)The lower bound on the system-wide average delay of a packetis then computed using the statistics of the exogenous arrival A ﬂow may pass through multiple links in . Among all theprocesses. We derive analytical expressions of the lower bounds ﬂows that pass through , let denote the maximum numberfor a large class of arrival processes. of links in the -bottleneck that are used by any single In this section, we ﬁrst characterize the bottlenecks in the ﬂow, i.e.,system. We then explain how to lower-bound the average delayof the packets of the ﬂows that pass through a given -bot- (7)tleneck. Our analysis justiﬁes the reduction of a -bottle-neck to a single-queue system fed by appropriate arrival pro-cesses. Finally, we present a greedy algorithm that takes as input Let denote the sum of queue lengths of the ﬁrsta system with ﬂows and possibly multiple bottlenecks and re- queues of ﬂow at time , i.e.,turns a lower bound on the system-wide average packet delay. (8)A. Characterizing Bottlenecks in the System Link interference causes certain bottlenecks to be formed in Summing (2) from to , we havethe system. Deﬁne a -bottleneck to be be a set of links such that no more than of its links can be scheduled (9)
4. 132 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 19, NO. 1, FEBRUARY 2011 Theorem 3.1: For a -bottleneck in the system, at any time , the sum of the queue lengths in , under any sched- uling policy is no smaller than that of the reduced system, i.e., . Proof: We prove the above theorem using the principle of mathematical induction. Base Case: The theorem holds true for since theFig. 2. Reducing a bottleneck-exclusive set in Fig. 9 to a G/D/1 queue. Notethat A (t), A (t), and A (t) are external arrivals to the original system, so system is initially empty.the arrivals to the reduced G/D/1 system are all external. Induction Hypothesis: Assume that the theorem holds at a time , i.e., . Induction Step: The following two cases arise. The sum of queues upstream of each link in at time is Case 1)given by and satisﬁes the following property: (13) Case 2) Using (10), we have the following: (10) (14) Now we consider the evolution of the queues under an ar-bitrary scheduling policy that is given by the following equation: Hence, the theorem holds for . Thus, by the principle of mathematical induction, the theorem (11) holds for all . Remarks: Note: By summing the queues upstream of the bottleneck and • The above analysis captures the combinatorial interferencedeﬁning , we are able to avoid correlation terms among constraints and reduces the bottleneck to a G/D/K systemthe arrival and service processes in the queue evolution equa- with appropriate inputs for the purpose of establishingtion of the system [(11)]. We obtain a lower bound on the value lower bounds. Such a system can be analyzed for a largeof in Theorem 3.1 by studying a reduced system. Using class of arrival trafﬁc.the result of Theorem 3.1, we obtain a lower bound on the ex- • Even when the arrival process is not amenable to anal-pected delay for the ﬂows passing through the bottleneck in ysis, the above reduction can be used to obtain sample pathCorollary 3.1. lower bound via simulation. For example, while evaluating Reduced System: Consider a system with a single server and a scheduling algorithm via trace-based simulator, we can as the input. The server serves at most packets from feed the arrival trace to the corresponding G/D/K systemthe queue. Let be the queue length of this system at to obtain a lower bound on its performance. Furthermore,time . The queue evolution of the reduced system is given by lower bound on important network-wide statistics couldthe following equation: also be obtained using the above technique along with the ﬂow partition technique described in Section III-D. (12) • The analysis here is very general and establishes a fun- damental lower bound even for the traditional wirelinewhere setting. if • We emphasize that can be computed from (5) and otherwise considers only the exogenous inputs to the system. Fur- thermore, the lower bound on the expected delay can be The reduction procedure is illustrated in Fig. 2 where we have computed using only the statistics of the exogenous arrivalreduced one of the bottlenecks in the grid example shown in process and not their sample paths.Fig. 9. Flows II, IV, and VI pass through an exclusive set using We note that a policy may achieve the above lower bound2, 3, and 2 hops of the exclusive set, respectively. The corre- on the sum of upstream queues if it schedules the samesponding G/D/1 system is fed by the exogenous arrival streams number of packets as the corresponding G/D/K system in every , , and . time-slot. However, this may not always be possible because of Without loss of generality, we can assume that both systems the interference caused by other ﬂows in the system. It is alsoare empty initially, i.e., . We now establish important to note that even if a scheme achieves the lower boundthat at all times , is smaller than . on , it does not imply that it would be delay-optimal (i.e.,
5. GUPTA AND SHROFF: DELAY ANALYSIS AND OPTIMALITY OF SCHEDULING POLICIES FOR MULTIHOP WIRELESS NETWORKS 133it will minimize the total number of packets in the system at Applying Little’s lawall times). We will provide an example of the clique networkin Section IV-A. We now discuss the derivation of an explicitlower bound on expected delay of the ﬂows passing through thebottleneck using this theorem.C. Bound on Expected Delay We now present a lower bound on the expected delay of the (19)ﬂows passing through the bottleneck as a simple function of theexpected delay of the reduced system. In the analysis, we useTheorem 3.1 to bound the queueing upstream of the bottleneckand a simple bound on the queueing downstream of the bottle- D. Flow Partitionneck. Applying Little’s law on the complete system, we derive alower bound on the expected delay of the ﬂows passing through Let be the set of ﬂows in the system. Let be a partitionthe bottleneck. on such that each element is a set of ﬂows passing Corollary 3.1: Let be the expected value of queuing through a common -bottleneck. The expected delaydelay for the G/D/1 system with input . Furthermore, let of the ﬂows in can be lower bounded using Corollary 3.1. A be the expected delay of the ﬂows passing through . system-wide lower bound on the expected delay of the packets,Then , can then be obtained by a straightforward application of the Little’s law (20) Proof: Let denote the queue length of the G/D/1system at time . Theorem 3.1 states that at all times Our objective is to compute a partition such that the lower bound on can be maximized. The optimal partition can be computed using a dynamic program, but the computation costs can be exponential in the number of ﬂows in the worst case. WeSince for all , , thus now present a greedy algorithm that computes a lower bound on the average delay for a system containing multiple bottlenecks. Assume that we have precomputed a list of -bottle- necks in the system. Algorithm 1 proceeds by greedily searching for a set of ﬂows and the corresponding -bot-Using (7), it follows that tleneck that yields the maximum lower bound. The value of the variable is incremented, and the ﬂows in are then removed from . The process is repeated until all the ﬂows are (15) removed. Thus, we obtain a decomposition of the wireless net- work into several single-queue systems and obtain a bound onand hence the expected delay. Algorithm 1: Greedy Partitioning Algorithm (16) 1: After crossing the bottleneck, a packet of ﬂow has to cross 2: 3: repeat hops. Since the links are of unit capacity, the delay at 4: Find the -bottleneck which maximizeseach of these hops is at least one unit. Thus, for all 5: (17) 6: 7: until Taking expectations on both sides of (16) and using (17), we 8: returnobtain The -bottlenecks correspond to cliques in the conﬂict graph [14]. Let be the largest number of links that inter- fere with a link . The time complexity to compute all (18) the -bottlenecks is exponential in in the worst case. In
6. 134 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 19, NO. 1, FEBRUARY 2011general, the time complexity to compute all the -bottle-necks is exponential. However, in our experiments we ﬁnd thatthe number of -bottlenecks in graphs with 2-hop interfer-ence is much smaller. To reduce the complexity of the problem,we can restrict ourselves to computing bottlenecks around theset of links where the several ﬂows converge. The lower bound analysis may be loose on account of thefollowing. First, inequality (15) is loose when the ﬂows passthrough different number of links in the same bottleneck.Second, the lower bound obtained by Algorithm 1 can capturethe effect of any ﬂow at only one bottleneck. Hence, it would Fig. 3. Example network (clique) with interference constraints such that onlyunderestimate the congestion caused by a ﬂow passing through one pair of nodes can communicate at any given time. Note that the packets maymultiple bottlenecks. Third, we assume that the queueing in still need to traverse multiple hops.each bottleneck is independent of each other, which may notbe possible because of interference among two bottlenecks. gained from the nature of the delay-optimal scheduling for theFinally, in the derivation of the lower bound by the reduction clique and tandem networks.technique, we have neglected the nonempty queue constraintsby grouping the arrivals into a single queue, and hence we A. Cliqueunderestimate the delay. We evaluate the impact of these relax- A clique network is one in which the interference constraintsations on the accuracy of the lower bound using simulations. allow only one link to be scheduled at any given time. SuchDespite these relaxations, we ﬁnd that the lower bound gives a a situation may arise, for example, in the downlink of a baseuseful estimate of the average delay in the system. station that employs relays to increase coverage and/or data rates (see, for e.g., [23]). Suppose there are ﬂows in the clique IV. DESIGN OF DELAY-EFFICIENT POLICIES network. An example network with six ﬂows is shown in Fig. 3. Every link lies in the interference range of the other, and hence We now address the important question of designing a delay- only one link can be scheduled at any given time.efﬁcient scheduler for general multihop wireless networks. We It is well known that the Shortest Remaining Timewill see that although delay-optimal policies can be derived for First (SRTF) policy is sample path optimal in a work-con-some simple networks like the clique and the tandem, deriving serving queue with preemption. Using this result, we cansuch policies in general is extremely complex. Intuitively, such design a scheduling policy that minimizes the total numbera scheduler must satisfy the following properties. of packets in the system at all times for every sequence of • Ensure high throughput: This is important because if the arrivals. This is also known as sample-path delay optimality. In scheduling policy does not guarantee high throughput, then particular, we will show that for the given network, scheduling the delay may become inﬁnite under heavy loading. the packet that is closest to its destination is optimal. • Allocate resources equitably: The network resources must Let be the maximum number of hops a ﬂow takes in the be shared among the ﬂows so as not to starve some of the clique network ﬂows. Also, noninterfering links in the network have to be scheduled such that certain links are not starved for service. (21) Starvation leads to an increase in the average delay in the system. For the sake of simplicity, we deﬁne an -dimensional The above properties are difﬁcult to achieve, given the dy- vector , which represents the state of the system:namics of the network and the lack of a priori information of ,the packet arrival process. In the light of the previous work [16], where .[18], we choose to investigate the back-pressure policy with Note that we do not distinguish packets of different ﬂows inﬁxed routing (Section IV-B). The back-pressure policy has been this description of the state. We only consider the distance ofwidely used to develop solutions for a variety of problems in a given packet from its destination. Let be the numberthe context of wireless networks [8], [18]; and the importance of of packets that are hops from their respective destinationsstudying the tradeoffs in stability, delay, and complexity of these at time . The following equation describes the evolution ofsolutions is now being realized by the research community. This when the activation vector is scheduled at time :policy tries to maintain the queues corresponding to each ﬂowin decreasing order of size from the source to the destination.This is achieved by using the value of differential backlog (dif-ference of backlogs at the two ends of a link) as the weight for ifthe link and scheduling the matching with the highest weight. (22)As a result, the policy is throughput-optimal. Henceforth, weshall refer to this policy as only the back-pressure policy. A scheduling operation schedules a packet from , pro- We ﬁrst study the delay-optimal policy for a clique network. vided that the queue is nonempty. is the number of exoge-We then modify the back-pressure policy using the intuition nous packets arriving to the system at time , which are hops
7. GUPTA AND SHROFF: DELAY ANALYSIS AND OPTIMALITY OF SCHEDULING POLICIES FOR MULTIHOP WIRELESS NETWORKS 135from their respective destinations. The optimal scheduling rule at all times, nor is it guaranteed to be delay-optimal. It is in- schedules teresting to note that the optimal policies for the clique net- work and the tandem network (see [24]) have the property that (23) they give priority to the packets closest to their destination. The rationale behind this choice is that such a decision drains out We begin with the proof of the sample-path optimality result. the system in minimum possible time (also called clearance Lemma 4.1: Consider the evolution of the system under the time [9], [10]), which is a necessary (but not sufﬁcient) condi-policy and an arbitrary policy . Let , be the tion for delay optimality. In the context of stochastic networks,queue length processes under and , respectively, when such policies are called the Last Buffer First Serve (LBFS) poli-the system starts from the same initial state under both policies. cies [19]. At the other end of the spectrum are the First BufferAssume that the number of arrivals in any slot is ﬁnite. For all First Serve (FBFS) policies [19], which incur maximum delay , we have among the class of work-conserving policies. We simulate these networks in Section V and observe that the average delay of the optimal policy is close to the lower bound. However, as shown in [10], it is impossible to design poli- Proof: The system is preemptive in that a different packet cies that minimize the total number of packets in the system atmay be scheduled in the next slot. The system is work-con- all times, even for a simple switch (equivalent to a wireless net-serving because, after each scheduling decision, the number of work with bipartite graph and node-exclusive interference). Wehops the packet needs to traverse decreases by 1. We will de- instead take the approach of modifying the back-pressure policyscribe the mapping of the problem to an equivalent work-con- by increasing the relative priority of packets that are withinserving system. 1 hop of their destinations. Assume a single-queue system with a single server. Each newarrival of a packet corresponding to a ﬂow in the clique network B. Back-Pressure Policymarks the arrival of a new job to the corresponding single-queue The back-pressure policy may lead to large delays since thesystem. The remaining service time of the job is equal to its backlogs are progressively larger from the destination to thedistance from the destination. Hence, the SRPT policy corre- source. The packets are routed only from a longer queue to asponds to scheduling the packet closest to the destination. In shorter queue, and certain links may have to remain idle untilother words, rule is optimal. this condition is met. Hence, it is likely that all the queues up- Note: Lemma 4.1 tells us that the policy is a sample-path stream of a bottleneck will grow long, leading to larger delays.delay-optimal policy since its optimality does not depend on the A common observation of the optimal policies for the cliquenature of the arrival process. We next show that any work-con- and the tandem network is that increasing the priority of packetsserving scheduling policy minimizes for the clique network close to the destination reduces the delay. In the context of(see (10) of Section III-B) on a sample path basis although it wireline (stochastic-processing) networks, this is known as themay not be minimize the sum of queue lengths at all times. LBFS rule studied in [19]. We introduce a new family of func- Lower Bound: Since no more than one link can be scheduled tions parametrized by for computing the differential backlogsin the clique network, it is a -bottleneck. Let us deﬁne of the links. Using the parameter , we control relative priority for the clique network (see (10) of Section III-B). It can be of links. This in turn inﬂuences the average delay of the system.shown that for the clique network Our simulations indicate that for certain topologies, for an appropriate choice of , the average delay in the above system can be reduced close to the fundamental lower bound. For a (IV.24) tandem queue network, as goes to zero, the delay performance of the back-pressure policy numerically coincides with that of We now show that any work-conserving policy (that sched- the delay-optimal policy proposed by [24] and also the lowerules a nonempty queue) will achieve the lower bound on , bound provided in this paper. We give a formal description ofi.e., at all times . Suppose is the activation vector the back-pressure policy.scheduled by the policy Let be a link of interest. Suppose that ﬂow passes through link and that nodes and are at a distance of and hops, respectively, from the source node . In our notation, . Deﬁne the differential backlog of ﬂow passing through a link as for some (26) For each link , the ﬂow with the maximum differential backlog (25) is chosen by the ﬂow-scheduling component (27) in Fig. 4. The link-scheduling component shown in Fig. 4 schedules the acti- From the above, we conclude that a policy that minimizes vation vector with the maximum weight at every time slot. A may not minimize the sum of queue lengths in the system packet of ﬂow is transmitted on link at time
8. 136 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 19, NO. 1, FEBRUARY 2011 We implemented an algorithm to compute all the exclusive sets of a graph under a given interference model. We also im- plement Algorithm 1 and the back-pressure policy described in Section IV-B. Cplex [12], an integer-programming solver, was used to compute the maximum weight matchings. Except for the Tandem Queue, the 2-hop interference model has been used in all other simulations. All the simulations have been run long enough for the 95% conﬁdence intervals to become small as shown in Fig. 8. Arrival Processes: The arrival stream at each source is a series of active and idle periods. During the active periods, the source injects one packet into the queue in every time slot. The length of the active periods (denoted by random variable ) is distributed according the Zipf law with power exponent 1.25 and ﬁnite sup-Fig. 4. Back-pressure policy with ﬁxed routing. port . Truncated heavy-tailed distributions like Zipf have been found to model the Internet trafﬁc [7]. During the active period, the source generates one packet every time slot.if ﬂow had the maximum differential backlog at link , link The idle periods are geometrically distributed with mean . Thewas present in the maximum weighted matching, and the corre- mean arrival rate of a source can be controlled by changing thesponding queue was nonempty. value of . The lower bounds were obtained using Algorithm 1. Let denote the Euclidean norm of vector . The We use the analysis in [6] to obtain the expected delay for thesystem is considered to be stable if is single-queue systems.bounded. If the system is stable, then the throughput of a givenﬂow is the same as the arrival rate. A throughput vector is A. Tandem Queueadmissible if there is some scheduling policy under which thesystem is stable when the arrival rate vector is . We denote We consider a stream of packets ﬂowing over the wirelessby the closure of the convex hull of the set of activation links in tandem, as shown in Fig. 5(a) under the 1-hop interfer-vectors , and by the interior of the convex hull. ence model. For this system, any two links that are adjacent to Let be the indicator variable indicating whether the each other form an exclusive set. Choosing the ﬁrst two linksﬂow passes through link . The sum of the rates of the ﬂows as the bottleneck maximizes the lower bound in Corollary 3.1sharing link is given by as it maximizes the value of . Note that is the same for all exclusive sets in the system. The lower bound for the above arrival process is given by (30)Let be the corresponding ﬂow rate vector. (31) It has been shown in [16] that if each arrival process is i.i.d. where is the arrival rate in packets/slot andin time, and that the ﬁrst two moments of all the arrival streams is the mean residual time in an active period. Simulation results are ﬁnite, then is a necessary condition for a in Fig. 6 show that this lower bound virtually coincides with thestabilizing scheduling policy to exist. It has also been shown that delay performance of the optimal scheme [24].the back-pressure policy with ﬁxed routing (with ) stabi- The value of in the back-pressure policy can be used tolizes the system for any arrival rate satisfying the preceding con- control the relative priority of links. For example, assume thatdition. It can be shown using the ﬂuid model techniques devel- the queue lengths at different nodes in the tandem queue are asoped in [4] and [22] that this policy with is stable when- shown in Fig. 5(a). For , the differential backlogsever the arrival processes satisfy a strong-law-of-large numbers through are 20, 30, 10, 5, 5, 12, 17, and 1, respectively.assumption and the ﬂow rate vector . For , the differential backlogs through are 0.035, 0.071, 0.033, 0.019, 0.022, 0.007, 0.335, and 1, respec- V. ILLUSTRATIVE EXAMPLES tively. Notice that as the value of decreases, the value of differ- We now demonstrate our methodology on a variety of exam- ential backlog between two nonempty queues becomes smaller.ples shown in Fig. 5. The bottleneck sets in each example have The differential backlog at the last hop becomes comparativelybeen highlighted in the corresponding ﬁgures. We also support large for small values of , thereby increasing the relative pri-the lower bound with results obtained from the simulations of ority of the last link. We observe, for small values of , mostthe back-pressure policy (Section IV-B) to show that the lower of the queueing takes place in the ﬁrst few hops of the ﬂow andbounds are indeed useful. Not only does the lower bound serve the average backlogs at the downstream links are very small.as a rough estimate, but it can also be used to gain understanding Intuitively, the scheme reduces to the delay-optimal scheme ason the back-pressure policy itself. Furthermore, we also com- conﬁrmed by the simulation results shown in Fig. 6.pare the performance of the back-pressure policy with the max- Indeed, the scheme in [24] is valid only for the tandem queueimal policy [3], [26] in Sections V-C and V-D. under 1-hop interference model. It has been suggested in the
9. GUPTA AND SHROFF: DELAY ANALYSIS AND OPTIMALITY OF SCHEDULING POLICIES FOR MULTIHOP WIRELESS NETWORKS 137Fig. 5. Illustration of the Lower Bound analysis (bottlenecks used in the analysis have been highlighted) (a) Tandem Queue (b) Dumbbell (c) Tree (d) Cycle. Fig. 7. Simulation results for clique shown in Fig. 3.Fig. 6. Simulation results for Tandem Queue.literature [15], [22] that the delay performance of single-hopsystems improves as goes to zero. We also observe a similarpattern for the tandem queue. However, as we will see later,this observation may not be generalizable to a multihop wirelessnetworks with several ﬂows since, for small values of , certainﬂows may be starved for resources.B. Clique We now consider the clique network with sixﬂows as shown in Fig. 3 with the load vector packets/slot. Fig. 8. Simulation results for dumbbell topology.We observe in Fig. 7 that the optimal policy (Last Buffer FirstServe) derived in Section IV is indeed closest to the lowerbound. The gap between the performance of LBFS and the C. Dumbbell Topologylower bound can be attributed to the fact that the ﬂows differ Consider a dumbbell topology [Fig. 5(b)] with multiplein path length, making inequality (15) loose. On the other ﬂows passing through a single link with the load vectorhand, we notice that the FBFS policy incurs the maximum packets/slot. The bottleneck-exclusivedelay among the policies simulated herein. Note that for this set has been highlighted in Fig. 5(b). The performance ofexample, every work conserving policy is throughput-optimal back-pressure policy is compared to that of the lowerand also minimizes at all times. However, the average bound in Fig. 8. This example shows that the back-pressuredelay performance is dependent on the relative priority of the policy is able to share the resources among the ﬂows in abuffers. It is evident from Fig. 7 that decreasing the value of manner such that the overall delay of the system is close toin the back-pressure policy improves its delay performance. the lower bound. We also note that the delay performance of
10. 138 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 19, NO. 1, FEBRUARY 2011Fig. 9. Grid network with multiple ﬂows. Some of the important bottleneckshave been highlighted. Fig. 11. Simulation results for cycle topology. We simulated the system with load vector packets/slot and observed that the lower bounds derived by the analysis of the G/D/2 system are tighter when . We also obtained lower bounds using the bottlenecks corresponding to exclusive sets {1, 2, 3} and {6, 7, 8} for ﬂows I and II, respectively. These bounds are nonetheless useful for light loads. Thus, it is possible to derive accurate lower bounds for the wireless system by considering appropriate bottlenecks. We would like to note that in this case, ﬂows I and II have the same source and destination nodes, and the two ﬂows interact closely with each other because of link interference. We foundFig. 10. Simulation results for tree topology. that for , ﬂow II is starved when the system is highly loaded . This is because, for small values of , the differential backlogs are very similar in value to each other, eventhe back-pressure policy is signiﬁcantly better than that of the if some of the queues are very long. Hence, the algorithm is notmaximal scheduling policy [3], [26]. able to preferentially schedule the longer queues in the system. For , the performance at lighter loads is, however, veryD. Tree Topology similar to that for , which is shown in Fig. 11. Consider a tree topology with three ﬂows convergingat the root of the tree shown in Fig. 5(c) with load vector F. Analysis of the Example in Fig. 9 packets/slot. Such a topology is often found In this example, we analyze the wireless grid with randomlyin sensor networks, wireless access networks, etc. For the case generated ﬂows described in Fig. 9. There are several bottle-simulated here, Algorithm 1 decomposes the system into two necks in this system that interfere with each other under thebottlenecks. Note that the inequality (15) would be loose since 2-hop interference model. We studied the system for several dif-ﬂows II and III pass through a different number of links in the ferent input load vectors . We ﬁnd that depending on the inputbottleneck. Also, Algorithm 1 underestimates the lower bound load vector, Algorithm 1 computes different partitions for theby neglecting the interference between the two bottlenecks. ﬂows in the system. We discuss two representative load vectorsAs shown in Fig. 10, the performance of back-pressure policy to evaluate the impact of the relaxations made in the analysis. is signiﬁcantly better than that of the maximal Case 1) packets/scheduling policy and is also close to the lower bound. This slot.suggests that the impact of the relaxations made in the analysis For the given load vector, Algorithm 1 computesis relatively small in this case. the partition {{II, VI}; {I, V}; {III}; {IV}; {VII}}. Note that ﬂow IV interferes with ﬂows II and VIE. Cycle Topology signiﬁcantly, but this effect is not captured by the We now illustrate the application of the lower bound analysis lower bound analysis. Also note that ﬂow I inter-to a -bottleneck. For the given cycle network in Fig. 5(d), feres with ﬂows IV and VI. The lower bound com-no more than two links can be scheduled at any time under puted by Algorithm 1 is 185.9 slots/packet. We also2-hop interference constraints, i.e., for , simulated the system under the back-pressure policy . It can be easily veriﬁed that the analysis presented for several different values of . The average delayin Section III-B can be used to reduce the system to a G/D/2 was found to be 308.0 slots/packet for .queue having arrivals and , respectively. For smaller values of , ﬂow VI was starved for
11. GUPTA AND SHROFF: DELAY ANALYSIS AND OPTIMALITY OF SCHEDULING POLICIES FOR MULTIHOP WIRELESS NETWORKS 139Fig. 12. Linear network with multiple short ﬂows and a single long ﬂow. resources, resulting in larger delays. The average delay for and was 323.3 and 476.2 slots/packet, respectively. Case 2) packets/ slot. In this case, we remove ﬂow IV from the system and keep all other arrivals rate the same. The lower bound computed by Algo- Fig. 13. Simulation results for the linear network. rithm 1 is 196.0 slots/packet. The average delay under the back-pressure policy was found to be 230.7 slots/packet for , which is in better up to link 1. Note that the packets do not have a common des- agreement with the lower bound as compared to the tination. Thus, once the link schedule is obtained, we schedule previous case, even though ﬂow I interferes with the ﬂow on the link for which the packet is closest to its desti- ﬂow VI. Interestingly, in this case, decreasing the nation; i.e., we schedule the short ﬂow in preference to the long value of causes an increase in the queues along ﬂow. ﬂow II, while increasing the value of causes an We conclude that the lower bound analysis presented here increase in the queues along ﬂow VI. The average can play an important role in obtaining insights into the de- delay for and was 254.7 and sign and evaluation of scheduling policies for multihop wireless 244.1 slots/packet, respectively. networks. Section VI provides a perspective on the research on These examples also show that it is nontrivial to predict the delay analysis in multihop wireless networks and the contribu-value of in the back-pressure policy that minimizes the av- tions made in this paper.erage delay in the system. Thus, a small value of is not sufﬁ-cient for the policy to be delay-efﬁcient. VI. DISCUSSION AND RELATED WORKG. Linear Network Much of the analysis [3], [8], [18] for multihop wireless Finally, we discuss an example of a line network with several networks has been limited to establishing the stability of the“short” ﬂows and a single “long” ﬂow as shown in Fig. 12. The system. Whenever there exists a scheme that can stabilizepacket arrival rate is the same for all the ﬂows in this network. the system for a given load, the back-pressure policy is alsoWe use a 1-hop interference model. In this example, it is very guaranteed to keep the system stable. Hence, it is referred to asdifﬁcult to allocate the resources equitably and at the same time a throughput-optimal policy. It also has the advantage of beingreduce the backlogs in the system. For example, by using a small a myopic policy in that it does not require knowledge of thevalue of , the short ﬂows (I–X) get much higher priority in arrival process. In this paper, we have taken an important stepcomparison to ﬂow XI. On the other hand, for a large value of , toward the expected delay analysis of these systems.the backlogs upstream of ﬂow (in the case of ﬂow XI) are large. The general research on the delay analysis of scheduling poli-Hence, it is not possible to reduce the delay simply by changing cies has progressed in the following main directions. as indicated by Fig. 13. We then implement the scheme pro- • Heavy trafﬁc regime using ﬂuid models: Fluid modelsposed in [2] to alleviate the problem of large backlogs associated have typically been used to either establish the stabilitywith back-pressure algorithm by using counters called shadow of the system or to study the workload process in thequeues to allocate service rates to each ﬂow on each link in an heavy trafﬁc regime. It has been shown in [5] that theadaptive fashion without knowing the set of packet arrival rates. maximum-pressure policy (similar to the back-pressureHowever, we ﬁnd that it does not reduce the queueing in the policy) minimizes the workload process for a stochasticsystem. Comparing the performance of these algorithms to the processing network in the heavy trafﬁc regime whenlower bound in Fig. 13, we concluded that there must be poli- processor splitting is allowed.cies that incur smaller delay in the system. We then design a • Stochastic bounds using Lyapunov drifts: This method isnew scheduling policy, which although is not guaranteed to be developed in [8], [17], [20], and [21] and is used to deriveoptimal, has much better delay performance. In fact, its perfor- upper bounds on the average queue length for these sys-mance is close to the lower bound as shown in Fig. 13. tems. However, these results are order results and provide The scheme is based on the observation that the packet closer only a limited characterization of the delay of the system.to the destination must be given higher priority. We implement For example, it has been shown in [21] that the maximalthe scheduling rule followed by Tassiulas’ optimal policy in matching policies achieve delay for networks with[24]. Thus, we schedule links beginning from link 10 and go single-hop trafﬁc when the input load is in the reduced
12. 140 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 19, NO. 1, FEBRUARY 2011 capacity region. This analysis, however, has not been ex- discussions in Sections V-E and V-F, some of the ﬂows may tended to the multihop trafﬁc case because of the lack of an be starved for resources when is small. Hence, the intuition analogous Lyapunov function for the back-pressure policy. from the single-hop case does not automatically generalize to • Large deviations: Large deviation results for cellular sys- the multihop case. tems have been obtained in [25], [28] to calculate queue- overﬂow probability. Similar analysis is much more dif- VII. CONCLUSION ﬁcult for the multihop wireless network considered here due to the complex interactions between the arrival, ser- The delay analysis of wireless networks is largely an open vice, and backlog process. problem. In fact, even in the wireline setting, obtaining analyt- Here, we have taken a different approach to reduce the wire- ical results on the delay beyond the product form types of net-less network to single queueing systems that are then analyzed works has posed great challenges. These are further exacerbatedto construct the lower bound. This technique captures the essen- in the wireless setting due to complexity of scheduling neededtial features of the wireless network and is useful since, in many to mitigate interference. Thus, new approaches are required tocases, we can also ﬁnd that the back-pressure policy performs address the delay problem in multihop wireless systems. To thisclose to the lower bound. Perhaps, the most important advantage end, we develop a new approach to reduce the bottlenecks inof the lower bound is that it can be used for analyzing a large a multihop wireless to single-queue systems to carry out lowerclass of arrival processes using known results in the queueing bound analysis.literature [6]. For a special class of wireless systems (cliques), we are able Our approach, however, depends on the efﬁcient computation to obtain a sample-path delay-optimal scheduling policy. Weof the bottlenecks in the system. A complete characterization of also obtain policies that minimize a function of queue lengthsthe bottlenecks in a multihop wireless network is an extremely at all times on a sample-path basis. Furthermore, for a tandemdifﬁcult problem. Exclusive sets characterized in [14] prove to queueing system, we show numerically that the expected delaybe a good beginning for delay analysis. However, they are not of a previously known delay-optimal policy coincides with theenough to obtain tight lower bounds, as shown in the case of a lower bound.cyclic network. The analysis is very general and admits a large class of ar- The design of a delay-optimal policy that achieves minimum rival processes. Also, the analysis can be readily extended topossible average delay of packets in the network for a given handle channel variations. The main difﬁculty, however is inrouting matrix has proved to be very challenging. Except for identifying the bottlenecks in the system. The lower bound nota delay-optimal scheduling scheme for the tandem queue under only helps us identify near-optimal policies, but may also helpthe node-exclusive interference model derived in [24], no result in the design of a delay-efﬁcient policy as indicated by the nu-is known for other topologies and interference models. merical studies. 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IEEE INFOCOM, 2008, pp. 6–10. nications and a Professor of electrical and computer engineering and computer [22] D. Shah and D. Wischik, “Heavy trafﬁc analysis of optimal scheduling science and engineering. His research interests span the areas of wireless and algorithms for switched networks,” 2008. wireline communication networks. He is especially interested in fundamental [23] K. Sundaresan and S. Rangarajan, “On exploiting diversity and spa- problems in the design, performance, pricing, and security of these networks. tial reuse in relay-enabled wireless networks,” in Proc. ACM MobiHoc, Dr. Shroff is an Editor for the IEEE/ACM TRANSACTIONS ON NETWORKING 2008, pp. 13–22. and Computer Networks, and a past editor of IEEE Communications Letters. He [24] L. Tassiulas and A. Ephremides, “Dynamic scheduling for minimum has served on the Technical and Executive Committees of several major con- delay in tandem and parallel constrained queueing models,” Ann. Oper. ferences and workshops. He was the Technical Program Co-Chair of the IEEE Res., vol. 48, pp. 333–355, 1993. INFOCOM 2003, the premier conference in communication networking. He [25] V. J. Venkataramanan and X. Lin, “Structural properties of LDP for was also the Conference Chair of the 14th Annual IEEE Computer Communi- queue-length based wireless scheduling algorithms,” in Proc. 45th cations Workshop (CCW 1999), the Program Co-Chair for the Symposium on Annu. Allerton Conf. Commun., Control, Comput., 2007, pp. 759–766. High-Speed Networks and IEEE GLOBECOM 2001, and the Panel Co-Chair [26] T. Weller and B. Hajek, “Scheduling nonuniform trafﬁc in a packet- for ACM MobiCom 2002. He was also a co-organizer of the National Science switching system with small propagation delay,” IEEE/ACM Trans. Foundation (NSF) workshop on Fundamental Research in Networking, held in Netw., vol. 5, no. 6, pp. 813–823, Dec. 1997. Arlie House, Warrenton, VA, in 2003. In 2008, he served as the Technical Pro- [27] Y. Xi and E. M. Yeh, “Optimal capacity allocation, routing, and con- gram Co-Chair of ACM MobiHoc. He received the Best Paper Award at IEEE gestion control in wireless networks,” in Proc. IEEE. ISIT, Jul. 2006, INFOCOM 2006 and 2008, the IEEE IWQoS 2006 Best Student Paper Award, pp. 2511–2515. the 2005 Best Paper of the Year Award for the Journal of Communications and [28] L. Ying, R. Srikant, A. Eryilmaz, and G. E. Dullerud, “A large devi- Networking, the 2003 Best Paper of the Year Award for Computer Networks, ations analysis of scheduling in wireless networks,” IEEE Trans. Inf. and the NSF CAREER Award in 1996. His INFOCOM 2005 paper was also Theory, vol. 52, no. 11, pp. 5088–5098, Nov. 2006. selected as one of two runner-up papers for the Best Paper Award.
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