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Admission control and routing in multi hop wireless networks


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  • 1. Proceedings of the WeA14.647th IEEE Conference on Decision and ControlCancun, Mexico, Dec. 9-11, 2008 Admission Control and Routing in Multi-Hop Wireless Networks Juan Jos´ Jaramillo and R. Srikant e Abstract— To enable services such as streaming multimedia The wireless channel is a shared resource, thus thereand voice in multi-hop wireless networks it is necessary to is resource contention among transmissions from differentdevelop algorithms that guarantee Quality of Service (QoS). nodes. As a result, even if a flow requests a pre-specifiedIn this paper, we consider the problem of optimal routingand admission control for flows which require a pre-specified bandwidth from the network, the load imposed by a flow onbandwidth from the network. We develop an admission control a node is a function of the topology of the network (forand routing algorithm whose performance is close to that of example, the number of neighbors and hidden terminals)an omniscient off-line algorithm that has complete a priori and the choice of the route itself. Hence, unlike a wirelineknowledge of the entire sequence (including the future) of flow network where the bandwidth consumed by a flow along aarrivals and their bandwidth requests. Our algorithm makesno statistical assumptions on the flow arrival pattern or other link is known at the arrival time of a flow, in a wirelessparameters of the arriving requests, and can be implemented network, the load imposed by a flow on a node can onlyin a distributed manner. be determined during route discovery. Therefore, it is not immediately obvious that the techniques for deriving optimal I. I NTRODUCTION QoS routing algorithms for wireline networks can be applied Future multi-hop wireless networks will carry a host of to wireless networks.multimedia services such as voice calls and video conferenc- In this paper, our contributions are as follows:ing. A common feature of such services is that they require • We first develop a model for QoS routing in multi-Quality of Service (QoS) guarantees; specifically we consider hop wireless networks which allows us to derive anservices that require a pre-specified bandwidth between the admission control and routing algorithm with provableendpoints of the flow. To support such services, the network performance guarantees using competitive analysis andmust be equipped with a protocol to decide whether or the work developed for wireline networks in [4].not to accept a new request, and to find a route with • We then show that no other algorithm performs bettersufficient bandwidth for an admitted flow. Optimal admission in an asymptotic sense to be described later. The proofcontrol and routing for pre-specified bandwidth flows has of this result is more involved and relies on the uniquebeen extensively studied for wireline networks. In the case characteristics of wireless networks, and uses the tech-of wireless networks, a number of papers have highlighted niques developed in [4].the difficulties in designing good QoS routing algorithms. • The optimal algorithm is not in a form that is amenableIn particular, the importance of taking contention count into to distributed implementation. Thus, an important con-consideration for available bandwidth estimation has been tribution is to convert the algorithm into a form thatrecognized in [1], [2] and the importance of load balancing to allows the use of standard shortest-path algorithms.maximize the number of admitted flows has been highlighted The rest of the paper is organized as follows. Section IIin [3]. However, no provably optimal algorithm has been presents an overview of previous related work. Competitivedeveloped to the best of our knowledge. analysis theory is presented in Section III. The network The idea of achieving provably good performance with- model and definitions are described in section IV, while inout any modeling assumptions on the arrival requests was section V we introduce our algorithm and use the competitiveproposed in [4]–[6], based on the concept of competitive analysis theory to prove performance guarantees. Section VIratio [7]–[10]. Informally, competitive ratio measures the proves that our algorithm is asymptotically optimal withperformance loss of a given algorithm caused by imperfect respect to the competitive ratio. We show how the algorithmdecisions due to the fact that it is oblivious of the future when can be implemented in a distributed fashion in section VII.compared against an off-line algorithm that has complete a Conclusions are presented in section VIII.priori knowledge of the sequence of requests, including thefuture, and can therefore make perfect decisions. For reasons II. R ELATED W ORKthat we will describe next, it is difficult to immediately adapt Finding algorithms to support quality of service in multi-the competitive ratio-based routing algorithms to wireless hop networks has been an active topic of research in thenetworks. last several years. Reference [2] studies the problem of Research supported by Motorola through the Motorola Center for Com- bandwidth estimation at a node while [1], [11] study themunication. problem of estimating the impact of contention in the avail- J. J. Jaramillo and R. Srikant are with the Coordinated Science Lab- able bandwidth in a multi-hop path. In [12], [13] someoratory, and the Department of Electrical and Computer Engineering,University of Illinois, Urbana, IL 61801 USA (e-mail:; heuristics are presented to support QoS but the effect contention in the admission process is ignored. The work978-1-4244-3124-3/08/$25.00 ©2008 IEEE 2356 Authorized licensed use limited to: Annamalai University. Downloaded on July 28,2010 at 05:32:00 UTC from IEEE Xplore. Restrictions apply.
  • 2. 47th IEEE CDC, Cancun, Mexico, Dec. 9-11, 2008 WeA14.6in [14] takes into account contention under the implicit mance of the best possible off-line algorithm for any requestassumption that the interference range of a node is equal sequence, and for a given performance its transmission range. IV. N ETWORK M ODEL In [15] the use of packet scheduling to guarantee QoSin multi-hop networks is studied. Some proposals rely on a The network is composed of a set N of N nodes, wherecentral algorithm to do admission control [16], [17], while node n ∈ N has capacity u(n). Without loss of generality,others have proposed strategies assuming a TDMA [18]– in the rest of the paper we will assume that[20] or CDMA over TDMA [21], [22] layer. The work in u(n) = 1 f or all n ∈ N . (1)[23] explores how to provide implicit synchronization inCSMA/CA networks to achieve TDM-like performance. The input to the algorithm is a set of flow requests F = A solution for multichannel multi-hop networks has been {f1 , f2 , . . . , fk }, where flow j is specified byproposed in [3] under the assumption that requests can be fj = sj , dj , rj (t), tS , tF , ρj . j jsplit; if requests are non-splittable a heuristic is introducedwhere requests are routed in the least-congested, minimum- Nodes sj and dj are the source and destination respec-hop path. To the best of our knowledge, our algorithm is tively of an unidirectional flow.1 Flow j requests a bandwidththe first provably optimal for general networks that allows a rj (t) at time t, where we define rj (t) = 0 for t ∈ [ tS , tF ). / j jdistributed implementation. Thus, tS and tF are the start and finish times of flow j. j j For simplicity, and without loss of generality, we assume III. C OMPETITIVE A NALYSIS that these times are integers. A profit of ρj is accrued if The concept of competitive ratio was first introduced by the flow is admitted into the network. Given fj ∈ F, our[7] and further developed by [8]–[10]. Here we will present algorithm will output a path Pj assigned for the request,a brief overview of this theory. with the understanding that Pj = ∅ if it is rejected. In many situations we must develop efficient algorithms Let λn (t) be the relative load on node n at time t, which iswhich have to deal with a sequence of tasks one at a time, a function of the flows currently admitted by our algorithm.where the efficiency of current decisions is affected by future Flow j’s holding time is denoted by Tj = tF − tS , where j jtasks. One classical example is the well-known ski rental we define the maximum holding timeproblem, where a ski enthusiast plans on skiing for several T = max {Tj } .days while weather permits. Our ski fan is faced with two joptions every day, either rent skis for the day at a price of r As mentioned in Section I, the wireless channel is a sharedor buy them at cost b. If we knew that the total number of resource and contention among transmissions from differentdays to ski is d, then the optimal decision algorithm would nodes implies that when admitting a flow we must takebe to buy skis if b < rd. Since we have no knowledge of into account the impact of a flow in the network. To betterthe future, we have to develop an algorithm that has to make understand this, let us use the simple scenario of Fig. 1.decisions on a day by day basis and still performs well. To Our linear network of 5 nodes is such that any node cando that we introduce the concepts of on-line and off-line only communicate with its nearest neighbors, and where thealgorithm. interference range for each node is illustrated as a concentric Definition 1: An on-line algorithm is an algorithm that circle around each node. There is a flow from node B to nodehas to deal with a sequence of requests one at a time, without D that requires a rate of r. Due to the exposed terminalhaving the entire sequence available from the beginning. problem, node A cannot transmit while B is transmitting, so Definition 2: An off-line algorithm is an algorithm that node A’s available capacity ishas complete a priori knowledge of the entire request se- (1)quence, including future requests, before it outputs its answer u(A) − r = 1 − r,to solve the problem at hand. (1) One way to measure the performance of an on-line algo- where = means that the equality follows from equationrithm is by comparing it against the best possible off-line (1). We use this notation throughout the paper.algorithm when both have to deal with the same set of re- Thus, when scheduling this flow, we must reserve a ratequests. The competitive ratio then measures the performance of r in node A for it to remain idle. Similarly, node B mustloss of an on-line algorithm caused by imperfect decisions transmit at a rate r and must remain silent while C is relayingwhen compared against an off-line algorithm that can make a packet for this flow, which implies a resource reservation ofperfect decisions since it has complete knowledge of the rate 2r at B. Because of the hidden terminal problem, noderequest sequence. E must remain idle while D is receiving a packet, so we Definition 3: The competitive ratio of an on-line algo- have to reserve a rate of r at this node. Following the linesrithm is the supremum over all possible request sequences of this argument, it can be checked that we must reserve aof the performance ratio between the best possible off-line rate of 2r at nodes C and D in order to be able to schedulealgorithm and the on-line algorithm. this flow. These values are shown above each node in Fig. 1. This means that if an on-line algorithm has a competitive 1 For the case of bidirectional flows, we simply need to split the requestratio of c then its performance is at least 1/c the perfor- in two unidirectional flows that need to be accepted/rejected simultaneously. 2357 Authorized licensed use limited to: Annamalai University. Downloaded on July 28,2010 at 05:32:00 UTC from IEEE Xplore. Restrictions apply.
  • 3. 47th IEEE CDC, Cancun, Mexico, Dec. 9-11, 2008 WeA14.6 Define QT (Pj ) to be the total impact of flow j (routed on path Pj ) on the network. Thus, r 2r 2r 2r r A B r C r D E QT (Pj ) = Qn (Pj ). (3) n∈N Additionally, define QT and Q as follows: QT = max {QT (Pj )} j Fig. 1. Example of resource contention among nodes. Q = max {Qn (Pj )} . (4) j,n We normalize the cost such that for any flow fj ∈ F and any time such that rj (t) > 0 3r A r B r C r D r E ρj t0 t1 t2 t0 1≤ ≤F (5) QT rj (t)Tj for F large enough. For example, if we have that rj (t) = rj for t ∈ [ tS , tF ) and the profit is defined to be proportional j j to the bandwidth-holding time product, i.e. throughput, thenFig. 2. Example of resource contention among nodes under the assumption we can make ρj = QT rj Tj and let F = 1.of perfect packet transmission scheduling. Finally, we assume that rate requirements are small enough compared to node capacity. Specifically, This example gives the intuition for the following defini- min {u(n)} (1) 1 ntion. Let Qn (Pj ) be the impact of flow j on node n if path rj (t) ≤ = (6) Q log µ Q log µPj is used. Formally, where Qn (Pj ) = I{n′ ∈Pj } I{n′ =dj } + I{n′ ∈HTn (Pj )} , µ = 2 (1 + QT T F ) , (7) n′ ∈Nn (2) and log means log2 .where Nn is the set of nodes within interference range of Later we will prove in Section VI that (6) is a necessarynode n (including node n itself), HTn (Pj ) is the set of nodes condition for any algorithm to achieve logarithmic competi-in Nn that need to receive a transmission for flow j that node tive ratio.n cannot sense –that is, hidden terminal transmissions–, andI{} is the indicator function defined as V. A LGORITHM 1 if statement = T RU E The main goal is to develop an admission control and I{statement} = . routing algorithm that enforces capacity constraints, that is 0 if statement = F ALSE As an example, in Fig. 1 we have that for path P = λn (t) ≤ 1 f or all t and n ∈ N . (8){B, C, D}, QA (P ) = 1, QC (P ) = 2, and QE (P ) = 1.It must be noted that this definition is only an upper bound and maximizes profit:on the actual impact of a flow in a given node, and can ρj .only be improved by assuming a specific, possibly ideal, j:Pj =∅transmission scheduling algorithm. To illustrate this, in Fig. 2we assume that there is a flow from node A to E and that Furthermore, the algorithm cannot rely on knowledgewe schedule node transmissions such that nodes A and D about future flows to make admission decisions and oncesimultaneously transmit at time t0 , node B transmits at t1 , a flow has been admitted no rerouting is allowed and noand C is scheduled to transmit at time t2 . In this case, the flow is to be interrupted.impact of the flow on node C is 3 instead of 4, as estimated To do that, it will sequentially analyze flows from F andusing Qn (Pj ). 2 decide whether to admit them or not. Finding methods for estimating Qn (Pj ) has been an active A. Definitiontopic of research. For related work, the reader is referred to[1], [11], [14]. The decision rule for admitting flow fj ∈ F and assigning a path is: 2 It must be highlighted that for the ease of explanation we have assumedthat due to the exposed terminal problem a node cannot transmit while 1) For all t ∈ [ tS , tF ), n ∈ N let the cost of node n at j janother in its interference neighborhood is transmitting. If a certain physical time t belayer technology does not preclude such transmissions, the definition of (1)Qn (Pj ) should be modified accordingly and the results in the rest of the cn (t) = u(n) µλn (t) − 1 = µλn (t) − 1. (9)paper still apply with minor modifications. 2358 Authorized licensed use limited to: Annamalai University. Downloaded on July 28,2010 at 05:32:00 UTC from IEEE Xplore. Restrictions apply.
  • 4. 47th IEEE CDC, Cancun, Mexico, Dec. 9-11, 2008 WeA14.6 2) If there exists a path Pj from node sj to dj such that Lemma 3: Let AOA be the set of indices of accepted rj (t) requests by the optimal off-line algorithm but rejected by Qn (Pj ) cn (t) ≤ ρj (10) the ACR algorithm. Let m = max AOA . Then u(n) n∈N tS ≤t<tF j j ρj ≤ cn (t, m). then route fj using Pj and set j∈AOA n t rj (t) (Due to lack of space, the proofs of the Lemmas will be λn (t) ← λn (t) + Qn (Pj ) (11) presented in a longer version of the paper.) u(n) Now, we are ready to prove the following: for all n ∈ N , tS ≤ t < tF . j j Theorem 1: The ACR algorithm enforces capacity con- Note that straints and achieves a competitive ratio of O(log(QT T F )). rj (t) Qn (Pj ) (Due to lack of space, the proof of the Theorem will be u(n) presented in a longer version of the paper.)is the fraction of node n’s capacity that would be used by Remark: It is important to highlight that for the lemmasflow j. Thus, the algorithm compares the link cost weighted and theorem of this section we only assume that Qn (Pj ) ≥ 1,by this quantity to the profit and admits the call if the cost but the precise definition given in (2), which depends on theis less than or equal to the profit. assumptions about the physical layer, is only used in the following sections.B. Performance Guarantees Now that the wireless model and the algorithm are defined, VI. O PTIMALITYwe can use the techniques developed for wireline networks in Now we will prove that no other algorithm can achieve[4]. We will first proceed to prove that our algorithm enforces a better competitive ratio than the ACR algorithm in ancapacity constraints, which from now on we will call the asymptotic sense and that assumption (6) is a necessaryAdmission Control and Routing (ACR) algorithm. In other condition to achieve a good competitive ratio. To do that, wewords, if the ACR algorithm decides to admit a flow request, will first show that even if flow rates are small enough, thethen there is sufficient available capacity. profit of the optimal off-line algorithm will exceed the profit Lemma 1: Capacity constraints are enforced by the ACR of any on-line algorithm that is oblivious to the future by aalgorithm. That is, factor of Ω(log(QT T F )). Finally, we will present stronger λn (t) ≤ 1 f or all t and n ∈ N . bounds for the case when flow rates are allowed to be(Due to lack of space, the proof of the Lemma will be relatively large, showing that the competitive ratio degradespresented in a longer version of the paper.) when we allow large rates. The proof of the competitiveness of the ACR algorithm The techniques used here are again similar to the onesis done in two steps. In Lemma 2 we prove that the total developed for wireline networks in [4], but rely on the uniquenetwork cost is at most within a factor of the accrued gain, characteristics of wireless networks, specially to find worstand in Lemma 3 we prove that the profit due to requests case scenarios in Lemmas 4 and 7.rejected by the ACR algorithm and accepted by the optimal Lemma 4: Any on-line algorithm has a competitive ratiooff-line algorithm is bounded by the total network cost. These of Ω(log QT ).two results then imply that the profit of the ACR algorithm is Lemma 5: Any on-line algorithm has a competitive ratiowithin a factor of the profit accrued by the off-line algorithm, of Ω(log T ).and hence the competitive ratio is bounded. Lemma 6: Any on-line algorithm has a competitive ratio For the following two lemmas, define λn (t, j) to be the of Ω(log F ).relative load on node n at time t when only the first j − 1 (Due to lack of space, the proofs of the Lemmas will beflow requests have been either admitted or rejected. That is, presented in a longer version of the paper.) (11) ri (t) (1) Hence, we have just proved the following. λn (t, j) = Qn (Pi ) = Qn (Pi )ri (t), (12) Theorem 2: Any on-line algorithm has a competitive ratio i<j u(n) i<j of Ω(log(QT T F )).with the understanding that Pi = ∅ if fi ∈ {f1 , f2 , . . . , fj−1 } Proof: It is a direct consequence of Lemmas 4, 5 andis rejected. 6. Similarly, and from (9), let cn (t, j) be defined as For the proof of Theorem 2 we assume that (6) holds. We will now proceed to prove that if this bound does cn (t, j) = µλn (t,j) − 1. (13) not hold then no on-line algorithm can achieve logarithmic Lemma 2: Let AACR be the set of indices of accepted competitive ratio. 1flows by the ACR algorithm and k be the index of the last Lemma 7: If we allow requests of rate up to 4k then the 1flow request in F. Then, competitive ratio is Ω(QT ) for any positive integer k. 4k 1 Lemma 8: If we allow requests of rate up to k then the 2 log µ ρj ≥ cn (t, k + 1). 1 j∈AACR t n competitive ratio is Ω(T k ) for any positive integer k. 2359 Authorized licensed use limited to: Annamalai University. Downloaded on July 28,2010 at 05:32:00 UTC from IEEE Xplore. Restrictions apply.
  • 5. 47th IEEE CDC, Cancun, Mexico, Dec. 9-11, 2008 WeA14.6 1 Lemma 9: If we allow requests of rate up to k then the is the cost of using edge e for routing flow fj . It must be 1competitive ratio is Ω(F k ) for any positive integer k. noted that (15) decouples the cost of an edge from the path(Due to lack of space, the proofs of the Lemmas will be cost, allowing a distributed implementation of a shortest pathpresented in a longer version of the paper.) algorithm to find the optimal route. Furthermore, for every Therefore, we have the following theorem. edge e ∈ E we only need to gather information from the 1 local set Ns(e) Nd(e) to find Ce (j). It must be noted that Theorem 3: If we allow requests of rate up to 4k then the 1 1competitive ratio is Ω(QT + T 4k + F 4k ) for any positive 4k 1 for any admission algorithm this is the minimal informationinteger k. that must be gathered and updated in order to check resource Proof: This follows from Lemmas 7, 8 and 9. availability, since the transmission in this link will affect the Thus, in order to achieve logarithmic competitive ratio we load of all nodes in the set Ns(e) Nd(e) .need to let k be greater than VIII. C ONCLUSIONS log(QT T F ) . 4 log(log(QT T F )) We have developed a model for QoS routing in multi-hop VII. D ISTRIBUTED I MPLEMENTATION wireless networks which allows us to derive an algorithm with provable performance guarantees using competitive In its present form, checking analysis. We proved that our algorithm has a performance rj (t) close to that of an omniscient off-line algorithm that has Qn (Pj ) cn (t) ≤ ρj u(n) complete a priori knowledge of the entire sequence of flow n∈N tS ≤t<tF j j arrivals (including the future) and their bandwidth the ACR algorithm requires first to specify a path Pj Our algorithm makes no statistical assumptions on the flowfrom source to destination and then its associated cost can be arrival pattern or other parameters of the arriving requests.calculated. Since we ideally would like to use the minimum We also proved that no other algorithm performs bettercost path, this means that it would be required to first identify in an asymptotic sense of the competitive ratio. Finally,all possible paths and then find the cost for each one of them. we showed that our algorithm is amenable to a distributed The contribution of this section is to show how this can implemented using a distributed algorithm that can findthe minimum cost path without the need to first identify all R EFERENCESpossible solutions, and where every node only needs to getaccess to information from a local neighborhood. [1] K. Sanzgiri, I. D. Chakeres, and E. M. Belding-Royer, “Pre-reply probe and route request tail: Approaches for calculation of intra- Define the directed graph G = (N , E), where N is the flow contention in multihop wireless networks,” Mobile Networks andset of nodes in the wireless network and E is the set of all Applications, vol. 11, no. 1, pp. 21–35, Feb. 2006.directed transmission edges. Formally, e ∈ E if and only if [2] I. D. Chakeres, E. M. Belding-Royer, and J. P. 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