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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 measure.to 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.
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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.
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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.
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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 requests.in 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 implementation.be 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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