We obtain upper bounds for the loss probability in a queue driven by an M/M/∞ source. The bound is compared with exact numerical results, and with bounds for two related arrivals models: superposed two state Markov fluids, and the Ornstein—Uhlenbeck process. The bounds are shown to behave continuously through approximation procedures relating the models

Daley, D. J.

Duffield, N. G.

English

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Bounds and comparisons of the loss ratio in queuesdriven by an M/M/oo source.N.G. Duffield* and D.J. DaleyJune 27, 1994AbstractWe obtain upper bounds for the loss probability in a queue driven by an M/M/csource. The bound is compared with exact numerical results, and with bounds fortwo related arrivals models: superposed two state Markov fluids, and the Ornstein—Uhienbeck process. The bounds are shown to behave continuously through approximation procedures relating the models.Keywords: Martingales, fluid flow models, heavy traffic limitsAMS 1991 Classifications: Primary 60K25; Secondary 68M20, 90B22, 68M20DIAS-APG-94-25School of Mathematical Sciences, Dublin City University, Dublin 9, Ireland; Dublin Institute for Advanced Studies, 10 Burlington Road, Dublin 4, Ireland. E-mail: thtffie1d.udcu. iet5tochastic Analysis Group, Australian National University, Canberra ACT 2600, Australia E-mail:dary1orac anu. ectu. au11 Introduction.There has been much interest recently in obtaining bounds for queue length distributionsin ATM multiplexers driven by Markovian traffic. Two fundamentalarrival processes havebeen investigated: superposed two-state Markov arrivals [5]; and the Ornstein—Uhlenbeckprocess [11]. In both cases, exponential bounds of the formP[queue> b] e_Sb(1)are obtained. The latter treatmentwas motivated in part as a heavy traffic approximationto the queue driven by an M/M/ccprocess, which in turn approximates superpositions ofMarkov fluid sources. Queues with Markov fluid arrivals were studied in [1]. The queue withM/M/oo arrivals has been investigated numerically in [6]. We mention alsoother recentwork on bounds for queues with Markovian arrivals [2, 7].The purpose of this note is to give explicit upper bounds of the form(1) directly for theM/M/co arrival process. Moreover, the bounds can be compared withexisting results ina number of directions. Firstly, bybounding the mean queue length we can compare withthe exact results of [6]. Secondly, we cancompare the bounds with those forqueues withfinitely superposed Markov fluid arrivals, and queues with Ornstein—Uhlenbeck arrivals. TheM/M/cc process can be regardedas intermediate between these: thearrival process of LMarkov fluid sources each with activity proportional to L’ convergesas L —* cc to theM/M/cc process, which in turn converges in a rescaled heavy traffic limit to the Ornstein—Uhlenbeck process. We show thatthe bounds we obtain for M/M/ccconverge in the samemanner. This is useful for the following reason. The prefactor of (1) found in [5]for L-foldsuperpositions of bursty sources is ofthe form for some < 1 depending only on theload of the system and the parameters of a single source. This demonstrates the economyof scale which is to be obtained bystatistical multiplexing. Our comparison of the boundsshows how this economy behaves through the approximation procedures described.2 The M/M/o process: boundsLet X = (X)a÷ denote the stationary M/M/co process. X is the birth-death process withPoissonian births at some rate \, each of which dies after an i.i.d. time which is exponentiallydistributed with mean Moreprecisely, X has sample paths in D+[O, cc), the spaceof non-negative integer valued paths which are right continuous andhave left limits. Thestationary distribution is Poissonianwith mean A/n. The corresponding Markov semigroupon t (the space of real sequences topologized with the supremum norm) has generator G2corresponding to the rate matrix (Gy),€z+ where(y=x—1),—(A±x) (x=y),A (y=x+l), (-)O (otherwise).The existence of such a closed G generating a Markov semigroup follows from standardconditions (see e.g. section IV.4 of [8]).The queueing process is as follows. X is the number of sources active at time —t. Eachactive source empties fluid into the buffer of a queue at rate a. Fluid is drained at rate sfrom the buffer. Defining the workload= f(aXti — s) dt’ (t 0), (3)the fluid remaining in the buffer at time 0 is (see e.g. §6—9 of [41)Q=supWt. (4)t>oThe offered load is p = E[aXo]/s = aA/(su).Let 11 denote the identity on % and H the number operator, densely defined in 40 by(J\fv)(x) = xv(x). For 8 E R let w(8) and v(.;&) be the maximal eigenvalue and corresponding eigenvector of the densely defined operator G(O) G + 8(aH — sI). DefineS = sup{8 w(8) = O}, and abbreviate v(.; 8) by v(.). Finally, let J’ be the canonicalfiltration generated by X.Lemma 1 M :=e5Tv(X) is an F-martingale.Proof: Consider the joint Markov process (Wi, X)>0 taking values in R x Z. Its generatorG is densely defined on C(R) 40 by(Og (5)Letting f(w, x) =e5’v(x) then Of = (II 0 G(8))f = 0, since G(8)v = 0. Consequently, byDynkin’s Theorem (see e.g. [10]), f(W,X) is an .F-martingale. CLet us find S and v. The eigenvector equation G(S)v = 0 becomes(v(x —1) + (Sa — )v(x)) + (Av(x +1) — (A + Ss)v(x)) = 0 (6)for x E N. This is solved by settingv(x+1) A+Ss_____==(7)v(x) A a—8a3for all n E N. The second equality in (7) yields (for p < 1)— a..\5 = = (1 — p)/a and hence r = p_I. (8).saTheorem 1 W7z.em p < 1 and b> 0,P{Q bJ ps/ae(1_P)(3/_Lb/a) (9)Proof: For any b > 0 define the F-stopping timer inf{t 0 I W > b}. Note{Q > b} = {r < }. By a mostly familiar argument involving Doob’s Optional SamplingTheorem (see e.g. [13}).E[11/Ioj = E[MTAkI E[Mr;7 < k] (10)and so by bounded convergence as k — c,E[Mo}E[M7lr<cjF{r<ooJ. (11)But if r < cc, then M e infr:>g v(x), because e5”= eSb and aXr s. (Since X isright-continuous, W is continuous and hence W,. =b. Similarly, if aX.,. <s, then W < b Oflsome interval (‘r, r + ), a contradiction). Hence_SbErW 1 ‘—SbEf —Xo1P[Q > bJe 1 01 = e 1P i p3/ae(1_P)(s/a_o/a) (12)lflfz:z>s/a v(x) inf:z>s1. pC3 Comparisons with finite superpositions.The M/M/oo process is itself a limit of a superposition of (rescaled) two state Markov fluidarrivals. This latter model was introduced in intoqueueing theory by Anick, Mitra andSondhi [1]. Specifically, consider the continuous time Markov chain with state space {0, 1}with transitions 0 —* 1 occurring at rate Ai and thereverse transition occurring at rate ,u.The Markov chain represents a fluid source: in stateI fluid arrives at rate a, in state 0 nofluid arrives. In an L-fold superposition, let X denote the number of sources active at time—t. X is the stationary Markov chain with generator corresponding to the rate matrix G”where for 0 <x,y < L(y=x—1),GL — —((L—x)AL+xL) (x=y), 13(L—x) (y=x+1),0 (otherwise).4Theorem 2 When PL < 1 andb> 0,1aL—3PLand TLPL aL—sXL converges to X in distribution. (See e.g.{10}). Furthermore, under (16)(14)The stationary distribution ofXL is binomial with mean LAL/(AL + au. The superpositionis served at constant rate s, and so the offered load is PL = aE[XJ/s = aLAL/(s(AL + ALL)).Let Q” denote the corresponding queue lengthat time 0. By repeating the steps of Theorem1 making the appropriate changes one proves:—Sb X0/ Lp[QL bJ e E{rL I =(\LTL + ILinf>31 \ L + /.LLwhere (1—pL)(AL+L)______(a — s/L)> 1. (15)By standard methods it is proved that under the scaling limitL — , L.AL \, —-f(16)6L8,rr(17)so that the bound in Theorem 2converges to that of Theorem 1.Daley and Ott [6] have performed exactnumerical calculations of the mean buffer occupationsin both cases. In Table 1 we present a comparison of the boundsfor heavy traffic where wechoose L = \/L, f-’L = 1u take ,u/a = 1 and vary c s/a. Herewe use the fact that sinceQ 0, P[Q > b e_Sb for b> 0 implies that E[Q] q/8. In Table 2 we investigate theaccuracy of the prefactor to theexponential bound, using the fact that P{Q > b] beimplies P[Q = 0] 1 — q5. In both tables, L = co corresponds to M/M/COarrivals.Table 1Stationary mean buffer content:bounds and exact evaluationsLPLt=6 ,=12c=24.96 .98 .99.96 .98 .99 .96 .98 .9950 Exact 16.34 35.62 74.30 11.33 25.6554.47 4.60 11.23 24.68Bound 19.36 38.77 77.52 14.44 28.97 57.90 6.75 13.64 27.23100 Exact 18.66 40.65 84.79 15.28 34.48 73.12 10.20 24.3953.14Bound 22.03 44.18 88.39 19.25 38.72 77.49 14.25 28.88 57.85250 Exact 20.12 43.83 91.41 17.94 40.4285.64 14.67 34.77 75.47Bound 23.72 47.59 95.25 22.47 45.25 90.62 20.08 40.73 81.70CO Exact 21.12 46.0295.97 19.84 44.64 94.54 18.11 42.73 92.54Bound 24.88 49.94 99.97 24.75 49.88 99.94 24.51 49.76 99.885Table 2P{QL. 0]: bounds and exact evaluationsic=6 c=12 c=24PL .96 .98 .99 .96 .98 .99 .96.98 .99L50 Exact .8876 .9432 .9714 .8415 .9194 .9593 .7498 .8706 .9343Bound .9944 .9986 .9997 .9872 .9968 .9992 .9637 .9908 .9977100 Exact .8905 .9447 .9722 .8508 .9243 .9618 .7865 .8905 .9446Bound .9948 .9987 .9997 .9889 .9972 .9993 .9746 .9936 .998400 Exact .8931 .9460 .9729 .8584 .9282 .9639 .8097 .9029 .9510Bound .9951 .9988 .9997 .9902 .9976 .9994 .9805 .9951 .99884 Comparison with Ornstein—Uhlenbeck arrivals.Finally, we investigate the relation of the bound of Theorem 1 to those found for theOrnstein—Uhlenbeck arrival process in [11]. The Ornstein—Uhlenbeck arrival process canbe seen as the last step in a chain of approximations: finite superpositions are approximatedby M/M/00 processes; the latter in turn by the Ornstein—Uhlenbeck process. To be precisewe take the limit, with a = = B’ and= (_1/2—(18)for positive constants s, j3 and v. This has the consequence that the load is p = 1— vi_h/2: weare dealing with a heavy traffic limit. For a particular value of i we denote the correspondingactivity process X of section 2 by X. Set Z =“2a(Xi —/1u). Then under the limits(18) Z’ converges to Z, the stationary diffusion which is the solution of the stochasticdifferential equationdZ = —Z±sdB. (19)where B is standard Brownian motion. (See [9] and [111). In other words, Z is a stationaryOrnstein—Uhlenbeck velocity process with mean 0 and variance The queue length for theprocess X’ is çtQ’ = sup J dt’(aX — s) = sup J dt’(Z — 3sz’). (20)t0 0 to 0In [11] the bound for the queue Q = supto f(Zt’ — sv)dt’ driven by Z and served at rate/35z) is found though martingale methods to beP[Q> bJ <e2/2e3. (21)6We remark now that the bound of Theorem 1 converges to the RHS of (21) under the limit(18). To see this note that under (18), 8 r43/s independent of ic, while the prefactor ofTheorem 1 has the limit1im(pe’) lim((l _1/2)e_vKhI)F=e2/2. (22)5 Conclusions.The bounds of Theorem 2 for finite superpositions contain a prefactor which is exponentialin L, the number of sources in the superposition. These determine the economies of scalewhich are to be found by multiplexing larger number of sources together at constant load,in the sense that they demonstrate how the usual effective bandwidth approximation F[Q >bl e_Sb overestimate the probability of loss. (See for example, [12] for discussion of theeffective bandwidth approximation, and [3] for a further discussion on economies of scale).The bounds obtain by successive approximations in Theorem 1 and equation (21) and theirconvergence in these approximations shows how the remnant of this economy survives in theheavy traffic limit.Acknowledgements. N.G.D. acknowledges a Visiting Fellowship from the Australian National University, where this work was started, and the hospitality of the Stochastic AnalysisGroup.7References[1] D. Anick, ID. Mitra and M.M. Sondhi (1982). Stochastic theory of a data-handlingsystem with multiple sources. Bell Sys. Tech. J., 61:1872—1894[2] S. Asmussen and T. Rolski (1993). Risk theory in a periodic environment: the Cramer—Lundberg approximation and Lundberg’s inequality. Preprint[3] D.D. Botvich and N.G. Duffield (1994). Large deviations, the shape of the loss curve,and economies of scale in large multiplexers.Preprint.[4] A.A. Borovkov (1976). Stochastic processes in queueing theory. Springer, New York.[5] E. Buffet and N.G. Duffleld (1992). Exponential upper bounds via martingales for multiplexers with Markovian arrivals. J. Appi. Prob., to appear.[6] D.J. Daley and T. Ott (1993). On a fluid model for a packet switch. Preprint.[7] N.G. Duffield (1993). Exponential bounds for queueswith Markovian arrivals. QueueingSystems , to appear.[8] S.N. Ethier and T.G. Kurtz (1986) Markov processes: characterization and convergence.Wiley, New York.[9] Iglehart, D.L. (1965) Limit diffusion approximations for the many server queue and therepairman problem. J. Appi. Prob. 2:429—441.[10] S. Karlin and JiM. Taylor (1981). A second coursein stochastic processes AcademicPress, New York.[11] V. Kulkarni and T. Rolski (1993). Fluid model driven by an Ornstein—Uhlenbeck process.Preprint.[12] W. Whitt (1993). Tall probabilities with statistical multiplexing and effective bandwidths in multi-class queues. Telecommunications Systems. 2:71—107[13] D. Williams (1991). Probability with martingales. CUP, Cambridge.8

Bounds and comparisons of the loss ratio in queues

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Bounds and comparisons of the loss ratio in queues driven by an M/M/∞ source.

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