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31
Logarithmic Asymptotics For SteadyState Tail Probabilities In A SingleServer Queue
, 1993
"... We consider the standard singleserver queue with unlimited waiting space and the firstin firstout service discipline, but without any explicit independence conditions on the interarrival and service times. We find conditions for the steadystate waitingtime distribution to have smalltail asympt ..."
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Cited by 150 (14 self)
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We consider the standard singleserver queue with unlimited waiting space and the firstin firstout service discipline, but without any explicit independence conditions on the interarrival and service times. We find conditions for the steadystate waitingtime distribution to have smalltail asymptotics of the form x  1 logP(W > x)  q * as x for q * > 0. We require only stationarity of the basic sequence of service times minus interarrival times and a Ga .. rtnerEllis condition for the cumulant generating function of the associated partial sums, i.e., n  1 log Ee qS n y(q) as n , plus regularity conditions on the decay rate function y. The asymptotic decay rate q * is the root of the equation y(q) = 0. This result in turn implies a corresponding asymptotic result for the steadystate workload in a queue with general nondecreasing input. This asymptotic result covers the case of multiple independent sources, so that it provides additional theoretical support for a concept of effective bandwidths for admission control in multiclass queues based on asymptotic decay rates.
Squeezing The Most Out Of ATM
, 1996
"... Even though ATM seems to be clearly the wave of the future, one performance analysis indicates that the combination of stringent performance requirements (e.g., 10  9 cell blocking probabilities), moderatesize buffers and highly bursty traffic will require that the utilization of the network be ..."
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Cited by 72 (10 self)
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Even though ATM seems to be clearly the wave of the future, one performance analysis indicates that the combination of stringent performance requirements (e.g., 10  9 cell blocking probabilities), moderatesize buffers and highly bursty traffic will require that the utilization of the network be quite low. That performance analysis is based on asymptotic decay rates of steadystate distributions used to develop a concept of effective bandwidths for connection admission control. However, we have developed an exact numerical algorithm that shows that the effectivebandwidth approximation can overestimate the target small blocking probabilities by several orders of magnitude when there are many sources that are more bursty than Poisson. The bad news is that the appealing simple connectionadmissioncontrol algorithm using effective bandwidths based solely on tailprobability asymptotic decay rates may actually not be as effective as many have hoped. The good news is that the statistical multiplexing gain on ATM networks may actually be higher than some have feared. For one example, thought to be realistic, our analysis indicates that the network actually can support twice as many sources as predicted by the effectivebandwidth approximation. That discrepancy occurs because for a large number of bursty sources the asymptotic constant in the tail probability exponential asymptote is extremely small. That in turn can be explained by the observation that the asymptotic constant decays exponentially in the number of sources when the sources are scaled to keep the total arrival rate fixed. We also show that the effectivebandwidth approximation is not always conservative. Specifically, for sources less bursty than Poisson, the asymptotic constant grows exponentially in the numbe...
Waitingtime tail probabilities in queues with longtail servicetime distributions
 QUEUEING SYSTEMS
, 1994
"... We consider the standard GI/G/1 queue with unlimited waiting room and the firstin firstout service discipline. We investigate the steadystate waitingtime tail probabilities P(W> x) when the servicetime distribution has a longtail distribution, i.e., when the servicetime distribution fails to ..."
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Cited by 55 (21 self)
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We consider the standard GI/G/1 queue with unlimited waiting room and the firstin firstout service discipline. We investigate the steadystate waitingtime tail probabilities P(W> x) when the servicetime distribution has a longtail distribution, i.e., when the servicetime distribution fails to have a finite moment generating function. We have developed algorithms for computing the waitingtime distribution by Laplace transform inversion when the Laplace transforms of the interarrivaltime and servicetime distributions are known. One algorithm, exploiting Pollaczek’s classical contourintegral representation of the Laplace transform, does not require that either of these transforms be rational. To facilitate such calculations, we introduce a convenient twoparameter family of longtail distributions on the positive half line with explicit Laplace transforms. This family is a Pareto mixture of exponential (PME) distributions. These PME distributions have monotone densities and Paretolike tails, i.e., are of order x − r for r> 1. We use this family of longtail distributions to investigate the quality of approximations based on asymptotics for P(W> x) as x → ∞. We show that the asymptotic approximations with these longtail servicetime distributions can be remarkably inaccurate for typical x values of interest. We also derive multiterm asymptotic expansions for the waitingtime tail probabilities in the M/G/1 queue. Even three terms of this expansion can be remarkably inaccurate for typical x values of interest. Thus, we evidently must rely on numerical algorithms for determining the waitingtime tail probabilities in this case. When working with servicetime data, we suggest using empirical Laplace transforms.
Sampling At Subexponential Times, With Queueing Applications
, 1998
"... We study the tail asymptotics of the r.v. X(T ) where fX(t)g is a stochastic process with a linear drift and satisfying some regularity conditions like a central limit theorem and a large deviations principle, and T is an independent r.v. with a subexponential distribution. We find that the tail of ..."
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Cited by 39 (4 self)
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We study the tail asymptotics of the r.v. X(T ) where fX(t)g is a stochastic process with a linear drift and satisfying some regularity conditions like a central limit theorem and a large deviations principle, and T is an independent r.v. with a subexponential distribution. We find that the tail of X(T ) is sensitive to whether or not T has a heavier or lighter tail than a Weibull distribution with tail e \Gamma p x . This leads to two distinct cases, heavytailed and moderately heavytailed, but also some results for the classical lighttailed case are given. The results are applied via distributional Little's law to establish tail asymptotics for steadystate queue length in GI/GI/1 queues with subexponential service times. Further applications are given for queues with vacations, and M/G/1 busy periods.
Exponential approximations for tail probabilities in queues, I: waiting times
 Oper. Res
, 1995
"... In this paper, we focus on simple exponential approximations for steadystate tail probabilities in G/GI/1 queues based on largetime asymptotics. We relate the largetime asymptotics for the steadystate waiting time, sojourn time and workload. We evaluate the exponential approximations based on th ..."
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Cited by 39 (20 self)
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In this paper, we focus on simple exponential approximations for steadystate tail probabilities in G/GI/1 queues based on largetime asymptotics. We relate the largetime asymptotics for the steadystate waiting time, sojourn time and workload. We evaluate the exponential approximations based on the exact asymptotic parameters and their approximations by making comparisons with exact numerical results for BMAP/GI/1 queues. Numerical examples show that the exponential approximations are remarkably accurate at the 90 th percentile and beyond. Key words: queues; approximations; asymptotics; tail probabilities; sojourn time and workload.
Loss Probability Calculations and Asymptotic Analysis for Finite Buffer Multiplexers
, 2001
"... In this paper, we propose an approximation for the loss probability, @ A, in a finite buffer system with buffer size. Our study is motivated by the case of a highspeed network where a large number of sources are expected to be multiplexed. Hence, by appealing to Central Limit Theorem type of argum ..."
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Cited by 28 (4 self)
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In this paper, we propose an approximation for the loss probability, @ A, in a finite buffer system with buffer size. Our study is motivated by the case of a highspeed network where a large number of sources are expected to be multiplexed. Hence, by appealing to Central Limit Theorem type of arguments, we model the input process as a general Gaussian process. Our result is obtained by making a simple mapping from the tail probability in an infinite buffer system to the loss probability in a finite buffer system. We also provide a strong asymptotic relationship between our approximation and the actual loss probability for a fairly large class of Gaussian input processes. We derive some interesting asymptotic properties of our approximation and illustrate its effectiveness via a detailed numerical investigation.
Exponential Bounds with Applications to Call Admission
, 1996
"... In this paper we develop a framework for computing upper and lower bounds of an exponential form for a large class of single resource systems with Markov additive inputs. Specifically, the bounds are on quantities such as backlog, queue length, and response time. Explicit or computable expressions f ..."
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Cited by 21 (10 self)
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In this paper we develop a framework for computing upper and lower bounds of an exponential form for a large class of single resource systems with Markov additive inputs. Specifically, the bounds are on quantities such as backlog, queue length, and response time. Explicit or computable expressions for our bounds are given in the context of queueing theory and numerical comparisons with other bounds are presented. The paper concludes with two applications to admission control in multimedia systems. Keywords: Tail distribution; Exponential bound; Large deviation principle; Ergodicity; Markov chain; Matrix analysis; Queues; Markov additive process; Effective bandwidth; Call admission control. P. Nain was supported in part by NSF under grant NCR9116183. This work was done when this author was visiting the University of Massachusetts in Amherst during the academic year 199394. y D. Towsley was supported in part by NSF under grant NCR9116183. 0 1 Introduction We are witnessing a ph...
Heavytraffic asymptotic expansions for the asymptotic decay rates
 in the BMAP/G/1 queue. Stochastic Models
, 1994
"... versatile Markovian point process, tail probabilities in queues, asymptotic decay rate, PerronFrobenius eigenvalue, asymptotic expansion, caudal characteristic curve, heavy traffic In great generality, the basic steadystate distributions in the BMAP / G /1 queue have asymptotically exponential tai ..."
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Cited by 15 (10 self)
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versatile Markovian point process, tail probabilities in queues, asymptotic decay rate, PerronFrobenius eigenvalue, asymptotic expansion, caudal characteristic curve, heavy traffic In great generality, the basic steadystate distributions in the BMAP / G /1 queue have asymptotically exponential tails. Here we develop asymptotic expansions for the asymptotic decay rates of these tail probabilities in powers of one minus the traffic intensity. The first term coincides with the decay rate of the exponential distribution arising in the standard heavytraffic limit. The coefficients of these heavytraffic expansions depend on the moments of the servicetime distribution and the derivatives of the PerronFrobenius eigenvalue δ(z) of the BMAP matrix generating function D(z) at z = 1. We give recursive formulas for the derivatives δ (k) ( 1). The asymptotic expansions provide the basis for efficiently computing the asymptotic decay rates as functions of the traffic intensity, i.e., the caudal characteristic curves. The asymptotic expansions also reveal what features of the model the asymptotic decay rates primarily depend upon. In particular, δ(z) coincides with the limiting timeaverage of the factorial cumulant generating function (the logarithm of the generating function) of the arrival counting process, and the derivatives δ (k) ( 1) coincide with the asymptotic factorial cumulants of the arrival counting process. This insight is important for admission control schemes in multiservice networks based in part on asymptotic decay rates. The interpretation helps identify appropriate statistics to compute from network traffic data in order to predict performance. 1.
Fundamental Characteristics of Queues with Fluctuating Load
 Proceedings of the ACM SIGMETRICS 2006 Conference on Management and Modeling of Computer Systems. Saint
, 2006
"... Systems whose arrival or service rates fluctuate over time are very common, but are still not well understood analytically. Stationary formulas are poor predictors of systems with fluctuating load. When the arrival and service processes fluctuate in a Markovian manner, computational methods, such as ..."
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Cited by 15 (10 self)
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Systems whose arrival or service rates fluctuate over time are very common, but are still not well understood analytically. Stationary formulas are poor predictors of systems with fluctuating load. When the arrival and service processes fluctuate in a Markovian manner, computational methods, such as Matrixanalytic and spectral analysis, have been instrumental in the numerical evaluation of quantities like mean response time. However, such computational tools provide only limited insight into the functional behavior of the system with respect to its primitive input parameters: the arrival rates, service rates, and rate of fluctuation. For example, the shape of the function that maps rate of fluctuation to mean response time is not well understood, even for an M/M/1 system. Is this function increasing, decreasing, monotonic? How is its shape affected by the primitive input parameters? Is there a simple closedform approximation for the shape of this curve? Turning to user experience: How is the performance experienced by a user arriving into a “high load ” period different from that of a user arriving into a “low load ” period, or simply a random user. Are there stochastic relations between these? In this paper, we provide the first answers to these fundamental questions.
Asymptotic Analysis Of Tail Probabilities Based On The Computation Of Moments
, 1995
"... Choudhury and Lucantoni recently developed an algorithm for calculating moments of a probability distribution by numerically inverting its moment generating function. They also showed that highorder moments can be used to calculate asymptotic parameters of the complementary cumulative distribution ..."
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Cited by 13 (7 self)
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Choudhury and Lucantoni recently developed an algorithm for calculating moments of a probability distribution by numerically inverting its moment generating function. They also showed that highorder moments can be used to calculate asymptotic parameters of the complementary cumulative distribution function when an asymptotic form is assumed, such as F c (x) ~ ax b e hx as x . Momentbased algorithms for computing asymptotic parameters are especially useful when the transforms are not available explicitly, as in models of busy periods or polling systems. Here we provide additional theoretical support for this momentbased algorithm for computing asymptotic parameters and new refined estimators for the case b 0. The new refined estimators converge much faster (as a function of moment order) than the previous estimators, which means that fewer moments are needed, thereby speeding up the algorithm. We also show how to compute all the parameters in a multiterm asymptote of the form F c (x) ~ k = 1 S m a k x b  k + 1 e hx . We identify conditions under which the estimators converge to the asymptotic parameters and we determine rates of convergence, focusing especially on the case b 0. Even when b = 0, we show that it is necessary to assume the asymptotic form for the complementary distribution function; the asymptotic form is not implied by convergence of the momentbased estimators alone. In order to get good estimators of the asymptotic decay rate h and the asymptotic power b when b 0, a multipleterm asymptotic expansion is required. Such asymptotic expansions typically hold when b 0, corresponding to the dominant singularity of the transform being a multiple pole (b a positive integer) or an algebraic singularity (branch point, b noninteger)...