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17
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...
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.
Asymptotics for steadystate tail probabilities in structured Markov queueing models
 Commun. Statist.Stoch. Mod
, 1994
"... In this paper we establish asymptotics for the basic steadystate distributions in a large class of singleserver queues. We consider the waiting time, the workload (virtual waiting time) and the steadystate queue lengths at an arbitrary time, just before an arrival and just after a departure. We s ..."
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Cited by 38 (10 self)
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In this paper we establish asymptotics for the basic steadystate distributions in a large class of singleserver queues. We consider the waiting time, the workload (virtual waiting time) and the steadystate queue lengths at an arbitrary time, just before an arrival and just after a departure. We start by establishing asymptotics for steadystate distributions of Markov chains of M/GI/1 type. Then we treat steadystate distributions in the BMAP/GI/1 queue, which has a batch Markovian arrival process (BMAP). The BMAP is equivalent to the versatile Markovian point process or Neuts (N) process; it generalizes the Markovian arrival process (MAP) by allowing batch arrivals. The MAP includes the Markovmodulated Poisson process (MMPP), the phasetype renewal process (PH) and independent superpositions of these as special cases. We also establish asymptotics for steadystate distributions in the MAP/MSP/1 queue, which has a Markovian service process (MSP). The MSP is a MAP independent of the arrival process generating service completions during the time the server is busy. In great generality (but not always), the basic steadystate distributions have asymptotically exponential tails in all these models. When they do, the asymptotic parameters of the different distributions are closely related. 1.
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.
Performance Analysis of a Digital Link with Heterogeneous Multislot Traffic
 IEEE Trans. on Comm
, 1995
"... We present a unified model to compute various performance measures when Poissonian and nonPoissonian (renewal) multislot traffic streams are offered to a digital link in a (double) loss system. We represent the nonPoissonian arrival process by a matrixexponential distribution, requiring only tha ..."
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Cited by 10 (8 self)
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We present a unified model to compute various performance measures when Poissonian and nonPoissonian (renewal) multislot traffic streams are offered to a digital link in a (double) loss system. We represent the nonPoissonian arrival process by a matrixexponential distribution, requiring only that the interarrival distribution has a rational Laplace transform. Several distributions are considered as the nonPoissonian traffic. The resulting model uses matrix algebraic techniques only,thus not requiring any complex and/or tedious transform techniques. We also incorporate various control policies in our modeling framework using acceptance functions. Through our computational studies, we conclude that the second and the third moments of the nonPoissonian traffic have significant impact on various performance measures.
A HeavyTraffic Expansion For Asymptotic Decay Rates Of Tail Probabilities In MultiChannel Queues
 RES. LETTERS
, 1992
"... We establish a heavytraffic asymptotic expansion (in powers of one minus the traffic intensity) for the asymptotic decay rates of queuelength and workload tail probabilities in stable infinitecapacity multichannel queues. The specific model has multiple independent heterogeneous servers, each wi ..."
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Cited by 9 (7 self)
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We establish a heavytraffic asymptotic expansion (in powers of one minus the traffic intensity) for the asymptotic decay rates of queuelength and workload tail probabilities in stable infinitecapacity multichannel queues. The specific model has multiple independent heterogeneous servers, each with i.i.d. service times, that are independent of the arrival process, which is the superposition of independent nonidentical renewal processes. Customers are assigned to the first available server in the order of arrival. The heavytraffic expansion yields relatively simple approximations for the tails of steadystate distributions and higher percentiles, yielding insight into the impact of the first three moments of the defining distributions.
Realtime delay estimation based on delay history
, 2007
"... Motivated by interest in making delay announcements to arriving customers who must wait in call centers and related service systems, we study the performance of alternative realtime delay estimators based on recent customer delay experience. The main estimators considered are: (i) the delay of the ..."
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Cited by 8 (4 self)
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Motivated by interest in making delay announcements to arriving customers who must wait in call centers and related service systems, we study the performance of alternative realtime delay estimators based on recent customer delay experience. The main estimators considered are: (i) the delay of the last customer to enter service (LES), (ii) the delay experienced so far by the customer at the head of the line (HOL), and (iii) the delay experienced by the customer to have arrived most recently among those who have already completed service (RCS). We compare these delayhistory estimators to the estimator based on the queue length (QL), which requires knowledge of the mean interval between successive service completions in addition to the queue length. We characterize performance by the mean squared error (MSE). We do analysis and conduct simulations for the standard GI/M/s multiserver queueing model, emphasizing the case of large s. We obtain analytical results for the conditional distribution of the delay given the observed HOL delay. An approximation to its mean value serves as a refined estimator. For all three candidate delay estimators, the MSE relative to the square of the mean is asymptotically negligible in the manyserver and classical heavytraffic limiting regimes.
An ODE for an overloaded X model involving a stochastic averaging principle. working paper
"... We study an ordinary differential equation (ODE) arising as the manyserver heavytraffic fluid limit of a sequence of overloaded Markovian queueing models with two customer classes and two service pools. The system, known as the X model in the callcenter literature, operates under the fixedqueue ..."
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Cited by 7 (4 self)
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We study an ordinary differential equation (ODE) arising as the manyserver heavytraffic fluid limit of a sequence of overloaded Markovian queueing models with two customer classes and two service pools. The system, known as the X model in the callcenter literature, operates under the fixedqueueratiowiththresholds (FQRT) control, which we proposed in a recent paper as a way for one service system to help another in face of an unanticipated overload. Each pool serves only its own class until a threshold is exceeded; then oneway sharing is activated with all customerserver assignments then driving the two queues toward a fixed ratio. For large systems, that fixed ratio is achieved approximately. The ODE describes system performance during an overload. The control is driven by a queuedifference stochastic process, which operates in a faster time scale than the queueing processes themselves, thus achieving a timedependent steady state instantaneously in the limit. As a result, for the ODE, the driving process is replaced by its longrun average behavior at each instant of time; i.e., the ODE involves a heavytraffic averaging principle (AP). 1. Introduction. We