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61
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.
The Asymptotic Efficiency Of Simulation Estimators
 Operations Research
, 1992
"... A decisiontheoretic framework is proposed for evaluating the efficiency of simulation estimators. The framework includes the cost of obtaining the estimate as well as the cost of acting based on the estimate. The cost of obtaining the estimate and the estimate itself are represented as realizations ..."
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Cited by 43 (14 self)
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A decisiontheoretic framework is proposed for evaluating the efficiency of simulation estimators. The framework includes the cost of obtaining the estimate as well as the cost of acting based on the estimate. The cost of obtaining the estimate and the estimate itself are represented as realizations of jointly distributed stochastic processes. In this context, the efficiency of a simulation estimator based on a given computational budget is defined as the reciprocal of the risk (the overall expected cost). This framework is appealing philosophically, but it is often difficult to apply in practice (e.g., to compare the efficiency of two different estimators) because only rarely can the efficiency associated with a given computational budget be calculated. However, a useful practical framework emerges in a large sample context when we consider the limiting behavior as the computational budget increases. A limit theorem established for this model supports and extends a fairly well known e...
Departures from Many Queues in Series
, 1990
"... We consider a series of n singleserver queues, each with unlimited waiting space and the firstin firstout service discipline. Initially, the system is empty; then k customers are placed in the first queue. The service times of all the customers at all the queues are i.i.d. with a general distribu ..."
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Cited by 43 (5 self)
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We consider a series of n singleserver queues, each with unlimited waiting space and the firstin firstout service discipline. Initially, the system is empty; then k customers are placed in the first queue. The service times of all the customers at all the queues are i.i.d. with a general distribution. We are interested in the time D(k, n) required for all k customers to complete service from all n queues. In particular, we investigate the limiting behavior of D(k, n) as n and/or k . There is a duality implying that D(k, n) is distributed the same as D(n , k) so that results for large n are equivalent to results for large k. A previous heavytraffic limit theorem implies that D(k, n) satisfies an invariance principle as n , converging after normalization to a functional of kdimensional Brownian motion. We use the subadditive ergodic theorem and a strong approximation to describe the limiting behavior of D(k n , n) where k n as n . The case of k n = xn corresponds to a hydrodyna...
Sample Path Large Deviations and Intree Networks
 Queueing Systems
, 1994
"... Using the contraction principle, in this paper we derive a set of closure properties for sample path large deviations. These properties include sum, reduction, composition and reflection mapping. Using these properties, we show that the exponential decay rates of the steady state queue length distri ..."
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Cited by 40 (8 self)
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Using the contraction principle, in this paper we derive a set of closure properties for sample path large deviations. These properties include sum, reduction, composition and reflection mapping. Using these properties, we show that the exponential decay rates of the steady state queue length distributions in an intree network with routing can be derived by a set of recursive equations. The solution of this set of equations is related to the recently developed theory of effective bandwidth for high speed digital networks, especially ATM networks. We also prove a conditional limit theorem that illustrates how a queue builds up in an intree network.
A Poisson Limit for Buffer Overflow Probabilities
 in Proceedings of IEEE INFOCOM
, 2002
"... Abstract — A key criterion in the design of highspeed networks is the probability that the buffer content exceeds a given threshold. We consider Ò independent identical traffic sources modelled as point processes, which are fed into a link with speed proportional to Ò. Under fairly general assumpti ..."
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Cited by 39 (1 self)
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Abstract — A key criterion in the design of highspeed networks is the probability that the buffer content exceeds a given threshold. We consider Ò independent identical traffic sources modelled as point processes, which are fed into a link with speed proportional to Ò. Under fairly general assumptions on the input processes we show that the steady state probability of the buffer content exceeding a threshold � � tends to the corresponding probability assuming Poisson input processes. We verify the assumptions for a large class of longrange dependent sources commonly used to model data traffic. Our results show that with superposition, significant multiplexing gains can be achieved for even smaller buffers than suggested by previous results, which consider Ç Ò buffer size. Moreover, simulations show that for realistic values of the exceedance probability and moderate utilisations, convergence to the Poisson limit takes place at reasonable values of the number of sources superposed. This is particularly relevant for highspeed networks in which the cost of highspeed memory is significant. Keywords—Longrange dependence, overflow probability, Poisson limit, heavy tails, point processes, multiplexing.
Fluid and diffusion limits for queues in slowly changing environments
 Stoch. Mod
, 1997
"... diffusion processes, fluid limit, heavy traffic. We consider an infinitecapacity sserver queue in a finitestate random environment, where the traffic intensity exceeds 1 in some environment states and the environment states change slowly relative to arrivals and service completions. Queues grow i ..."
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Cited by 17 (4 self)
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diffusion processes, fluid limit, heavy traffic. We consider an infinitecapacity sserver queue in a finitestate random environment, where the traffic intensity exceeds 1 in some environment states and the environment states change slowly relative to arrivals and service completions. Queues grow in unstable environment states, so that it is useful to look at the system in the time scale of mean environmentstate sojourn times. As the mean environmentstate sojourn times grow, the queuelength and workload processes grow. However, with appropriate normalizations, these processes converge to fluid processes and diffusion processes. The diffusion process in a random environment is a refinement of the fluid process in a random environment. We show how the scaling in these limits can help explain numerical results for queues in slowly changing random environments. For that purpose, we apply recently developed numericaltransforminversion algorithms for the MAP/G/1 queue and the piecewisestationary Mt/Gt/1 queue. 1.
Heavytraffic limits for waiting times in manyserver queues with abandonments
, 2008
"... In this online supplement we provide results that we have omitted from the main paper. First, in Appendix A, we give a proof of Lemma 2.1. In Appendix B we give a proof of Theorem 6.1 using the technique described in [2]. Finally, in Appendix C, we give an alternative proof of Theorem 5.2 using stop ..."
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Cited by 16 (8 self)
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In this online supplement we provide results that we have omitted from the main paper. First, in Appendix A, we give a proof of Lemma 2.1. In Appendix B we give a proof of Theorem 6.1 using the technique described in [2]. Finally, in Appendix C, we give an alternative proof of Theorem 5.2 using stopped arrival processes as in the proof of Theorem 6.3.
Limit Theorems For Continuous Time Random Walks With Infinite Mean Waiting Times
 Journal of Applied Probability
, 2003
"... A continuous time random walk is a simple random walk subordinated to a renewal process, used in physics to model anomalous di#usion. In this paper we show that, when the time between renewals has infinite mean, the scaling limit is an operator Levy motion subordinated to the hitting time process of ..."
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Cited by 15 (9 self)
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A continuous time random walk is a simple random walk subordinated to a renewal process, used in physics to model anomalous di#usion. In this paper we show that, when the time between renewals has infinite mean, the scaling limit is an operator Levy motion subordinated to the hitting time process of a classical stable subordinator. Density functions for the limit process solve a fractional Cauchy problem, the generalization of a fractional partial di#erential equation for Hamiltonian chaos. We also establish a functional limit theorem for random walks with jumps in the strict generalized domain of attraction of a full operator stable law, which is of some independent interest. 1.
Functional Large Deviation Principles for Waiting and Departure Processes
 Prob. Engin. Info. Sci
, 1998
"... We establish functional large deviation principles (FLDPs) for waiting and departure processes in singleserver queues with unlimited waiting space and the firstin firstout service discipline. We apply the extended contraction principle to show that these processes obey FLDPs in the function space ..."
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Cited by 13 (4 self)
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We establish functional large deviation principles (FLDPs) for waiting and departure processes in singleserver queues with unlimited waiting space and the firstin firstout service discipline. We apply the extended contraction principle to show that these processes obey FLDPs in the function space D with one of the nonuniform Skorohod topologies whenever the arrival and service processes obey FLDPs and the rate function is finite for appropriate discontinuous functions. We apply our previous FLDPs for inverse processes to obtain an FLDP for the waiting times in a queue with a superposition arrival process. We obtain FLDPs for queues within acyclic networks by showing that FLDPs are inherited by processes arising from the network operations of departure, superposition and random splitting. For this purpose, we also obtain FLDPs for split point processes. For the special cases of deterministic arrival processes and deterministic service processes, we obtain convenient explicit express...