Results 1  10
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93
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 189 (15 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.
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 diffusion. 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 o ..."
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Cited by 73 (33 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 diffusion. 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 differential 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.
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 60 (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...
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 57 (19 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...
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 55 (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.
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 43 (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.
Bistability in Communication Networks
 DISORDER IN PHYSICAL SYSTEMS
, 1990
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Regular variation for measures on metric spaces
 Publ. Inst. Math. (Beograd) (N.S
"... Abstract. The foundations of regular variation for Borel measures on a complete separable space S, that is closed under multiplication by nonnegative real numbers, is reviewed. For such measures an appropriate notion of convergence is presented and the basic results such as a Portmanteau theorem, a ..."
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Cited by 31 (5 self)
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Abstract. The foundations of regular variation for Borel measures on a complete separable space S, that is closed under multiplication by nonnegative real numbers, is reviewed. For such measures an appropriate notion of convergence is presented and the basic results such as a Portmanteau theorem, a mapping theorem and a characterization of relative compactness are derived. Regular variation is defined in this general setting and several statements that are equivalent to this definition are presented. This extends the notion of regular variation for Borel measures on the Euclidean space Rd to more general metric spaces. Some examples, including regular variation for Borel measures on Rd, the space of continuous functions C and the Skorohod space D, are provided. 1.
Fair dynamic routing in largescale heterogeneousserver systems: Technical appendix
, 2008
"... In a call center, there is a natural tradeoff between minimizing customer wait time and fairly dividing the workload amongst agents of different skill levels. The relevant control is the routing policy; that is, the decision concerning which agent should handle an arriving call when more than one a ..."
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Cited by 24 (4 self)
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In a call center, there is a natural tradeoff between minimizing customer wait time and fairly dividing the workload amongst agents of different skill levels. The relevant control is the routing policy; that is, the decision concerning which agent should handle an arriving call when more than one agent is available. We formulate an optimization problem for a call center with two heterogeneous agent pools, one that handles calls at a faster speed than the other, and a single customer class. The objective is to minimize steadystate expected customer wait time subject to a “fairness ” constraint on the workload division. The optimization problem we formulate is difficult to solve exactly. Therefore, we solve the diffusion control problem that arises in the manyserver heavytraffic QED limiting regime. The resulting routing policy is a threshold policy that prioritizes faster agents when the number of customers in the system exceeds some threshold level and otherwise prioritizes slower agents. We prove our proposed threshold routing policy is nearoptimal as the number of agents increases, and the system’s load approaches its maximum processing capacity. We further show simulation results that evidence that our proposed threshold routing policy outperforms a common routing policy used in call centers (that routes to the agent that has been idle the longest) in terms of the steadystate expected customer waiting time for identical desired workload divisions.
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 23 (9 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.