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Positive recurrence of piecewise OrnsteinUhlenbeck processes and common quadratic Lyapunov functions. Annals of Applied Probability
, 2012
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Fluid models of manyserver queues with abandonment
, 2011
"... We study manyserver queues with abandonment in which customers have general service and patience time distributions. The dynamics of the system are modeled using measurevalued processes, to keep track of the residual service and patience times of each customer. Deterministic fluid models are establ ..."
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Cited by 8 (2 self)
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We study manyserver queues with abandonment in which customers have general service and patience time distributions. The dynamics of the system are modeled using measurevalued processes, to keep track of the residual service and patience times of each customer. Deterministic fluid models are established to provide firstorder approximation for this model. The fluid model solution, which is proved to uniquely exists, serves as the fluid limit of the manyserver queue, as the number of servers becomes large. Based on the fluid model solution, firstorder approximations for various performance quantities are proposed. Key words and phrases: manyserver queue, abandonment, measure valued process, quality driven, efficiency driven, quality and efficiency driven. 1
State Space Collapse in ManyServer Diffusion Limits of Parallel Server Systems
, 2006
"... We consider a class of queueing systems that consist of server pools in parallel and multiple customer classes. Customer service times are assumed to be exponentially distributed. We study the asymptotic behavior of these queueing systems in a heavy traffic regime that is known as the Halfin and Wh ..."
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We consider a class of queueing systems that consist of server pools in parallel and multiple customer classes. Customer service times are assumed to be exponentially distributed. We study the asymptotic behavior of these queueing systems in a heavy traffic regime that is known as the Halfin and Whitt manyserver asymptotic regime. Our main contribution is a general framework for establishing state space collapse results in this regime for parallel server systems. In our work, state space collapse refers to a decrease in the dimension of the processes tracking the number of customers in each class waiting for service and the number of customers in each class being served by various server pools. We define and introduce a “state space collapse ” function, which governs the exact details of the state space collapse. We show that a state space collapse result holds in manyserver heavy traffic if a corresponding deterministic hydrodynamic model satisfies a similar state space collapse condition. Our methodology is similar in spirit to that in Bramson [10], which focuses on the conventional heavy traffic regime. We illustrate the applications of our results by establishing state space collapse results in manyserver diffusion limits of staticbufferpriority Vparallel server systems, Nmodel parallel server systems, and minimumexpecteddelay–fasterserverfirst distributed server pools systems. We show for these systems that the condition on the hydrodynamic model can easily be checked using the standard tools for fluid models.
Diffusion models and steadystate approximations for exponentially ergodic markovian queues
, 2013
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Hazard Rate Scaling for the GI/M/n +GI Queue
, 2009
"... We obtain a heavytraffic limit for the GI/M/n+GI queue which includes the entire abandonment distribution. Our main approach is to scale the hazard rate function in an appropriate way such that our resulting diffusion approximation contains the entire hazard rate function. We then show through num ..."
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We obtain a heavytraffic limit for the GI/M/n+GI queue which includes the entire abandonment distribution. Our main approach is to scale the hazard rate function in an appropriate way such that our resulting diffusion approximation contains the entire hazard rate function. We then show through numerical studies that for various key performance measures, our approximations outperform those commonly used in practice which only involved the abandonment distribution through the value of its density at the origin. The robustness of our results is also demonstrated by applying them to solving constraint satisfaction problems arising in the context of telephone call centers. 1
DIFFUSION MODELS AND STEADYSTATE APPROXIMATIONS FOR EXPONENTIALLY ERGODIC MARKOVIAN QUEUES BY ITAI GURVICH Northwestern University
"... Motivated by queues with manyservers, we study Brownian steadystate approximations for continuous time Markov chains (CTMCs). Our approximations are based on diffusion models (rather than a diffusion limit) whose steadystate, we prove, approximates well that of the Markov chain. Strong approximat ..."
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Motivated by queues with manyservers, we study Brownian steadystate approximations for continuous time Markov chains (CTMCs). Our approximations are based on diffusion models (rather than a diffusion limit) whose steadystate, we prove, approximates well that of the Markov chain. Strong approximations provide such “limitless ” approximations for process dynamics. Our focus here is on steadystate distributions and the diffusion model that we propose is tractable relative to strong approximations. Within an asymptotic framework, in which a scale parameter n is taken large, a uniform (in the scale parameter) Lyapunov condition is proved to guarantee that the gap between steadystate moments of the diffusion and those of the properly centered and scaled CTMCs, shrinks at a rate of √ n. The uniform Lyapunov requirement is satisfied, in particular, if the scaled and centered sequence converges to a diffusion limit for which a Lyapunov condition is satisfied. Our proofs build on gradient estimates for the solutions of the Poisson equations associated with the (sequence of) diffusion models together with elementary Martingale arguments. As a by product of our analysis, we explore connections between Lyapunov functions for the Fluid Model, the Diffusion Model and the CTMC. 1. Introduction. Fluid
©2010 INFORMS Customer Abandonment in ManyServer Queues
, 2009
"... We study G/G/n+GI queues in which customer patience times are independent, identically distributed following a general distribution. When a customer’s waiting time in queue exceeds his patience time, the customer abandons the system without service. For the performance of such a system, we focus on ..."
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We study G/G/n+GI queues in which customer patience times are independent, identically distributed following a general distribution. When a customer’s waiting time in queue exceeds his patience time, the customer abandons the system without service. For the performance of such a system, we focus on the abandonment process and the queue length process. We prove that under some conditions, a deterministic relationship between the two stochastic processes holds asymptotically under the diffusion scaling when the number of servers n goes to infinity. These conditions include a minor assumption on the arrival processes that can be timenonhomogeneous and a key assumption that the sequence of diffusionscaled queue length processes, indexed by n, is stochastically bounded. We also establish a comparison result that allows one to verify the stochastic boundedness by studying a corresponding sequence of systems without customer abandonment. Key words: multiserver queues; customer abandonment; manyserver heavy traffic; HalfinWhitt regime; quality and efficiencydriven regime
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"... Nonnegativity of solutions to the basic adjoint relationship for some diffusion processes ..."
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Nonnegativity of solutions to the basic adjoint relationship for some diffusion processes
Stochastic Systems arXiv: 1104.0347 MANYSERVER QUEUES WITH CUSTOMER ABANDONMENT: NUMERICAL ANALYSIS OF THEIR DIFFUSION MODELS∗
"... We use multidimensional diffusion processes to approximate the dynamics of a queue served by many parallel servers. The queue is served in the firstinfirstout (FIFO) order and the customers waiting in queue may abandon the system without service. Two diffusion models are proposed in this paper. ..."
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We use multidimensional diffusion processes to approximate the dynamics of a queue served by many parallel servers. The queue is served in the firstinfirstout (FIFO) order and the customers waiting in queue may abandon the system without service. Two diffusion models are proposed in this paper. They differ in how the patience time distribution is built into them. The first diffusion model uses the patience time density at zero and the second one uses the entire patience time distribution. To analyze these diffusion models, we develop a numerical algorithm for computing the stationary distribution of such a diffusion process. A crucial part of the algorithm is to choose an appropriate reference density. Using a conjecture on the tail behavior of a limit queue length process, we propose a systematic approach to constructing a reference density. With the proposed reference density, the algorithm is shown to converge quickly in numerical experiments. These experiments also show that the diffusion models are good approximations for manyserver queues, sometimes for queues with as few as twenty servers. 1. Introduction. The