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13
Manyserver queues with customer abandonment: numerical analysis of their diffusion models
, 2011
"... The performance of a call center is sensitive to customer abandonment. In this survey paper, we focus on / /G GI n GI parallelserver queues that serve as a building block to model call center operations. Such a queue has a general arrival process (the G), independent and identically distributed ..."
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The performance of a call center is sensitive to customer abandonment. In this survey paper, we focus on / /G GI n GI parallelserver queues that serve as a building block to model call center operations. Such a queue has a general arrival process (the G), independent and identically distributed (iid) service times with a general distribution (the first GI), and iid patience times with a general distribution (the GI ). Following the squareroot safety staffing rule, this queue can be operated in the quality and efficiencydriven (QED) regime, which is characterized by large customer volume, the waiting times being a fraction of the service times, only a small fraction of customers abandoning the system, and high server utilization. Operational efficiency is the central target in a system whose staffing costs dominate other expenses. If a moderate fraction of customer abandonment is allowed, such a system should be operated in an overloaded regime known as the efficiencydriven (ED) regime. We survey recent results on the manyserver queues that are operated in the QED and ED regimes. These results include the performance insensitivity to patience time distributions and diffusion and fluid approximate models as practical tools for performance analysis.
From Local to Global Stability in Stochastic Processing Networks through Quadratic Lyapunov Functions”, preprint
, 2012
"... Abstract We construct a generic, simple, and efficient scheduling policy for stochastic processing networks, and provide a general framework to establish its stability. Our policy is randomized and prioritized: with high probability it prioritizes jobs which have been least routed through the netwo ..."
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Abstract We construct a generic, simple, and efficient scheduling policy for stochastic processing networks, and provide a general framework to establish its stability. Our policy is randomized and prioritized: with high probability it prioritizes jobs which have been least routed through the network. We show that the network is globally stable under this policy if there exists an appropriate quadratic 'local' Lyapunov function that provides a negative drift with respect to nominal loads at servers. Applying this generic framework, we obtain stability results for our policy in many important examples of stochastic processing networks: open multiclass queueing networks, parallel server networks, networks of inputqueued switches, and a variety of wireless network models with interference constraints. Our main novelty is the construction of an appropriate 'global' Lyapunov function from quadratic 'local' Lyapunov functions, which we believe to be of broader interest.
Diffusion models and steadystate approximations for exponentially ergodic markovian queues
, 2013
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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
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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
2 From Local to Global Stability in Stochastic Processing Networks through Quadratic Lyapunov Functions
, 2014
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Stochastic Systems TIGHTNESS OF STATIONARY DISTRIBUTIONS OF A FLEXIBLESERVER SYSTEM IN THE HALFINWHITT ASYMPTOTIC REGIME
"... We consider a largescale flexible service system with two large server pools and two types of customers. Servers in pool 1 can only serve type 1 customers, while server in pool 2 are flexible – they can serve both types 1 and 2. (This is a socalled “Nsystem. ” Our results hold for a more general ..."
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We consider a largescale flexible service system with two large server pools and two types of customers. Servers in pool 1 can only serve type 1 customers, while server in pool 2 are flexible – they can serve both types 1 and 2. (This is a socalled “Nsystem. ” Our results hold for a more general class of systems as well.) The service rate of a customer depends both on its type and the pool where it is served. We study a priority service discipline, where type 2 has priority in pool 2, and type 1 prefers pool 1. We consider the HalfinWhitt asymptotic regime, where the arrival rate of customers and the number of servers in each pool increase to infinity in proportion to a scaling parameter n, while the overall system capacity exceeds its load by O( n). For this system we prove tightness of diffusionscaled stationary distributions. Our approach relies on a single common Lyapunov function G(x), defined on the entire state space as a functional of the driftbased fluid limits (DFL). Specifically, G(x) = 0 g(y(t))dt, where y(·) is the DFL starting at x, and g(·) is a “distance ” to the origin. The key part of the analysis is the study of the (first and second) derivatives of the DFLs and function G(x). Our approach, as well as many parts of the analysis, seem quite generic and may be of independent interest. 1. Introduction. In this paper