Article

# Many-server queues with customer abandonment: numerical analysis of their diffusion models

04/2011; DOI: 10.1214/11-SSY029
Source: arXiv

ABSTRACT

We use multidimensional diffusion processes to approximate the dynamics of a
queue served by many parallel servers. The queue is served in the
first-in-first-out (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 many-server queues, sometimes
for queues with as few as twenty servers.

### Full-text

Available from: Shuangchi He, May 29, 2014
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• "Comparing the form of Σ above to (2.24) of [14] confirms that it is positive definite. Thus √ Σ exists. "
##### Article: Stein's method for steady-state diffusion approximations of $M/Ph/n+M$ systems
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ABSTRACT: We consider $M/Ph/n+M$ queueing systems. We prove the rate of convergence for approximating the stationary distribution of the normalized system size process by that of a piecewise Ornstein-Uhlenbeck (OU) process. We prove that for a large class of functions, the difference of the expectation under the stationary measure of the piecewise OU process and the expectation under the stationary measure of the system size is at most $C/\lambda^{1/4}$, where the constant $C$ is independent of the arrival rate $\lambda$ and the number of servers $n$ as long as they are in the Halfin-Whitt parameter regime. For the proof, we develop a modular framework that is based on Stein's method. The framework has three components: Poisson equation, generator coupling, and state space collapse. The framework, with further refinement, is likely applicable to steady-state diffusion approximations for other stochastic systems.
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• "Their approximations are rooted in many-server heavy traffic limit theorems proved in Dai et al. (2010). The numerical examples in Dai and He (2013) demonstrate that the steady-state performance of the diffusion model provides a remarkably accurate estimate for the steady-state performance of the corresponding queueing system, even when the number of servers is moderate. "
##### Article: Validity of heavy-traffic steady-state approximations in many-server queues with abandonment
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ABSTRACT: We consider GI/Ph/n+M parallel-server systems with a renewal arrival process, a phase-type service time distribution, n homogenous servers, and an exponential patience time distribution with positive rate. We show that in the Halfin-Whitt regime, the sequence of stationary distributions corresponding to the normalized state processes is tight. As a consequence, we establish an interchange of heavy traffic and steady state limits for GI/Ph/n+M queues.
Queueing Systems 06/2013; 78(1). DOI:10.1007/s11134-014-9394-x · 0.84 Impact Factor
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• "Such approximations are rooted in many-server heavy traffic limits proved in Puhalskii and Reiman (2000) and Dai et al. (2010). These approximations are remarkably accurate in predicting system performance measures, sometimes for systems with as few as 20 servers, see He and Dai (2011). For a diffusion approximation to work, it is critical to know whether the approximating diffusion process has a unique stationary distribution. "
##### Article: Positive recurrence of piecewise Ornstein-Uhlenbeck processes and common quadratic Lyapunov functions
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ABSTRACT: We study the positive recurrence of piecewise Ornstein-Uhlenbeck (OU) diffusion processes, which arise from many-server queueing systems with phase-type service requirements. These diffusion processes exhibit different behavior in two regions of the state space, corresponding to "overload" (service demand exceeds capacity) and "underload" (service capacity exceeds demand). The two regimes cause standard techniques for proving positive recurrence to fail. Using and extending the framework of common quadratic Lyapunov functions from the theory of control, we construct Lyapunov functions for the diffusion approximations corresponding to systems with and without abandonment. With these Lyapunov functions, we prove that piecewise OU processes have a unique stationary distribution.
The Annals of Applied Probability 07/2011; 23(4). DOI:10.1214/12-AAP870 · 1.45 Impact Factor