Article

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

04/2011;
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.

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##### 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; · 0.44 Impact Factor
• ##### Article: On the steady-state probability of delay and large negative deviations for the $GI/GI/n$ queue in the Halfin-Whitt regime
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ABSTRACT: We consider the FCFS $GI/GI/n$ queue in the Halfin-Whitt heavy traffic regime, and prove bounds for the steady-state probability of delay (s.s.p.d.) for generally distributed processing times. We prove that there exists $\epsilon > 0$, depending on the inter-arrival and processing time distributions, such that the s.s.p.d. is bounded from above by $\exp\big(-\epsilon B^2\big)$ as the associated excess parameter $B \rightarrow \infty$; and by $1 - \epsilon B$ as $B \rightarrow 0$. We also prove that the tail of the steady-state number of idle servers has a Gaussian decay, and use known results to show that our bounds are tight (in an appropriate sense). \\\indent Our main proof technique is the derivation of new stochastic comparison bounds for the FCFS $GI/GI/n$ queue, which are of a structural nature, hold for all $n$ and times $t$, and build on the recent work of \citet{GG.10c}.
06/2013;
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##### Article: A one-dimensional diffusion model for overloaded queues with customer abandonment
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ABSTRACT: We use an Ornstein--Uhlenbeck (OU) process to approximate the queue length process in a $GI/GI/n+M$ queue. This one-dimensional diffusion model is able to produce accurate performance estimates in two overloaded regimes: In the first regime, the number of servers is large and the mean patience time is comparable to or longer than the mean service time; in the second regime, the number of servers can be arbitrary but the mean patience time is much longer than the mean service time. Using the diffusion model, we obtain Gaussian approximations for the steady-state queue length and the steady-state virtual waiting time. Numerical experiments demonstrate that the approximate distributions are satisfactory for queues in these two regimes. To mathematically justify the diffusion model, we formulate the two overloaded regimes into an asymptotic framework by considering a sequence of queues. The mean patience time goes to infinity in both asymptotic regimes, whereas the number of servers approaches infinity in the first regime but does not change in the second. The OU process is proved to be the diffusion limit for the queue length processes in both regimes. A crucial tool for proving the diffusion limit is a functional central limit theorem for the superposition of time-scaled renewal processes. We prove that the superposition of $n$ independent, identically distributed stationary renewal processes, after being centered and scaled in both space and time, converges in distribution to a Brownian motion as $n$ goes to infinity.
12/2013;