102 research outputs found
Critically loaded multi-server queues with abandonments, retrials, and time-varying parameters
In this paper, we consider modeling time-dependent multi-server queues that
include abandonments and retrials. For the performance analysis of those, fluid
and diffusion models called "strong approximations" have been widely used in
the literature. Although they are proven to be asymptotically exact, their
effectiveness as approximations in critically loaded regimes needs to be
investigated. To that end, we find that existing fluid and diffusion
approximations might be either inaccurate under simplifying assumptions or
computationally intractable. To address that concern, this paper focuses on
developing a methodology by adjusting the fluid and diffusion models so that
they significantly improve the estimation accuracy. We illustrate the accuracy
of our adjusted models by performing a number of numerical experiments
Large deviations analysis for the queue in the Halfin-Whitt regime
We consider the FCFS queue in the Halfin-Whitt heavy traffic
regime. It is known that the normalized sequence of steady-state queue length
distributions is tight and converges weakly to a limiting random variable W.
However, those works only describe W implicitly as the invariant measure of a
complicated diffusion. Although it was proven by Gamarnik and Stolyar that the
tail of W is sub-Gaussian, the actual value of was left open. In subsequent work, Dai and He
conjectured an explicit form for this exponent, which was insensitive to the
higher moments of the service distribution.
We explicitly compute the true large deviations exponent for W when the
abandonment rate is less than the minimum service rate, the first such result
for non-Markovian queues with abandonments. Interestingly, our results resolve
the conjecture of Dai and He in the negative. Our main approach is to extend
the stochastic comparison framework of Gamarnik and Goldberg to the setting of
abandonments, requiring several novel and non-trivial contributions. Our
approach sheds light on several novel ways to think about multi-server queues
with abandonments in the Halfin-Whitt regime, which should hold in considerable
generality and provide new tools for analyzing these systems
Many-server diffusion limits for queues
This paper studies many-server limits for multi-server queues that have a
phase-type service time distribution and allow for customer abandonment. The
first set of limit theorems is for critically loaded queues, where
the patience times are independent and identically distributed following a
general distribution. The next limit theorem is for overloaded
queues, where the patience time distribution is restricted to be exponential.
We prove that a pair of diffusion-scaled total-customer-count and
server-allocation processes, properly centered, converges in distribution to a
continuous Markov process as the number of servers goes to infinity. In the
overloaded case, the limit is a multi-dimensional diffusion process, and in the
critically loaded case, the limit is a simple transformation of a diffusion
process. When the queues are critically loaded, our diffusion limit generalizes
the result by Puhalskii and Reiman (2000) for queues without customer
abandonment. When the queues are overloaded, the diffusion limit provides a
refinement to a fluid limit and it generalizes a result by Whitt (2004) for
queues with an exponential service time distribution. The proof
techniques employed in this paper are innovative. First, a perturbed system is
shown to be equivalent to the original system. Next, two maps are employed in
both fluid and diffusion scalings. These maps allow one to prove the limit
theorems by applying the standard continuous-mapping theorem and the standard
random-time-change theorem.Comment: Published in at http://dx.doi.org/10.1214/09-AAP674 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
The snowball effect of customer slowdown in critical many-server systems
Customer slowdown describes the phenomenon that a customer's service
requirement increases with experienced delay. In healthcare settings, there is
substantial empirical evidence for slowdown, particularly when a patient's
delay exceeds a certain threshold. For such threshold slowdown situations, we
design and analyze a many-server system that leads to a two-dimensional Markov
process. Analysis of this system leads to insights into the potentially
detrimental effects of slowdown, especially in heavy-traffic conditions. We
quantify the consequences of underprovisioning due to neglecting slowdown,
demonstrate the presence of a subtle bistable system behavior, and discuss in
detail the snowball effect: A delayed customer has an increased service
requirement, causing longer delays for other customers, who in turn due to
slowdown might require longer service times.Comment: 23 pages, 8 figures -- version 3 fixes a typo in an equation. in
Stochastic Models, 201
Heavy-traffic limits for waiting times in many-server queues with abandonment
We establish heavy-traffic stochastic-process limits for waiting times in
many-server queues with customer abandonment. If the system is asymptotically
critically loaded, as in the quality-and-efficiency-driven (QED) regime, then a
bounding argument shows that the abandonment does not affect waiting-time
processes. If instead the system is overloaded, as in the efficiency-driven
(ED) regime, following Mandelbaum et al. [Proceedings of the Thirty-Seventh
Annual Allerton Conference on Communication, Control and Computing (1999)
1095--1104], we treat customer abandonment by studying the limiting behavior of
the queueing models with arrivals turned off at some time . Then, the
waiting time of an infinitely patient customer arriving at time is the
additional time it takes for the queue to empty. To prove stochastic-process
limits for virtual waiting times, we establish a two-parameter version of
Puhalskii's invariance principle for first passage times. That, in turn,
involves proving that two-parameter versions of the composition and inverse
mappings appropriately preserve convergence.Comment: Published in at http://dx.doi.org/10.1214/09-AAP606 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Convergence to Equilibrium States for Fluid Models of Many-server Queues with Abandonment
Fluid models have become an important tool for the study of many-server
queues with general service and patience time distributions. The equilibrium
state of a fluid model has been revealed by Whitt (2006) and shown to yield
reasonable approximations to the steady state of the original stochastic
systems. However, it remains an open question whether the solution to a fluid
model converges to the equilibrium state and under what condition. We show in
this paper that the convergence holds under a mild condition. Our method builds
on the framework of measure-valued processes developed in Zhang (2013), which
keeps track of the remaining patience and service times
Fluid Models of Many-server Queues with Abandonment
We study many-server queues with abandonment in which customers have general
service and patience time distributions. The dynamics of the system are modeled
using measure- valued processes, to keep track of the residual service and
patience times of each customer. Deterministic fluid models are established to
provide first-order approximation for this model. The fluid model solution,
which is proved to uniquely exists, serves as the fluid limit of the
many-server queue, as the number of servers becomes large. Based on the fluid
model solution, first-order approximations for various performance quantities
are proposed
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