16 research outputs found
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
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
Asymptotically optimal idling in the GI/GI/N+GI queue
We formulate a control problem for a GI/GI/N+GI queue, whose objective is to trade off the long-run average operational costs with server utilization costs. To solve the control problem, we consider an asymptotic regime in which the arrival rate and the number of servers grow large. The solution to an associated fluid control problem motivates that non-idling service disciplines are not in general optimal, unless some arrivals are turned away. We propose an admission control policy designed to ensure that servers have sufficient idle time, which we show is asymptotically optimal
Fluid limits of many-server queues with reneging
This work considers a many-server queueing system in which impatient
customers with i.i.d., generally distributed service times and i.i.d.,
generally distributed patience times enter service in the order of arrival and
abandon the queue if the time before possible entry into service exceeds the
patience time. The dynamics of the system is represented in terms of a pair of
measure-valued processes, one that keeps track of the waiting times of the
customers in queue and the other that keeps track of the amounts of time each
customer being served has been in service. Under mild assumptions, essentially
only requiring that the service and reneging distributions have densities, as
both the arrival rate and the number of servers go to infinity, a law of large
numbers (or fluid) limit is established for this pair of processes. The limit
is shown to be the unique solution of a coupled pair of deterministic integral
equations that admits an explicit representation. In addition, a fluid limit
for the virtual waiting time process is also established. This paper extends
previous work by Kaspi and Ramanan, which analyzed the model in the absence of
reneging. A strong motivation for understanding performance in the presence of
reneging arises from models of call centers.Comment: Published in at http://dx.doi.org/10.1214/10-AAP683 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org