THESIS
2015
Abstract
Fluid models, in particular their equilibrium states, have become an important tool
for the study of many-server queues with general service and patience time distributions.
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 Chapter I that the convergence
holds under some conditions. Our method builds on the framework of measure-valued
processes, which keeps track of the remaining patience and service times.
We extend the measure-valued
fluid model in Chapter II, which tracks residuals of
patience and service times, to allow for time-varying arrivals. The
fluid model can be
characterized by a one-dimensional convolution equation involving both the patience and
service time dist...[
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Fluid models, in particular their equilibrium states, have become an important tool
for the study of many-server queues with general service and patience time distributions.
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 Chapter I that the convergence
holds under some conditions. Our method builds on the framework of measure-valued
processes, which keeps track of the remaining patience and service times.
We extend the measure-valued
fluid model in Chapter II, which tracks residuals of
patience and service times, to allow for time-varying arrivals. The
fluid model can be
characterized by a one-dimensional convolution equation involving both the patience and
service time distributions. We also make an interesting connection to the measure-valued
fluid model tracking the elapsed waiting and service times. Our analysis shows that the two
fluid models are actually characterized by the same one-dimensional convolution equation.
In Chapter III of this thesis, we study a multiclass many-server system with renewal
arrivals and generally distributed service and patience times under a family of nonpreemptive
allocation policies. The status of the system is described by pairs of measure-valued
processes to track residual service and patience times of customers in each class. We establish
fluid approximations and study the long-term behavior of the
fluid model. The
equilibrium state of the
fluid model connects to a nonlinear program which helps to identify
a lower bound of the long-run expected total holding and abandonment costs, and design
an allocation policy to achieve such a lower bound.
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