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...[ Read more ]

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.