Many-Server Queues with Time-Varying Arrivals, Customer Abandonment, and non-Exponential Distributions

Many-Server Queues with Time-Varying Arrivals, Customer A bandonment and Non-Exponential Distributions Yunan Liu This thesis develops deterministic heavy-traffic fluid appr oximations for many-server stochastic queueing models. The queueing models, with many homogenous servers working independently in parallel, are intended to model largescale service systems such as call centers and health care systems. Such models also have been e mploy d to study communication, computing and manufacturing systems. The heavytraffic approximations yield relatively simple formulas for quantities describing syst em performance, such as the expected number of customers waiting in the queue. The new performance approximations are valuable because, i n the generality considered, these complex systems are not amenable to exact mathem atical analysis. Since the approximate performance measures can be computed quite rap idly, they usefully complement more cumbersome computer simulation. Thus these heavy -traffic approximations can be used to improve capacity planning and operational contro l. More specifically, the heavy-traffic approximations here ar e fo large-scale service systems, having many servers and a high arrival rate. The main fo cus is on systems that have time-varying arrival rates and staffing functions. The system is considered under the assumption that there are alternating periods of overloadi ng and underloading, which commonly occurs when service providers are unable to adjust the staffing frequently enough to economically meet demand at all times. The models also allow the realistic features of customer aba ndonment and non-exponential probability distributions for the service times and the tim es customers are willing to wait before abandoning. These features make the overall stochas tic model non-Markovian and thus very difficult to analyze directly. This thesis provide s effective algorithms to compute approximate performance descriptions for these complex sy stems. These algorithms are based on ordinary differential equations and fixed point equ ations associated with contraction operators. Simulation experiments are conducted to ve rify that the approximations are effective. This thesis consists of four pieces of work, each presented i n one chapter. The first chapter (Chapter 2) develops the basic fluid approximation f or a non-Markovian manyserver queue with time-varying arrival rate and staffing. Th e second chapter (Chapter 3) extends the fluid approximation to systems with complex netw ork structure and Markovian routing to other queues of customers after completing servi c from each queue. The extension to open networks of queues has important applications. For one example, in hospitals, patients usually move among different units such as emergen cy rooms, operating rooms, and intensive care units. For another example, in manufactu ring systems, individual products visit different work stations one or more times. The ope n n twork fluid model has multiple queues each of which has a time-varying arrival rat e and staffing function. The third chapter (Chapter 4) studies the large-time asympt otic dynamics of a single fluid queue. When the model parameters are constant, converg ence to the steady state as time evolves is established. When the arrival rates are pe riodic functions, such as in service systems with daily or seasonal cycles, the existenc e of a periodic steady state and the convergence to that periodic steady state as time evolve s are established. Conditions are provided under which this convergence is exponentially f st. The fourth chapter (Chapter 5) uses a fluid approximation to g ain insight into nearly periodic behavior seen in overloaded stationary many-serv er queues with customer abandonment and nearly deterministic service times. Determini stic service times are of applied interest because computer-generated service times, such a automated messages, may well be deterministic, and computer-generated service is becom ing ore prevalent. With deterministic service times, if all the servers remain busy fo r a long interval of time, then the times customers enter service assumes a periodic behavi or throughout that interval. In overloaded large-scale systems, these intervals tend to pe rsist for a long time, producing nearly periodic behavior. To gain insight, a heavy-traffic limit theorem is establishe d showing that the fluid model arises as the many-server heavy-traffic limit of a sequence o f appropriately scaled queueing models, all having these deterministic service times. S imulation experiments confirm that the transient behavior of the limiting fluid model provi des a useful description of the transient performance of the queueing system. However, unl ike the asymptotic loss of memory results in the previous chapter for service times wit h densities, the stationary fluid model with deterministic service times does not approach st eady state as time evolves independent of the initial conditions. Since the queueing mod el with deterministic service times approaches a proper steady state as time evolves, this model with deterministic service times provides an example where the limit interchange ( limiting steady state as time evolves and heavy traffic as scale increases) is not valid.