Efficient Simulation and Performance Stabilization for Time-Varying Single-Server Queues

Edición de la obra Efficient Simulation and Performance Stabilization for Time-Varying Single-Server Queues

This thesis develops techniques to evaluate and to improve the performance of single-server service systems with time-varying arrivals. The performance measures considered are the time-varying expected length of the queue and the expected customer waiting time. Time varying arrival rates are considered because they often occur in service systems. For example, arrival rates often vary significantly over the hours of each day and over the days of each week. Stochastic textbook methods do not apply to models with time-varying arrival rates. Hence new techniques are needed to provide high quality of service when stationary steady-state analysis is not appropriate. In contrast to the extensive recent literature on many-server queues with time-varying arrival rates, we focus on single-server queues with time-varying arrival rates. Single-server queues arise in real applications where there is no flexibility in the number of service facilities (servers). Different analysis techniques are required for single-server queues, because the two kinds of models exhibit very different performance. Many-server models are more tractable because methods for highly tractable infinite-server models can be applied. In contrast, single-server models are more complicated because it takes a long time to respond to a build up of workload when there is only one server. The thesis is divided into two parts: simulation algorithms for performance evaluation and service-rate controls for performance stabilization. The first part of the thesis develops algorithms to efficiently simulate the single-server time-varying queue. For the generality considered, no explicit mathematical formulas are available for calculating performance measures, so simulation experiments are needed to calculate and evaluate system performance. Efficient algorithms for both standard simulation and rare-event simulation are developed. The second part of the thesis develops service-rate controls to stabilize performance in the time-varying single-server queue. The performance stabilization problem aims to minimize fluctuations in mean waiting times for customers coming at different times even though the arrival rate is time-varying. A new service rate control is developed, where the service rate at each time is a function of the arrival rate function. We show that a specific service rate control can be found to stabilize performance. In turn, that service rate control can be used to provide guidance for real applications on optimal changes in staffing, processing speed or machine power status over time. Both the simulation experiments to evaluate performance of alternative service-rate controls and the simulation search algorithm to find the best parameters for a damped time-lag service-rate control are based on efficient performance evaluation algorithms in the first part of the thesis. In Chapter Two, we present an efficient algorithm to simulate a general non-Poisson non-stationary point process. The general point process can be represented as a time transformation of a rate-one base process and by exploiting a table of the inverse cumulative arrival rate function outside of simulation, we can efficiently convert the simulated rate-one process into the simulated general point process. The simulation experiments can be conducted in linear time subject to small error bounds. Then we can apply this efficient algorithm to generate the arrival process, the service process and thus to calculate performance measures for the G_t/G_t/1 queues, which are single-server queues with time-varying arrival rates and service rates. Service models are constructed for this purpose where time-varying service rates are specified separately from the rate-one service requirement process, and service times are determined by equating service requirements with integrals of service rates over a time period equal to the service time. In Chapter Three, we develop rare-event simulation algorithms in periodic GI_t/GI/1 queues and further in GI_t/GI_t/1 queues to estimate probabilities of rare but important events as a sanity check of the system, for example, estimating the probability that the waiting time is very long. Importance sampling, specifically exponential tilting, is required to estimate rare-event probabilities because in standard simulation, the number of experiments may blow up to achieve a targeted relative error and for each experiment, it may take a very long time to determine that the rare event does not happen. To extend the rare-event simulation algorithm to periodic queues, we derive a convenient expression for the periodic steady-state virtual waiting time. We apply this expression to establish bounds between the periodic workload and the steady-state workload in stationary queues, so that we can prove that the exponential tilting algorithm with the same parameter efficient in stationary queues is efficient in the periodic setting as well, which has a bounded relative