A fluid queue with a finite buffer and subexponential input

We consider a fluid model similar to that of Kella and Whitt [33], but with a buffer having finite capacity K. The connections between the infinite buffer fluid model and the G/G/1 queue established in [33] are extended to the finite buffer case. It is shown that the stationary distribution of the buffer content is related to the stationary distribution of the finite dam. We also derive a number of new results for the latter model. In particular, an asymptotic expansion for the loss fraction is given for the case of subexponential service times. The stationary buffer content distribution of the fluid model is also related to that of the corresponding model with infinite buffer size, by showing that the two corresponding probability measures are proportional on [0,K) if the silence periods are exponentially distributed. These results are applied to obtain large buffer asymptotics for the loss fraction and the mean buffer content when the fluid queue is fed by N on-off sources with subexponential on-periods. The asymptotic results show a significant influence of heavy-tailed input characteristics on the performance of the fluid queue

Queueing Systems with Heavy Tails

VI+227hlm.;24c

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

This thesis develops deterministic heavy-traffic fluid approximations for many-server stochastic queueing models. The queueing models, with many homogeneous servers working independently in parallel, are intended to model large-scale service systems such as call centers and health care systems. Such models also have been employed to study communication, computing and manufacturing systems. The heavy-traffic approximations yield relatively simple formulas for quantities describing system performance, such as the expected number of customers waiting in the queue. The new performance approximations are valuable because, in the generality considered, these complex systems are not amenable to exact mathematical analysis. Since the approximate performance measures can be computed quite rapidly, they usefully complement more cumbersome computer simulation. Thus these heavy-traffic approximations can be used to improve capacity planning and operational control. More specifically, the heavy-traffic approximations here are for large-scale service systems, having many servers and a high arrival rate. The main focus 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 overloading 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 abandonment and non-exponential probability distributions for the service times and the times customers are willing to wait before abandoning. These features make the overall stochastic model non-Markovian and thus thus very difficult to analyze directly. This thesis provides effective algorithms to compute approximate performance descriptions for these complex systems. These algorithms are based on ordinary differential equations and fixed point equations associated with contraction operators. Simulation experiments are conducted to verify that the approximations are effective. This thesis consists of four pieces of work, each presented in one chapter. The first chapter (Chapter 2) develops the basic fluid approximation for a non-Markovian many-server queue with time-varying arrival rate and staffing. The second chapter (Chapter 3) extends the fluid approximation to systems with complex network structure and Markovian routing to other queues of customers after completing service 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 emergency rooms, operating rooms, and intensive care units. For another example, in manufacturing systems, individual products visit different work stations one or more times. The open network fluid model has multiple queues each of which has a time-varying arrival rate and staffing function. The third chapter (Chapter 4) studies the large-time asymptotic dynamics of a single fluid queue. When the model parameters are constant, convergence to the steady state as time evolves is established. When the arrival rates are periodic functions, such as in service systems with daily or seasonal cycles, the existence of a periodic steady state and the convergence to that periodic steady state as time evolves are established. Conditions are provided under which this convergence is exponentially fast. The fourth chapter (Chapter 5) uses a fluid approximation to gain insight into nearly periodic behavior seen in overloaded stationary many-server queues with customer abandonment and nearly deterministic service times. Deterministic service times are of applied interest because computer-generated service times, such as automated messages, may well be deterministic, and computer-generated service is becoming more prevalent. With deterministic service times, if all the servers remain busy for a long interval of time, then the times customers enter service assumes a periodic behavior throughout that interval. In overloaded large-scale systems, these intervals tend to persist for a long time, producing nearly periodic behavior. To gain insight, a heavy-traffic limit theorem is established showing that the fluid model arises as the many-server heavy-traffic limit of a sequence of appropriately scaled queueing models, all having these deterministic service times. Simulation experiments confirm that the transient behavior of the limiting fluid model provides a useful description of the transient performance of the queueing system. However, unlike the asymptotic loss of memory results in the previous chapter for service times with densities, the stationary fluid model with deterministic service times does not approach steady state as time evolves independent of the initial conditions. Since the queueing model 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

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 error. We apply this algorithm to compute the periodic steady-state distribution of reflected periodic Brownian motion with support of a heavy-traffic limit theorem and to calculate the periodic steady-state distribution and moments of the virtual waiting time. This algorithm's advantage in calculating these distributions and moments is that it can directly estimate them at a specific position of the cycle without simulating the whole queueing process until steady state is reached for the whole cycle. In Chapter Four, we conduct simulation experiments to validate performance of four service-rate controls: the rate-matching control, which is directly proportional to the arrival rate, two square-root controls related to the square root staffing formula and the square-root control based on the mean stationary waiting time. Simulations show that the rate-matching control stabilizes the queue length distribution but not the virtual waiting time. This is consistent with established theoretical results, which follow from the observation that with rate-matching control, the queueing process becomes a time transformation of the stationary queueing process with constant arrival rates and service rates. Simulation results also show that the two square-root controls analogous to the server staffing formula are not effective in stabilizing performance. On the other hand, the alternative square-root service rate control based on the mean stationary waiting time approximately stabilizes the virtual waiting time when the cycle is long so that the arrival rate changes slowly enough. In Chapter Five, since we are mostly interested in stabilizing waiting times in more common scenarios when the traffic intensity is not close to one or when the arrival rate does not change slowly, we develop a damped time-lag service-rate control that performs fairly well for this purpose. This control is a modification of the rate-matching control involving a time lag and a damping factor. To find the best parameters for this control, we search over reasonable intervals for the most time-stable performance measures, which are computed by the extended rare-event simulation algorithm in GI_t/GI_t/1 queue. We conduct simulation experiments to validate that this control is effective for stabilizing the expected steady-state virtual waiting time (and its distribution to a large extent). We also establish a heavy-traffic limit with periodicity in the fluid scale to provide theoretical support for this control. We also show that there is a time-varying Little's law in heavy-traffic, which implies that this control cannot stabilize the queue length and the waiting time at the same time