185 research outputs found
Performance analysis of time-dependent queueing systems: survey and classification
Many queueing systems are subject to time-dependent changes in system parameters, such as the arrival
rate or number of servers. Examples include time-dependent call volumes and agents at inbound call
centers, time-varying air traffic at airports, time-dependent truck arrival rates at seaports, and cyclic message volumes in computer systems.There are several approaches for the performance analysis of queueing systems with deterministic parameter changes over time. In this survey, we develop a classification scheme that groups these approaches according to their underlying key ideas into (i) numerical and analytical solutions,(ii)approaches based on models with piecewise constant parameters, and (iii) approaches based on mod-ified system characteristics. Additionally, we identify links between the different approaches and provide a survey of applications that are categorized into service, road and air traffic, and IT systems
Large deviations analysis for the queue in the Halfin-Whitt regime
We consider the FCFS queue in the Halfin-Whitt heavy traffic
regime. It is known that the normalized sequence of steady-state queue length
distributions is tight and converges weakly to a limiting random variable W.
However, those works only describe W implicitly as the invariant measure of a
complicated diffusion. Although it was proven by Gamarnik and Stolyar that the
tail of W is sub-Gaussian, the actual value of was left open. In subsequent work, Dai and He
conjectured an explicit form for this exponent, which was insensitive to the
higher moments of the service distribution.
We explicitly compute the true large deviations exponent for W when the
abandonment rate is less than the minimum service rate, the first such result
for non-Markovian queues with abandonments. Interestingly, our results resolve
the conjecture of Dai and He in the negative. Our main approach is to extend
the stochastic comparison framework of Gamarnik and Goldberg to the setting of
abandonments, requiring several novel and non-trivial contributions. Our
approach sheds light on several novel ways to think about multi-server queues
with abandonments in the Halfin-Whitt regime, which should hold in considerable
generality and provide new tools for analyzing these systems
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Staffing and Scheduling to Differentiate Service in Many-Server Service Systems
This dissertation contributes to the study of a queueing system with a single pool of multiple homogeneous servers to which multiple classes of customers arrive in independent streams. The objective is to devise appropriate staffing and scheduling policies to achieve specified class-dependent service levels expressed in terms of tail probability of delays. Here staffing and scheduling are concerned with specifying a time-varying number of servers and assigning newly idle servers to a waiting customer from one of K classes, respectively. For this purpose, we propose new staffing-and-scheduling solutions under the critically-loaded and overloaded regimes. In both cases, the proposed solutions are both time dependent (coping with the time variability in the arrival pattern) and state dependent (capturing the stochastic variability in service and arrival times). We prove heavy-traffic limit theorems to substantiate the effectiveness of our proposed staffing and scheduling policies. We also conduct computer simulation experiments to provide engineering confirmation and practical insight
Analysis of buffer allocations in time-dependent and stochastic flow lines
This thesis reviews and classifies the literature on the Buffer Allocation Problem under steady-state conditions and on performance evaluation approaches for queueing systems with time-dependent parameters. Subsequently, new performance evaluation approaches are developed. Finally, a local search algorithm for the derivation of time-dependent buffer allocations is proposed. The algorithm is based on numerically observed monotonicity properties of the system performance in the time-dependent buffer allocations. Numerical examples illustrate that time-dependent buffer allocations represent an adequate way of minimizing the average WIP in the flow line while achieving a desired service level
A computational approach to steady-state convergence of fluid limits for Coxian queuing networks with abandonment
Many-server queuing networks with general service and abandonment times have proven to be a realistic model for scenarios such as call centers and health-care systems. The presence of abandonment makes analytical treatment difficult for general topologies. Hence, such networks are usually studied by means of fluid limits. The current state of the art, however, suffers from two drawbacks. First, convergence to a fluid limit has been established only for the transient, but not for the steady state regime. Second, in the case of general distributed service and abandonment times, convergence to a fluid limit has been either established only for a single queue, or has been given by means of a system of coupled integral equations which does not allow for a numerical solution. By making the mild assumption of Coxian-distributed service and abandonment times, in this paper we address both drawbacks by establishing convergence in probability to a system of coupled ordinary differential equations (ODEs) using the theory of Kurtz. The presence of abandonments leads in many cases to ODE systems with a global attractor, which is known to be a sufficient condition for the fluid and the stochastic steady state to coincide in the limiting regime. The fact that our ODE systems are piecewise affine enables a computational method for establishing the presence of a global attractor, based on a solution of a system of linear matrix inequalities
Stochastic Systems ACHIEVING RAPID RECOVERY IN AN OVERLOAD CONTROL FOR LARGE-SCALE SERVICE SYSTEMS
We consider an automatic overload control for two large service systems modeled as multi-server queues, such as call centers. We assume that the two systems are designed to operate independently, but want to help each other respond to unexpected overloads. The proposed overload control automatically activates sharing (sending some customers from one system to the other) once a ratio of the queue lengths in the two systems crosses an activation threshold (with ratio and activation threshold parameters for each direction). To prevent harmful sharing, sharing is allowed in only one direction at any time. In this paper, we are primarily concerned with ensuring that the system recovers rapidly after the overload is over, either (i) because the two systems return to normal loading or (ii) because the direction of the overload suddenly shifts in the opposite direction. To achieve rapid recovery, we introduce lower thresholds for the queue ratios, below which one-way sharing is released. As a basis for studyin
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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
New stochastic processes to model interest rates : LIBOR additive processes
In this paper, a new kind of additive process is proposed. Our main goal is to define,
characterize and prove the existence of the LIBOR additive process as a new stochastic process.
This process will be de.ned as a piecewise stationary process with independent increments,
continuous in probability but with discontinuous trajectories, and having "càdlàg" sample paths.
The proposed process is specifically designed to derive interest-rates modelling because it
allows us to introduce a jump-term structure as an increasing sequence of Lévy measures. In
this paper we characterize this process as a Markovian process with an infinitely divisible,
selfsimilar, stable and self-decomposable distribution. Also, we prove that the Lévy-Khintchine
characteristic function and Lévy-Itô decomposition apply to this process. Additionally we
develop a basic framework for density transformations. Finally, we show some examples of
LIBOR additive processes
Sequential Tracking of a Hidden Markov Chain Using Point Process Observations
We study finite horizon optimal switching problems for hidden Markov chain
models under partially observable Poisson processes. The controller possesses a
finite range of strategies and attempts to track the state of the unobserved
state variable using Bayesian updates over the discrete observations. Such a
model has applications in economic policy making, staffing under variable
demand levels and generalized Poisson disorder problems. We show regularity of
the value function and explicitly characterize an optimal strategy. We also
provide an efficient numerical scheme and illustrate our results with several
computational examples.Comment: Key words and phrases. Markov Modulated Poisson processes, optimal
switchin
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