Stationary distributions and optimal control of queues with batch Markovian arrival process under multiple adaptive vacations

We consider an infinite-buffer single server queue with batch Markovian arrival process (BMAP) and exhaustive service discipline under multiple adaptive vacation policy. That is, the server serves until system emptied and after that server takes a random maximum number H different vacations until either he finds at least one customer in queue or the server have exhaustively taken all the vacations. The maximum number H of vacations taken by the server is a discrete random variable. We obtain queue-length distributions at various epochs such as, service completion/vacation termination, pre-arrival, arbitrary, post-departure and pre-service. The proposed analysis is based on the use of matrix-analytic procedure to obtain queue-length distribution at a post-departure epoch. Later we use supplementary variable method and simple algebraic manipulations to obtain the queue-length distribution at other epochs using queue-length distribution at post-departure epoch. Some important performance measures, like mean queue lengths and mean waiting times have been obtained. Several other vacation queueing models can be obtained as a special case of our model, e.g., single-, multiple-vacation model and queues with exceptional first vacation time. Finally, the total expected cost function per unit time is considered to determine a locally optimal multiple adaptive vacation policy at a minimum cost.     

Analysis of a finite-buffer bulk-service queue under Markovian arrival process with batch-size-dependent service

We consider a finite capacity single server queue in which the customers arrive according to a Markovian arrival process. The customers are served in batches following a ‘general bulk service rule’. The service times, which depend on the size of the batch, are generally distributed. We obtain, in steady-state, the joint distribution of the random variables of interest at various epochs. Efficient computational procedures in the case of phase type services are presented. An illustrative numerical example to bring out the qualitative nature of the model is presented.     

Performance analysis of a finite-buffer discrete-time queue with bulk arrival,bulk service and vacations

This paper considers a finite-buffer discrete-time Geo^X/G^Y/1/K+B queue with multiple vacations that has a wide range of applications including high-speed digital telecommunication systems and various related areas. The main purpose of this paper is to present a performance analysis of this system. For this purpose, we first derive a set of linear equations to compute the steady-state departure-epoch probabilities based on the embedded Markov chain technique. Next, we present numerically stable relationships for the steady-state probabilities of the queue lengths at three different epochs: departure, random, and arrival. Finally, based on these relationships, we present various useful performance measures of interest such as the moments of the number of packets in the queue at three different epochs, the mean delay in the queue of a packet, the loss probability and the probability that server is busy with computational experiences.     

Discrete-time multiserver queues with priorities

We analyze the performance of discrete-time multiserver queues that use a priority structure in the scheduling of departures. Such systems are often encountered in the context of ATM and in voice-data multiplexing. The arrival process is iid from slot to slot with a general distribution for the batch sizes. The analysis of both system contents and delay makes extensive use of generating functions and yields simple formulas to evaluate important performance measures such as mean values, variances and tail probabilities of these random variables. A numerical example illustrates the results of the analysis and shows that using priorities and sharing resources can lead to a considerable improvement of system performance.     

Markovian bulk service queue with delayed vacations

An M/M(a, b)/1 queueing system with multiple vacations is studied, in which if the number of customers in the queue is a - 1 either at a service completion epoch or at a vacation completion point, the server will wait for an exponential time in the system which is called the changeover time. During this changeover time if there is an arrival the server will start service immediately, otherwise at the end of the changeover time the server will go for a vacation. The duration of vacation is also exponential. This paper is concerned with the determination of the stationary distribution of the number of customers in the queue and the waiting time distribution of an arriving customer. The expected queue length is also obtained. Sample numerical illustrations are given.     

Analysis of queues with hyperexponential arrival distributions

We study H₂/H₂/1, H₂/M/1, and M/H₂/1 queueing systems with hyperexponential arrival distributions for the purpose of finding a solution for the mean waiting time in the queue. To this end we use the spectral decomposition method for solving the Lindley integral equation. For practical application of the obtained results, we use the method of moments. Since the hyperexponential distribution law has three unknown parameters, it allows to approximate arbitrary distributions with respect to the first three moments. The choice of this distribution law is due to its simplicity and the fact that in the class of distributions with coefficients of variation greater than 1, such as log-normal, Weibull, etc., only the hyperexponential distribution makes it possible to obtain an analytical solution.     

A closed-form solution on a level-dependent Markovian arrival process with queueing application

Abstract This paper reports a closed-form solution of the arrival events for a particular level-dependent Markovian arrival process (MAP). We apply the Baker–Hausdorff Lemma to the matrix expression of the number of arrival events in (0, t]. The successful derivation depends on the fact that the matrices representing the MAP have a specific structure. We report the results of numerical experiments indicating that the closed-form solution is less time-consuming than the uniformization technique for large values of t. As an application, we consider a finite-capacity, multi-server queueing model with impatient customers for possible use in automatic call distribution (ACD) systems. Our primary interest lies in performance measures related to customer waiting time, and we demonstrate how the closed-form solution is applicable to performance analysis.     

Computational analysis of a Markovian queueing system with geometric mean-reverting arrival process

In the queueing literature, an arrival process with random arrival rate is usually modeled by a Markov-modulated Poisson process (MMPP). Such a process has discrete states in its intensity and is able to capture the abrupt changes among different regimes of the traffic source. However, it may not be suitable for modeling traffic sources with smoothly (or continuously) changing intensity. Moreover, it is less parsimonious in that many parameters are involved but some are lack of interpretation. To cope with these issues, this paper proposes to model traffic intensity by a geometric mean-reverting (GMR) diffusion process and provides an analysis for the Markovian queueing system fed by this source. In our treatment, the discrete counterpart of the GMR arrival process is used as an approximation such that the matrix geometric method is applicable. A conjecture on the error of this approximation is developed out of a recent theoretical result, and is subsequently validated in our numerical analysis. This enables us to calculate the performance measures with high efficiency and precision. With these numerical techniques, the effects from the GMR parameters on the queueing performance are studied and shown to have significant influences.     

Queue length analysis of batch arrival queues under bilevel threshold control with early set-up

The queue length process of M^X/G/1 queues under bilevel threshold control and early set-up withlwithout server vacations is outlined. The stationary probability generating functions of the queue length distributions under a unified framework are also derived. Also derived are the mean queue length of each system.     

Optimal balking strategies in single-server queues with general service and vacation times

In many service systems arising in OR/MS applications, the servers may be temporarily unavailable, a fact that affects the sojourn time of a customer and his willingness to join. Several studies that explore the balking behavior of customers in Markovian models with vacations have recently appeared in the literature. In the present paper, we study the balking behavior of customers in the single-server queue with generally distributed service and vacation times. Arriving customers decide whether to enter the system or balk, based on a linear reward-cost structure that incorporates their desire for service, as well as their unwillingness to wait. We identify equilibrium strategies and socially optimal strategies under two distinct information assumptions. Specifically, in a first case, the customers make individual decisions without knowing the system state. In a second case, they are informed about the server’s current status. We examine the influence of the information level on the customers’ strategic response and we compare the resulting equilibrium and socially optimal strategies.     

Equilibrium customer strategies in Markovian queues with partial breakdowns

We consider equilibrium analysis of a single-server Markovian queueing system with working breakdowns. The system may become defective at any point of time when it is in operation. However, when the system is defective, instead of stopping service completely, the service continues at a slower rate. We assume that the arriving customers decide whether to join the system or balk based on a natural reward-cost structure. With considering waiting cost and reward, the balking behavior of customers is investigated and the corresponding Nash equilibrium strategies are derived. The effects of the information level on the equilibrium behavior are illustrated further via numerical experiments.     

Discrete-time queues with generally distributed service times and renewal-type server interruptions

In this contribution, we investigate a discrete-time single-server queue subjected to server interruptions. Server interruptions are modeled as an on/off process with geometrically distributed on-periods and generally distributed off-periods. As message lengths can exceed one time-slot, different operation modes are considered, depending on whether service of an interrupted message continues, partially restarts or completely restarts after an interruption. For all alternatives, we establish expressions for the steady-state probability generating functions (pgf) of the buffer contents at message departure times and random slot boundaries, of the unfinished work at random slot boundaries, the message delay, and the lengths of the idle and busy periods. From these results, closed-form expressions for various performance measures, such as mean and variance of the buffer occupancy and message delay, can be established. As an application, we show that this model is able to assess performance of a multi-class priority scheduling system. We then illustrate our approach with some numerical examples.     

时滞离散马尔可夫跳跃系统的鲁棒故障检测

研究具有状态时滞离散马尔可夫跳跃系统的鲁棒故障检测问题．基于依赖于系统模态的滤波器构造残差产生系统,利用H∞控制理论将故障检测滤波器的设计归结为H∞滤波问题,应用线性矩阵不等式技术得到了此类系统的故障检测滤波器存在的充分条件．数值仿真表明所提方法是可行的．     

Loss behavior in space priority queue with batch Markovian arrival process — discrete-time case

This paper applies matrix-analytic approach to the examination of the loss behavior of a space priority queue. In addition to the evaluation of the long-term high-priority and low-priority packet loss probabilities, we examine the bursty nature of packet losses by means of conditional statistics with respect to critical and non-critical periods that occur in an alternating manner. The critical period corresponds to having more than a certain number of packets in the buffer; non-critical corresponds to the opposite. Hence there is a threshold buffer level that splits the state space into two. By such a state-space decomposition, two hypothesized Markov chains are devised to describe the alternating renewal process. The distributions of various absorbing times in the two hypothesized Markov chains are derived to compute the average durations of the two periods and the conditional high-priority packet loss probability encountered during a critical period. These performance measures greatly assist the space priority mechanism for determining a proper threshold. The overall complexity of computing these performance measures is of the order O(K²m₁³m₂³), where K is the buffer capacity, and m₁ and m₂ are the numbers of phases of the underlying Markovian structures for the high-priority and low-priority packet arrival processes, respectively. Thus the results obtained are computationally tractable and numerical results show that, by choosing a proper threshold, a space priority queue not only can maintain the quality of service for the high-priority traffic but also can provide the near-optimum utilization of the capacity for the low-priority traffic.     

Optimal importance sampling for Markovian systems with applications to tandem queues

Importance sampling is a change-of-measure technique for speeding up the simulation of rare events in stochastic systems. In this paper we establish a number of properties characterizing optimal importance sampling measures for Markovian systems. We use these properties to develop a new method for computing the optimal measure and give specific results for a tandem queueing system. Optimal measures, though as diffcult to compute as the rare event probability itself, give useful insight into the characteristics of importance sampling measures. Our approach has no immediate computational advantage over other methods, but it suggests a number of heuristic approximations which may lead to computationally attractive methods.     

Product-form discrete-time queues in series with batch transition: late arrival case

This note presents a product-form equilibrium distribution for the specific type of queues in series, in which group transitions are allowed both in arrival and in service. Only the late arrival case is considered here. The results are valid for the discrete-time queueing network with geometric arrivals and the truncated geometric service time.     

Retrial queuing system with Markovian arrival flow and phase-type service time distribution

We consider a multi-server queuing system with retrial customers to model a call center. The flow of customers is described by a Markovian arrival process (MAP). The servers are identical and independent of each other. A customer’s service time has a phase-type distribution (PH). If all servers are busy during the customer arrival epoch, the customer moves to the buffer with a probability that depends on the number of customers in the system, leaves the system forever, or goes into an orbit of infinite size. A customer in the orbit tries his (her) luck in an exponentially distributed arbitrary time. During a waiting period in the buffer, customers can be impatient and may leave the system forever or go into orbit. A special method for reducing the dimension of the system state space is used. The ergodicity condition is derived in an analytically tractable form. The stationary distribution of the system states and the main performance measures are calculated. The problem of optimal design is solved numerically. The numerical results show the importance of considering the MAP arrival process and PH service process in the performance evaluation and capacity planning of call centers.     

Stability and Stabilization for Discrete-time Markovian Jump Stochastic Systems with Piecewise Homogeneous Transition Probabilities

International Journal of Control, Automation and Systems - In this paper, the stability and stabilization problems for discrete-time Markovian jump stochastic systems with time-varying transition...