On a batch arrival Poisson queue with a random setup time and vacation period

The single server queue with vacation has been extended to include several types of extensions and generalisations, to which attention has been paid by several researchers (e.g. see Doshi, B. T., Single server queues with vacations — a servey. Queueing Systems, 1986, 1, 29–66; Takagi, H., Queueing Analysis: A Foundation of Performance evaluation, Vol. 1, Vacation and Priority systems, Part. 1. North Holland, Amsterdam, 1991; Medhi, J., Extensions and generalizations of the classical single server queueing system with Poisson input. J. Ass. Sci. Soc., 1994, 36, 35–41, etc.). The interest in such types of queues have been further enhanced in resent years because of their theoretical structures as well as their application in many real life situations such as computer, telecommunication, airline scheduling as well as production/inventory systems. This paper concerns the model building of such a production/inventory system, where machine undergoes extra operation (such as machine repair, preventive maintenance, gearing up machinery, etc.) before the processing of raw material is to be started. To be realistic, we also assume that raw materials arrive in batch. This production system can be formulated as an M^x/M/1 queues with a setup time. Further, from the utility point of view of idle time this model can also be formulated as a case of multiple vacation model, where vacation begins at the end of each busy period. Besides, the production/inventory systems, such a model is generally fitted to airline scheduling problems also. In this paper an attempt has been made to study the steady state behavior of such an M^x/M/1 queueing system with a view to provide some system performance measures, which lead to remarkable simplification when solving other similar types of queueing models.This paper deals with the steady state behaviour of a single server batch arrival Poisson queue with a random setup time and a vacation period. The service of the first customer in each busy period is preceded by a random setup period, on completion of which service starts. As soon as the system becomes empty the server goes on vacation for a random length of time. On return from vacation, if he finds customer(s) waiting, the server starts servicing the first customer in the queue. Otherwise it takes another vacation and so on. We study the steady state behaviour of the queue size distribution at random (stationary) point of time as well as at departure point of time and try to show that departure point queue size distribution can be decomposed into three independent random variables, one of which is the queue size of the standard M^x/M/1 queue. The interpretation of the other two random variables will also be provided. Further, we derive analytically explicit expressions for the system state (number of customers in the system) probabilities and provide their appropriate interpretations. Also, we derive some system performance measures. Finally, we develop a procedure to find mean waiting time of an arbitrary customer.     

Analytic and numerical aspects of batch service queues with single vacation

This paper deals with an M/G/1 batch service queue where customers are served in batches of maximum size b with a minimum threshold value a. The server takes a single vacation when he finds less than a customers after the service completion. The vacation time of the server is arbitrarily distributed. Using the supplementary variable method we obtain the probability generating functions of the queue length distributions at various epochs. We also obtain relations among queue length distributions at arbitrary, service (vacation) termination epochs. Further their evaluation is also discussed. Finally, some numerical results and graphs are presented.     

A batch arrival queue under randomised multi-vacation policy with unreliable server and repair

This article examines an M^[x]/G/1 queueing system with an unreliable server and a repair, in which the server operates a randomised vacation policy with multiple available vacations. Upon the system being found to be empty, the server immediately takes a vacation. If there is at least one customer found waiting in the queue upon returning from a vacation, the server will be activated for service. Otherwise, if no customers are waiting for service at the end of a vacation, the server either remains idle with probability p or leaves for another vacation with probability 1???p. When one or more customers arrive when the server is idle, the server immediately starts providing service for the arrivals. It is possible that an unpredictable breakdown may occur in the server, in which case a repair time is requested. For such a system, we derive the distributions of several important system characteristics, such as the system size distribution at a random epoch and at a departure epoch, the system size distribution at the busy period initiation epoch, and the distribution of the idle and busy periods. We perform a numerical analysis for changes in the system characteristics, along with changes in specific values of the system parameters. A cost effectiveness maximisation model is constructed to show the benefits of such a queueing system.     

On a single server queue with two-stage heterogeneous service and deterministic server vacations

We analyse a single server queue with Poisson arrivals, two stages of heterogeneous service with different general (arbitrary) service time distributions and binomial schedule server vacations with deterministic (constant) vacation periods. After first-stage service the server must provide the second stage service. However, after the second stage service, he may take a vacation or may decide to stay on in the system. For convenience, we designate our model as M/G ₁, G ₂/D/1 queue. We obtain steady state probability generating function of the queue length for various states of the server. Results for some particular cases of interest such as M/E_{k 1} , E_{k 2} /D/1, M/M ₁, M ₂/D/1, M/E _k /D/1 and M/G ₁, G ₂/1 have been obtained from the main results and some known results including M/E_k /1 and M/G/1 have been derived as particular cases of our particular cases.     

Maximum entropy analysis of the M[x]/M/1 queueing system with multiple vacations and server breakdowns

We consider a single unreliable sever in an M^[x]/M/1 queueing system with multiple vacations. As soon as the system becomes empty, the server leaves the system for a vacation of exponential length. When he returns from the vacation, if there are customers waiting in the queue, he begins to serve the customers; otherwise, another vacation is taken. Breakdown times and repair times of the server are assumed to obey a negative exponential distribution. Arrival rate varies according to the server’s status: vacation, busy, or breakdown. Using the maximum entropy principle, we develop the approximate formulae for the probability distributions of the number of customers in the system which is used to obtain various system performance measures. We perform a comparative analysis between the exact results and the maximum entropy results. We demonstrate, through the maximum entropy results, that the maximum entropy principle approach is accurate enough for practical purposes.     

The analysis of the M/M/1 queue with two vacation policies (M/M/1/SWV+MV)

We consider an M/M/1 queue with two vacation policies which comprise single working vacation and multiple vacations, denoted by M/M/1/SMV+MV. Using two methods (called R-matrix method and G-matrix method), we obtain the stationary distribution of queue length (including the customer being in service) and make further analysis on the stationary numbers of customers in the working vacation and vacation period, respectively. The stochastic decomposition results of stationary queue length and the sojourn time of a customer are also derived. Meanwhile, we show that a simple and direct method of decomposition developed in Liu et al. [Stochastic decompositions in the M/M/1 queue with working vacations, Oper. Res. Lett. 35 (2007), pp. 595–600] is also applicable to our model. Furthermore, busy period is analysed by the limiting theorem of alternative renewal process. Finally, some boundary properties and numerical analysis on performance measures are presented.     

Optimal policies for a controlled queueing system with removable server under a random vacation circumstance

In this paper, we consider two stochastic models for a controlled single-server M/G/1 queueing system under a random vacation circumstance. The relevant expected costs are formulated under two control strategies and approximated by means of a familiar diffusion approximation method. Numerical calculations are devoted to examine the accuracy of the proposed approximation method. The results show that the diffusion approximation method is reasonably good for the performance evaluation on the typical queueing control systems, especially, for the M/M/1 single-server queue.     

A discrete-time single-server queue with a modified N -policy

We consider a discrete-time single-server queue where the idle server waits for reaching a level N in the queue size to start a batch service of N messages, although the following arrivals during the busy period receive single services. We find the stationary distributions of the queue and system lengths as well as some performance measures. The vacation and busy periods of the system and the number of messages served during a busy period are also analyzed. The stationary distributions of the time spent waiting in the queue and in the system are studied too. Finally, a total expected cost function is developed to determine the optimal operating N-policy at minimum cost.     

带启动时间和可修服务台的M/M/1/N工作休假排队系统

分析带有启动时间、服务台可故障的M/M/1/N单重工作休假排队系统.在该系统中,服务台在休假期间不是完全停止工作,而是处于低速服务状态.假定服务台允许出现故障且当出现故障时,服务台停止为顾客服务且立即进行修理.服务台的失效时间和修理时间均服从指数分布,且工作休假期和正规忙期具有不同的取值;同时,从关闭期到正规忙期有服从指数分布的启动时间.建立此工作休假排队系统的有限状态拟生灭过程(QBD),使用矩阵几何方法得到QBD的各稳态概率相互依赖的率阵,从而求得稳态概率向量.通过有限状态QBD的最小生成元和稳态概率向量得到系统的基本阵和协方差矩阵,求解出系统方差、系统稳态可用度、系统吞吐率、系统稳态队长、系统稳态故障频度等系统性能.数值分析体现了所提出方法的有效性和实用性,通过敏感性分析将各参数对系统性能的影响进行了初探,为此模型的实际应用提供了很好的理论依据.     

An MX/G/1 queue with randomized working vacations and at most J vacations

This paper treats a bulk arrival queue with randomized working vacation policy. Whenever the system becomes empty, the server takes a vacation. During the vacation period, customers are to be served at a lower rate. Once the vacation ends, the server will return to the normal working state and begin to serve the customers in the system if any. Otherwise, the server either remains idle with probability p or leaves for another vacation with probability 1?p. This pattern continues until the number of vacations taken reaches J. If the system is empty at the end of the Jth vacation, the server will wait idly for a new arrival. By using supplementary variable technique, we derive the system size distribution at arbitrary epoch, at departure epoch and at busy period initial epoch, as well as some important system characteristics. Numerical examples are provided to illustrate the influence of system parameters on several performance measures.     

Optimal control of the M/G/1 queue with repeated vacations of theserver

An M/G/1 queue where the server may take repeated vacations is considered. Whenever a busy period terminates, the server takes a vacation of random duration. At the end of each vacation, the server may either take a new vacation or resume service; if the queue is found empty, the server always takes a new vacation. The cost structure includes a holding cost per unit of time and per customer in the system and a cost each time the server is turned on. One discounted cost criterion and two average cost criteria are investigated. It is shown that the vacation policy that minimizes the discounted cost criterion over all policies (randomized, history dependent, etc.) converges to a threshold policy as the discount factor goes to zero. This result relies on a nonstandard use of the value iteration algorithm of dynamic programming and is used to prove that both average cost problems are minimized by a threshold policy     

Entropy maximization and NT vacation M/G/1 model with a startup and unreliable server: comparative analysis on the first two moments of system size

We consider the control policy of an M/G/1 queueing system with a startup and unreliable server, in which the length of the vacation period is controlled either by the number of arrivals during the idle period, or by a timer. After all the customers have been served in the queue, the server immediately takes a vacation and operates an NT vacation policy: the server reactivates as soon as the number of arrivals in the queue reaches a predetermined threshold N or when the waiting time of the leading customer reaches T units. In such a variant vacation system, the steady-state probabilities cannot be obtained explicitly. Thus, the maximum entropy principle is used to derive the approximate formulas for the steady-state probability distributions of the queue length. A comparitive analysis of two approximation approaches, using the first and the second moments of system size, is studied. Both solutions are compared with the exact results under several service time distributions with specific parameter values. Our numerical investigations demonstrate that the use of the second moment of system size for the available information is, in general, sufficient to obtain more accurate estimations than that of the first moment.     

Optimal state-free, size-aware dispatching for heterogeneous -type systems

We consider a cluster of heterogeneous servers, modeled as M/G/1 first-come first-serve queues with different processing speeds. A dispatcher that assigns jobs to the servers takes as input only the size of the arriving job and the overall job-size distribution. This general model captures the behavior of a variety of real systems, such as web server clusters. Our goal is to identify assignment strategies that the dispatcher can perform to minimize expected completion time and waiting time. We show that there exist optimal strategies that are deterministic, fixing the server to which jobs of particular sizes are always sent. We prove that the optimal strategy for systems with identical servers assigns a non-overlapping interval range of job sizes to each server. We then prove that when server processing speeds differ, it is necessary to assign each server a distinct set of intervals of job sizes in order to minimize expected waiting or response times.     

带工作休假和工作故障的M/M/1/N排队系统性能分析

本文在可修M/M/1/N排队系统中引入了启动时间、工作休假和工作故障策略.在该系统中,服务台在休假期间不是完全停止工作,而是处于低速服务状态.设定服务台在任何时候均可发生故障,当故障发生时立刻进行维修.且当服务台在正规忙期出现故障时,服务台仍以较低的服务速率为顾客服务.服务台的寿命时间和修理时间均服从指数分布,且在不同的时期有不同的取值.同时,从关闭期到正规忙期有服从指数分布的启动时间.本文建立此模型的有限状态拟生灭过程(QBD),使用矩阵几何方法得到系统的稳态概率向量,并应用基本阵和协方差矩阵理论,计算出系统稳态可用度、系统方差、系统吞吐率、系统稳态队长及各系统稳态概率等系统性能指标.同时,通过数值实验对各系统参数对系统性能的影响进行了初探.文中的敏感性分析体现了这种方法的有效性和可用性.实验表明,文中提出的模型,可有效改善仅带有工作休假或工作故障策略排队模型的系统性能.     

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.     

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.     

On the number of customers served in the M/G/1 retrial queue: first moments and maximum entropy approach

In this paper we present general results on the number of customers, I, served during the busy period in an M/G/1 retrial system. Its analysis in terms of Laplace transforms has been previously discussed in the literature. However, this solution presents important limitations in practice; in particular, the moments of I cannot be obtained by direct differentiation. We propose a direct method of computation for the second moment of I and also for the probability of k,k⩽4, customers being served in a busy period. Then, the maximum entropy principle approach is used to estimate the true distribution of I according to the available information.Scope and purposeWe consider an M/G/1 queue with retrials. Retrial queueing systems are characterized by the fact that, an arriving customer who finds the server busy is obliged to leave the service area and return later to repeat his request after some random time. We deal with I, the number of customers served during the busy period of a retrial queue, and obtain closed expressions for its main characteristics, which will be employed in order to estimate the true distribution of this random variable.     

An algorithmic analysis of multi-server vacation model with service interruptions

Motivated by the trade-off between reliability and utilization level of a stochastic service system, we considers a Markovian multi-server vacation queueing system with c unreliable servers. In such a system, some servers may not be available due to either planned stoppage (vacations) or unplanned service interruptions (server failures). The vacations are controlled by a threshold policy. With this policy, at a service completion instant, if d (?c) servers become idle, they take a vacation together and will keep taking vacations until they find at least c − d + 1 customers are in the system at a vacation completion instant, and then they return to serve the queue. In addition, all on-duty servers are subject to failures and can be repaired within a random period of time. We formulate a quasi-birth–death (QBD) process, establish the stability condition, and develop a computational algorithm to obtain the stationary performance measures of the system. Numerical examples are presented to show the performance evaluation and optimization of such a system. The insights gained from this model help practitioners make capacity and operating decisions for this type of waiting line systems.     

Cost-minimization analysis of a working vacation queue with N-policy and server breakdowns

This paper investigates the N-policy M/M/1 queueing system with working vacation and server breakdowns. As soon as the system becomes empty, the server begins a working vacation. The server works at a lower service rate rather than completely stopping service during a vacation period. The server may break down with different breakdown rates during the idle, working vacation, and normal busy periods. It is assumed that service times, vacation times, and repair times are all exponentially distributed. We analyze this queueing model as a quasi-birth–death process. Furthermore, the equilibrium condition of the system is derived for the steady state. Using the matrix-geometric method, we find the matrix-form expressions for the stationary probability distribution of the number of customers in the system and system performance measures. The expected cost function per unit time is constructed to determine the optimal values of the system decision variables, including the threshold N and mean service rates. We employ the particle swarm optimization algorithm to solve the optimization problem. Finally, numerical results are provided, and an application example is given to demonstrate the applicability of the queueing model.