Analysis of discrete-time batch service renewal input queue with multiple working vacations

This paper investigates a discrete-time single server batch service queue with multiple working vacations wherein arrivals occur according to a discrete-time renewal process. The server works with a different service rate rather than completely stopping during the vacation period. The service is performed in batches and the server takes a vacation when the system does not have any waiting customers at a service completion epoch or a vacation completion epoch. We present a recursive method, using the supplementary variable technique to obtain the steady-state queue-length distributions at pre-arrival, arbitrary and outside observer’s observation epochs. The displacement operator method is used to solve simultaneous non-homogeneous difference equations. Some performance measures and waiting-time distribution in the system have also been discussed. Finally, numerical results showing the effect of model parameters on key performance measures are presented.     

Impatient customers in an M/M/1 queue with single and multiple working vacations

We consider an M/M/1 queue with impatient customers and two different types of working vacations. The working vacation policy is the one in which the server serves at a lower rate during a vacation period rather than completely stop serving. The customer’s impatience is due to its arrival during a working vacation period, in which the customer service rate is lower than the normal busy period. We analyze the queue for two different working vacation termination policies, a multiple working vacation policy and a single working vacation policy. Closed-form solutions and various performance measures like, the mean queue lengths and the mean waiting times are derived. The stochastic decomposition properties are verified for both multiple and single working vacation cases. A comparison of both the models is carried out to capture their performances with the change in system parameters.     

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 replacement policy for a repairable system with multiple vacations and imperfect fault coverage

The present investigation deals with the reliability analysis of a repairable system consisting of single repairman who can take multiple vacations. The system failure may occur due to two types of faults termed as major and minor. When the system has failed due to minor faults, it is perfectly recovered by the repairman. If the system failure is due to major faults, there are some recovery levels/procedures that recover the faults imperfectly with some probability. However, the system cannot be repaired in ‘as good as new’ condition. It is assumed that the repairman can perform some other tasks when either the system is idle or waiting for recovery from the faults. The life time of the system and vacation time of the repairman are assumed to be exponential distributed while the repair time follows the general distribution. By assuming the geometric process for the system working/vacation time, the supplementary variable technique and Laplace transforms approach are employed to derive the reliability indices of the system. We propose the replacement policy to maximize the expected profit after a long run time. The validity of the analytical results is justified by taking numerical illustrations.     

带启动时间和工作故障的M/M/1/N排队系统性能分析

在M/M/1/N可修排队系统中引入了工作故障和启动时间.服务台在忙期允许出现故障,且在故障期间不是完全停止服务而是以较低的服务速率为顾客服务.同时,从关闭期到正规忙期有服从指数分布的启动时间.通过分析此模型的二维连续时间Markov过程,求解出系统平稳方程,建立此系统的有限状态拟生灭过程(QBD).根据系统参数,求解出水平相依的子率阵,从而得到系统稳态概率向量的矩阵几何表示形式.在系统稳态概率向量的基础上,求解出系统吞吐率、系统稳态可用度、系统稳态队长及系统处于各个状态的概率等性能指标的解析表达式.文中的敏感性分析体现了这种方法的有效性和可用性,同时,对系统各性能受系统参数的影响进行了探索.实验表明,文中提出模型的稳定性较好,且更贴近实际服务过程,因此这种模型将被广泛应用于各种实际服务中.     

The GI/M/1 queue and the GI/Geo/1 queue both with single working vacation

We first consider the continuous-time GI/M/1 queue with single working vacation (SWV). During the SWV, the server works at a different rate rather than completely stopping working. We derive the steady-state distributions for the number of customers in the system both at arrival and arbitrary epochs, and for the FIFO sojourn time for an arbitrary customer. We then consider the discrete-time GI/Geo/1/SWV queue by contrasting it with the GI/M/1/SWV queue.     

Steady state analysis and computation of the GI/M/1/L queue with multiple working vacations and partial batch rejection

This paper analyzes a finite-buffer bulk-arrival bulk-service queueing system with multiple working vacations and partial batch rejection in which the inter-arrival and service times are, respectively, arbitrarily and exponentially distributed. Using the supplementary variable and the embedded Markov chain techniques, we obtain the waiting queue-length distributions at pre-arrival and arbitrary epochs. We also present Laplace–Stiltjes transform of the actual waiting-time distribution in the queue. Finally, several performance measures and a variety of numerical results in the form of tables and graphs are discussed.     

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...     

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     

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.     

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.     

Analysis of the discrete-time Geo/G/1 working vacation queue and its application to network scheduling

In this paper we present an exact steady-state analysis of a discrete-time Geo/G/1 queueing system with working vacations, where the server can keep on working, but at a slower speed during the vacation period. The transition probability matrix describing this queuing model can be seen as an M/G/1-type matrix form. This allows us to derive the probability generating function (PGF) of the stationary queue length at the departure epochs by the M/G/1-type matrix analytic approach. To understand the stationary queue length better, by applying the stochastic decomposition theory of the standard M/G/1 queue with general vacations, another equivalent expression for the PGF is derived. We also show the different cases of the customer waiting to obtain the PGF of the waiting time, and the normal busy period and busy cycle analysis is provided. Finally, we discuss various performance measures and numerical results, and an application to network scheduling in the wavelength division-multiplexed (WDM) system illustrates the benefit of this model in real problems.     

Note on the article: Maximum entropy analysis of the M/M/1 queueing system with multiple vacations and server breakdowns

Wang et al. [Wang, K. H., Chan, M. C., & Ke, J. C. (2007). Maximum entropy analysis of the M^[x]/M/1 queueing system with multiple vacations and server breakdowns. Computers & Industrial Engineering, 52, 192–202] elaborate on an interesting approach to estimate the equilibrium distribution for the number of customers in the M^[x]/M/1 queueing model with multiple vacations and server breakdowns. Their approach consists of maximizing an entropy function subject to constraints, where the constraints are formed by some known exact results. By a comparison between the exact expression for the expected delay time and an approximate expected delay time based on the maximum entropy estimate, they argue that their maximum entropy estimate is sufficiently accurate for practical purposes. In this note, we show that their maximum entropy estimate is easily rejected by simulation. We propose a minor modification of their maximum entropy method that significantly improves the quality of the estimate.     

G/M/1 queue study of buffer behavior in the decoding system of computer communication

This paper deals with the buffer behavior at the decoding center of a computer communication system in which the messages are in the Huffman code of English text. It is assumed that the arrival of messages has an arbitrary distribution, with the message lengths having negative exponential distribution. The situation is well described by the G/M/1 model of queue theory. The waiting time model is simulated on the EC-1030 computer, assuming the HP2100A computer is the decoding machine. The simulation results are used for estimation of buffer size in a character-oriented system and block-oriented system for a very low overflow probability.     

Analytic and computational analysis of the discrete-time GI/D-MSP/1 queue using roots

This paper presents a simple closed-form analysis for evaluating system-length distributions at various epochs of the discrete-time GI/D-MSP/1 queue. The proposed analysis is based on roots of the associated characteristic equation of the vector-generating function of system-length distribution at prearrival epochs. We provide the steady-state system-length distribution at random epoch by using the classical argument based on Markov renewal theory. The queueing-time distribution has also been investigated. Numerical aspects have been tested for a variety of interarrival- and service-time distributions and a sample of numerical outputs is presented.     

An approximate transient analysis of the M(t)/M/1 queue

An approach, based on recent work by Stern [56], is described for obtaining the approximate transient behavior of both the M/M/1 and M(t)/M/1 queues, where the notation M(t) indicates an exponential arrival process with time-varying parameter λ(t). The basic technique employs an M/M/1K approximation to the M/M/1 queue to obtain a spectral representation of the time-dependent behavior for which the eigen values and eigenvectors are real.Following a general survey of transient analysis which has already been accomplished, Stern's M/M/1/K approximation technique is examined to determine how best to select a value for K which will yield both accurate and computationally efficient results. It is then shown how the approximation technique can be extended to analyze the M(t)/M/1 queue where we assume that the M(t) arrival process can be approximated by a discretely time-varying Poisson process.An approximate expression for the departure process of the M/M/1 queue is also proposed which implies that, for an M(t)/M/1 queue whose arrival process is discretely time-varying, the departure process can be approximated as discretely time-varying too (albeit with a different time-varying parameter).In all cases, the techniques and approximations are examined by comparison with exact analytic results, simulation or alternative discrete-time approaches.     

A matrix approach for the transient solution of an M/M/1/N queue with discouraged arrivals and reneging

This paper illustrates a computable matrix technique that can be used to derive explicit expressions for the transient state probabilities of a finite waiting space single-server queue, namely (M/M/1/N), having discouraged arrivals and reneging. The discipline is the classical first-come, first-served (FCFS). We obtain the transient solution of the system, with results in terms of the eigenvalues of a symmetric tridiagonal matrix. Finally, numerical calculations are given to illustrate the effectiveness of this technique and system behaviour.