Batch arrival queue with N-policy and at most J vacations

This paper studies the operating characteristics of an M^[x]/G/1 queueing system with N-policy and at most J vacations. The server takes at most J vacations repeatedly until at least N customers returning from a vacation are waiting in the queue. If no customer arrives by the end of the Jth vacation, the server becomes idle in the system until the number of arrivals in the queue reaches N. We derive the system size distribution at a random epoch and departure epoch, as well as various system characteristics.     

Operating characteristic analysis on the M/G/1 system with a variant vacation policy and balking

This paper studies the operating characteristics of an M^[x]/G/1 queueing system under a variant vacation policy, where the server leaves for a vacation as soon as the system is empty. The server takes at most J vacations repeatedly until at least one customer is found waiting in the queue when the server returns from a vacation. If the server is busy or on vacation, an arriving batch balks (refuses to join) the system with probability 1 − b. We derive the system size distribution at different points in time, as well as the waiting time distribution in the queue. Finally, important system characteristics are derived along with some numerical illustration.     

Two thresholds of a batch arrival queueing system under modified T vacation policy with startup and closedown

This paper studies the operating characteristics of the variant of an M^[x]/G/1 vacation queue with startup and closedown times. After all the customers are served in the system exhaustively, the server shuts down (deactivates) by a closedown time, and then takes at most J vacations of constant time length T repeatedly until at least one customer is found waiting in the queue upon returning from a vacation. If at least one customer is present in the system when the server returns from a vacation, then the server reactivates and requires a startup time before providing the service. On the other hand, if no customers arrive by the end of the J th vacation, the server remains dormant in the system until at least one customer arrives. We will call the vacation policy modified T vacation policy. We derive the steady‐state probability distribution of the system size and the queue waiting time. Other system characteristics are also investigated. The long‐run average cost function per unit time is developed to determine the suitable thresholds of T and J that yield a minimum cost. Copyright © 2007 John Wiley & Sons, Ltd.     

Optimal strategy policy in batch arrival queue with server breakdowns and multiple vacations

This paper considers a like-queue production system in which server vacations and breakdowns are possible. The decision-maker can turn a single server on at any arrival epoch or off at any service completion. We model the system by an M^[x]/M/1 queueing system with N policy. The server can be turned off and takes a vacation with exponential random length whenever the system is empty. If the number of units waiting in the system at any vacation completion is less than N, the server will take another vacation. If the server returns from a vacation and finds at least N units in the system, he immediately starts to serve the waiting units. It is assumed that the server breaks down according to a Poisson process and the repair time has an exponential distribution. We derive the distribution of the system size through the probability generating function. We further study the steady-state behavior of the system size distribution at random (stationary) point of time as well as the queue size distribution at departure point of time. Other system characteristics are obtained by means of the grand process and the renewal process. Finally, the expected cost per unit time is considered to determine the optimal operating policy at a minimum cost. The sensitivity analysis is also presented through numerical experiments.     

The performance measures and randomized optimization for an unreliable server M/G/1 vacation system

This paper examines an M^[x]/G/1 queueing system with a randomized vacation policy and at most J vacations. Whenever the system is 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 immediately 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. This pattern continues until the number of vacations taken reaches J. If the system is empty by the end of the Jth vacation, the server becomes idle in the system. Whenever one or more customers arrive at server idle state, the server immediately starts providing service for the arrivals. Assume that the server may meet an unpredictable breakdown according to a Poisson process and the repair time has a general distribution. For such a system, we derive the distributions of important system characteristics, such as system size distribution at a random epoch and at a departure epoch, system size distribution at busy period initiation epoch, the distributions of idle period, busy period, etc. Finally, a cost model is developed to determine the joint suitable parameters (p^∗, J^∗) at a minimum cost, and some numerical examples are presented for illustrative purpose.     

The randomized vacation policy for a batch arrival queue

This paper examines an M^[x]/G/1${\text{M}}^{[\text{x}]}/\text{G}/1$ queueing system with a randomized vacation policy and at most J vacations. Whenever the system is 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 immediately 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$1-p$. This pattern continues until the number of vacations taken reaches J. If the system is empty by the end of the J  th vacation, the server is dormant idly in the system. If there is one or more customers arrive at server idle state, the server immediately starts his services for the arrivals. For such a system, we derive the distributions of important characteristics, such as system size distribution at a random epoch and at a departure epoch, system size distribution at busy period initiation epoch, idle period and busy period, etc. Finally, a cost model is developed to determine the joint suitable parameters (p^∗,J^∗)$(p^{\ast }\text{,}{J}^{\ast })$ at a minimum cost, and some numerical examples are presented for illustrative purpose.     

An M/G/1 queue under hysteretic vacation policy with an early startup and un-reliable server

This paper considers the bi-level control of an M/G/1 queueing system, in which an un-reliable server operates N policy with a single vacation and an early startup. The server takes a vacation of random length when he finishes serving all customers in the system (i.e., the system is empty). Upon completion of the vacation, the server inspects the number of customers waiting in the queue. If the number of customers is greater than or equal to a predetermined threshold m, the server immediately performs a startup time; otherwise, he remains dormant in the system and waits until m or more customers accumulate in the queue. After the startup, if there are N or more customers waiting for service, the server immediately begins serving the waiting customers. Otherwise the server is stand-by in the system and waits until the accumulated number of customers reaches or exceeds N. Further, it is assumed that the server breaks down according to a Poisson process and his repair time has a general distribution. We obtain the probability generating function in the system through the decomposition property and then derive the system characteristics     

On the discrete-time Geo/G/1 queue with randomized vacations and at most J vacations

This paper examines a discrete-time Geo/G/1 queue, where the server may take at most J − 1 vacations after the essential vacation. In this system, messages arrive according to Bernoulli process and receive corresponding service immediately if the server is available upon arrival. When the server is busy or on vacation, arriving messages have to wait in the queue. After the messages in the queue are served exhaustively, the server leaves for the essential vacation. At the end of essential vacation, the server activates immediately to serve if there are messages waiting in the queue. Alternatively, the server may take another vacation with probability p or go into idle state with probability (1 − p) until the next message arrives. Such pattern continues until the number of vacations taken reaches J. This queueing system has potential applications in the packet-switched networks. By applying the generating function technique, some important performance measures are derived, which may be useful for network and software system engineers. A cost model, developed to determine the optimum values of p and J at a minimum cost, is also studied.     

Batch arrival queues under vacation policies with server breakdowns and startup/closedown times

This paper studies the operating characteristics of an M^[x]/G/1 queueing system under vacation policies with startup/closedown times, where the vacation time, the startup time, and the closedown time are generally distributed. When all the customers are served in the system exhaustively, the server shuts down (deactivates) by a closedown time. After shutdown, the server operates one of (1) multiple vacation policy and (2) single vacation policy. When the server reactivates since shutdown, he needs a startup time before providing the service. If a customer arrives during a closedown time, the service is immediately started without a startup time. The server may break down according to a Poisson process while working and his repair time has a general distribution. We analyze the system characteristics for the vacation models.     

Discrete NT-policy single server queue with Markovian arrival process and phase type service

We consider a discrete time single server queueing system in which arrivals are governed by the Markovian arrival process. During a service period, all customers are served exhaustively. The server goes on vacation as soon as he/she completes service and the system is empty. Termination of the vacation period is controlled by two threshold parameters N and T, i.e. the server terminates his/her vacation as soon as the number waiting reaches N or the waiting time of the leading customer reaches T units. The steady state probability vector is shown to be of matrix-geometric type. The average queue length and the probability that the server is on vacation (or idle) are obtained. We also derive the steady state distribution of the waiting time at arrivals and show that the vacation period distribution is of phase type.     

The Discrete-Time GI/Geo/1 Queue with Multiple Vacations

We study a discrete-time GI/Geo/1 queue with server vacations. In this queueing system, the server takes vacations when the system does not have any waiting customers at a service completion instant or a vacation completion instant. This type of discrete-time queueing model has potential applications in computer or telecommunication network systems. Using matrix-geometric method, we obtain the explicit expressions for the stationary distributions of queue length and waiting time and demonstrate the conditional stochastic decomposition property of the queue length and waiting time in this system.     

Min(N,V)--策略休假的M/G/1排队系统分析

在一个M／G／1休假排队系统中，同时考虑N-策略和多重休假策略，休假终止准则为任一个条件满足，我们称其为Min（N，V）-策略。本文给出了在此策略下的排队系统的稳态队长、忙期分布等基本指标。首次使用条件等待时间方法得到稳态等待时间的LST（Laplace-Stieltjes transform），同时还列举了一个应用的实例。最后指出本文模型是几个已研究模型的推广。     

An MX/G/1 queueing system with a setup period and a vacation period

This paper deals with an M^X/G/1 queueing system with a vacation period which comprises an idle period and a random setup period. The server is turned off each time when the system becomes empty. At this point of time the idle period starts. As soon as a customer or a batch of customers arrive, the setup of the service facility begins which is needed before starting each busy period. In this paper we study the steady state behaviour of the queue size distributions at stationary (random) point of time and at departure point of time. One of our findings is that the departure point queue size distribution is the convolution of the distributions of three independent random variables. Also, we drive analytically explicit expressions for the system state probabilities and some performance measures of this queueing system. Finally, we derive the probability generating function of the additional queue size distribution due to the vacation period as the limiting behaviour of the M^X/M/1 type queueing system. This revised version was published online in June 2006 with corrections to the Cover Date.     

带启动时间的多级适应性休假的M/G/1排队

本研究带启动时间的多级适应性休假的M/G/1间排队。给出稳态队长分布和母函数、等待时间分布和其LST及其随机分解结果，推导出忙期、假期和启动期的母函数。带有启动时间的单重休假和多重休假是本中模型的两个极端情况。     

Queues with hysteretic control by vacation and post-vacation periods

The paper investigates the queueing process in stochastic systems with bulk input, batch state dependent service, server vacations, and three post-vacation disciplines. The policy of leaving and entering busy periods is hysteretic, meaning that, initially, the server leaves the system on multiple vacation trips whenever the queue falls below r (⩾1), and resumes service when during his absence the system replenishes to N or more customers upon one of his returns. During his vacation trips, the server can be called off on emergency, limiting his trips by a specified random variable (thereby encompassing several classes of vacation queues, such as ones with multiple and single vacations). If by then the queue has not reached another fixed threshold M (⩽ N), the server enters a so-called “post-vacation period” characterized by three different disciplines: waiting, or leaving on multiple vacation trips with or without emergency. For all three disciplines, the probability generating functions of the discrete and continuous time parameter queueing processes in the steady state are obtained in a closed analytic form. The author uses a semi-regenerative approach and enhances fluctuation techniques (from his previous studies) preceding the analysis of queueing systems. Various examples demonstrate and discuss the results obtained. This revised version was published online in June 2006 with corrections to the Cover Date.     

Steady state analysis of an M/G/1 queue with linear retrial policy and two phase service under Bernoulli vacation schedule

We consider a single server queueing system with two phases of heterogeneous service and Bernoulli vacation schedule which operate under the so called linear retrial policy. This model extends both the classical M/G/1 retrial queue with linear retrial policy as well as the M/G/1 queue with two phases of service and Bernoulli vacation model. We carry out an extensive analysis of the model.     

Analysis of the M x /G/1 Queue with a Random Setup Time

Abstract

We consider an M ^x /G/1 queueing system with a random setup time, where the service of the first unit at the commencement of each busy period is preceded by a random setup time, on completion of which service starts. For this model, the queue size distributions at a random point of time as well as at a departure epoch and some important performance measures are known [see Choudhury, G. An M ^x /G/1 queueing system with setup period and a vacation period. Queueing Sys. 2000, 36, 23–38]. In this paper, we derive the busy period distribution and the distribution of unfinished work at a random point of time. Further, we obtain the queue size distribution at a departure epoch as a simple alternative approach to Choudhury4 Choudhury, G. 2000. An M^x/G/1 queueing system with setup period and a vacation period. Queueing Syst., 36: 23–38. [CROSSREF][Crossref], [Web of Science ®] , [Google Scholar]. Finally, we present a transform free method to obtain the mean waiting time of this model.     

A batch arrival retrial queue with two phases of service and Bernoulli vacation schedule

We consider an M X /G/1 queueing system with two phases of heterogeneous service and Bernoulli vacation schedule which operate under a linear retrial policy. In addition, each individual customer is subject to a control admission policy upon the arrival. This model generalizes both the classical M/G/1 retrial queue with arrivals in batches and a two phase batch arrival queue with a single vacation under Bernoulli vacation schedule. We will carry out an extensive stationary analysis of the system , including existence of the stationary regime, embedded Markov chain, steady state distribution of the server state and number of customer in the retrial group, stochastic decomposition and calculation of the first moment.     

Stationary Distributions of GI/M/c Queue with PH Type Vacations

We study a GI/M/c type queueing system with vacations in which all servers take vacations together when the system becomes empty. These servers keep taking synchronous vacations until they find waiting customers in the system at a vacation completion instant.The vacation time is a phase-type (PH) distributed random variable. Using embedded Markov chain modeling and the matrix geometric solution methods, we obtain explicit expressions for the stationary probability distributions of the queue length at arrivals and the waiting time. To compare the vacation model with the classical GI/M/c queue without vacations, we prove conditional stochastic decomposition properties for the queue length and the waiting time when all servers are busy. Our model is a generalization of several previous studies.