On a BMAP/G/1 G-queue with setup times and multiple vacations

In this paper, we consider a BMAP/G/1 G-queue with setup times and multiple vacations. Arrivals of positive customers and negative customers follow a batch Markovian arrival process (BMAP) and Markovian arrival process (MAP) respectively. The arrival of a negative customer removes all the customers in the system when the server is working. The server leaves for a vacation as soon as the system empties and is allowed to take repeated (multiple) vacations. By using the supplementary variables method and the censoring technique, we obtain the queue length distributions. We also obtain the mean of the busy period based on the renewal theory.     

An MAP/G/1 G-queues with preemptive resume and multiple vacations

In this paper, we consider an MAP/G/1 G-queues with possible preemptive resume service discipline and multiple vacations wherein the arrival process of negative customers is Markovian arrival process (MAP). The arrival of a negative customer may remove the customer being in service. The server leaves for a vacation as soon as the system empties and is allowed to take repeated (multiple) vacations. The service and vacation times are arbitrarily distributed. We obtain the queue length distributions with the method of supplementary variables, combined with the matrix-analytic method and censoring technique. We also obtain the mean of the busy period based on the renewal theory. Finally we provide expressions for a special case.     

The infinite-buffer single server queue with a variant of multiple vacation policy and batch Markovian arrival process

We consider an infinite-buffer single server queue where arrivals occur according to a batch Markovian arrival process (BMAP). The server serves until system emptied and after that server takes a vacation. The server will take a maximum number H of vacations until either he finds at least one customer in the queue or the server has exhaustively taken all the vacations. We obtain queue length distributions at various epochs such as, service completion/vacation termination, pre-arrival, arbitrary, departure, etc. Some important performance measures, like mean queue lengths and mean waiting times, etc. have been obtained. Several other vacation queueing models like, single and multiple vacation model, queues with exceptional first vacation time, etc. can be considered as special cases of our model.     

Analysis of the MAP/G/1/N queue with multiple vacations

This paper deals with a batch service queue and multiple vacations. The system consists of a single server and a waiting room of finite capacity. Arrival of customers follows a Markovian arrival process (MAP). The server is unavailable for occasional intervals of time called vacations, and when it is available, customers are served in batches of maximum size ‘b’ with a minimum threshold value ‘a’. We obtain the queue length distributions at various epochs along with some key performance measures. Finally, some numerical results have been presented.     

带有止步和中途退出的成批到达的M~x/M/R/N同步休假排队系统的性能分析

研究了带有止步和中途退出的Mx/M/R/N同步休假排队系统.顾客成批到达.到达的顾客如果看到服务员正在休假或者全忙,他或者以概率b决定进入队列等待服务,或者以概率1-b止步(不进入系统).系统根据一定的原则以概率nk在未止步的k个顾客中选择n个进入系统.在系统中排队等待服务的顾客可能因为等待的不耐烦而在没有接受服务的情况下离开系统(中途退出).系统中一旦没有顾客,R个服务员立即进行同步多重休假.首先,利用马尔科夫过程理论建立了系统稳态概率满足的方程组.其次,在证明了相关矩阵可逆性的基础上,利用矩阵解法求出了系统稳态概率的明显表达式,并得到了系统的平均队长、平均等待队长及顾客的平均损失率等性能指标.     

带有负顾客且具有Bernoulli反馈的M/M/1工作休假排队

本文研究带反馈的具有正、负两类顾客的M/M/1工作休假排队模型.工作休假策略为空竭服务多重工作休假.负顾客一对一抵消队尾的正顾客(若有),若系统中无正顾客时,到达的负顾客自动消失,负顾客不接受服务.完成服务的正顾客以概率p(0相似文献   

带RCE抵消策略的负顾客M/M/1工作休假排队系统

考虑服务员在休假期间不是完全停止工作,而是以相对于正常工作时低些的速率服务顾客的M/M/1工作休假排队模型.在此模型基础上,笔者针对现实的M/M/1排队模型中可能出现的外来干扰因素,提出了带RCE(Removal of Customers at the End)抵消策略的负顾客M/M/1工作休假排队这一新的模型.服务规则为先到先服务.工作休假策略为空竭服务多重工作休假.抵消原则为负顾客一对一抵消队尾的正顾客,若系统中无正顾客时,到达的负顾客自动消失,负顾客不接受服务.使用拟生灭过程和矩阵几何解方法给出了系统队长的稳态分布,证明了系统队长和等待时间的随机分解结果并给出稳态下系统中正顾客的平均队长和顾客在系统中的平均等待时间.     

A mixed priority retrial queue with negative arrivals,unreliable server and multiple vacations

A retrial queue accepting two types of positive customers and negative arrivals, mixed priorities, unreliable server and multiple vacations is considered. In case of blocking the first type customers can be queued whereas the second type customers leave the system and try their luck again after a random time period. When a first type customer arrives during the service of a second type customer, he either pushes the customer in service in orbit (preemptive) or he joins the queue waiting to be served (non-preemptive). Moreover negative arrivals eliminate the customer in service and cause server’s abnormal breakdown, while in addition normal breakdowns may also occur. In both cases the server is sent immediately for repair. When, upon a service or repair completion, the server finds no first type customers waiting in queue remains idle and activates a timer. If timer expires before an arrival of a positive customer the server departs for multiple vacations. For such a system the stability conditions and the system state probabilities are investigated both in a transient and in a steady state. A stochastic decomposition result is also presented. Interesting applications are also discussed. Numerical results are finally obtained and used to investigate system performance.     

带有负顾客且具有反馈的M/M/1/N工作休假排队

研究带反馈的且具有正、负两类顾客的M/M/1/N工作休假排队模型.工作休假策略为空竭服务多重工作休假.负顾客一对一抵消队首正在接受服务的正顾客(若有),若系统中无正顾客时,到达的负顾客自动消失,负顾客不接受服务.完成服务的正顾客以概率p(0相似文献   

The queue length distributions in the finite buffer bulk-service MAP/G/1 queue with multiple vacations

We consider a finite buffer batch service queueing system with multiple vacations wherein the input process is Markovian arrival process (MAP). The server leaves for a vacation as soon as the system empties and is allowed to take repeated (multiple) vacations. The service- and vacation- times are arbitrarily distributed. We obtain the queue length distributions at service completion, vacation termination, departure, arbitrary and pre-arrival epochs. Finally, some performance measures such as loss probability, average queue lengths are discussed. Computational procedure has been given when the service- and vacation- time distributions are of phase type (PH-distribution).     

Optimal Hysteretic Control for the BMAP/G/ 1 System with Single and Group Service Modes

In this paper, we consider a single server queuing model with an infinite buffer in which customers arrive according to a batch Markovian arrival process (BMAP). The services are offered in two modes. In mode 1, the customers are served one at a time and in mode 2 customers are served in groups of varying sizes. Various costs for holding, service and switching are imposed. For a given hysteretic strategy, we derive an expression for the cost function from which an optimal hysteretic control can be obtained. Illustrative numerical examples are presented.     

带有止步和中途退出的成批到达的Mx/M/1/N多重休假排队系统的性能分析

本文研究了带有止步和中途退出的M^x／M/1／N多重休假排队系统。顾客成批到达，到达后每批中的顾客，或者以概率b决定进入队列等待服务，或者以概率1-b止步（不进入系统）。顾客进入系统后可能因为等待的不耐烦而在没有接受服务的情况下离开系统（中途退出）。系统中一旦没有顾客，服务员立即进行多重休假。首先，利用马尔科夫过程理论建立了系统稳态概率满足的方程组。其次，在利用高等代数相关知识证明了相关矩阵可逆性的基础上，利用矩阵解法求出了稳态概率的矩阵解，并得到了系统的平均队长、平均等待队长以及顾客的平均损失率等性能指标。     

带负顾客启动期N策略的Geom/Geom/1工作休假排队

将带RCH抵消策略的负顾客、启动期和N策略引入离散时间排队.休假策略为空竭服务多重工作休假.负顾客一对一抵消队首正在接受服务的正顾客,若系统中无正顾客时,到达的负顾客自动消失,负顾客不接受服务.利用拟生灭过程和矩阵几何解方法,给出了稳态队长分布及其随机分解.通过数值例子表现了启动率和负顾客到达率对稳态队长的影响.     

The Periodic BMAP/PH/c Queue

In queueing theory, most models are based on time-homogeneous arrival processes and service time distributions. However, in communication networks arrival rates and/or the service capacity usually vary periodically in time. In order to reflect this property accurately, one needs to examine periodic rather than homogeneous queues. In the present paper, the periodic BMAP/PH/c queue is analyzed. This queue has a periodic BMAP arrival process, which is defined in this paper, and phase-type service time distributions. As a Markovian queue, it can be analysed like an (inhomogeneous) Markov jump process. The transient distribution is derived by solving the Kolmogorov forward equations. Furthermore, a stability condition in terms of arrival and service rates is proven and for the case of stability, the asymptotic distribution is given explicitly. This turns out to be a periodic family of probability distributions. It is sketched how to analyze the periodic BMAP/M _t /c queue with periodically varying service rates by the same method.     

Analyzing the finite buffer batch arrival queue under Markovian service process: GIX/MSP/1/N

We consider a batch arrival finite buffer single server queue with inter-batch arrival times are generally distributed and arrivals occur in batches of random size. The service process is correlated and its structure is presented through Markovian service process (MSP). The model is analyzed for two possible customer rejection strategies: partial batch rejection and total batch rejection policy. We obtain steady-state distribution at pre-arrival and arbitrary epochs along with some important performance measures, like probabilities of blocking the first, an arbitrary, and the last customer of a batch, average number of customers in the system, and the mean waiting times in the system. Some numerical results have been presented graphically to show the effect of model parameters on the performance measures. The model has potential application in the area of computer networks, telecommunication systems, manufacturing system design, etc.      

The GI/M/1 queue with phase-type working vacations and vacation interruption

Consider a GI/M/1 queue with phase-type working vacations and vacation interruption where the vacation time follows a phase-type distribution. The server takes the original work at the lower rate during the vacation period. And, the server can come back to the normal working level at a service completion instant if there are customers at this instant, and not accomplish a complete vacation. From the PH renewal process theory, we obtain the transition probability matrix. Using the matrix-analytic method, we obtain the steady-state distributions for the queue length at arrival epochs, and waiting time of an arbitrary customer. Meanwhile, we obtain the stochastic decomposition structures of the queue length and waiting time. Two numerical examples are presented lastly.     

Analysis of the finite source MAP/PH/N retrial G-queue operating in a random environment

Finite source retrial G-queues are good mathematical models of communication systems and networks, so their investigation is important for theory and applications. In this paper, we analyze the MAP/PH/N retrial queue with finite number of sources and MAP arrivals of negative customers operating in a finite state Markovian random environment. The arrival of a negative customer with equal probability goes to any busy server to remove the customer being in service. The multi-dimensional Markov chain describing the behavior of the system is investigated. The algorithms for calculating the stationary state probabilities are elaborated. Main performance measures are obtained. Illustrative numerical examples are presented.     

Analyzing discrete-time D-BMAP/G/1/N queue with single and multiple vacations

This paper analyzes a single-server finite-buffer vacation (single and multiple) queue wherein the input process follows a discrete-time batch Markovian arrival process (D-BMAP). The service and vacation times are generally distributed and their durations are integral multiples of a slot duration. We obtain the state probabilities at service completion, vacation termination, arbitrary, and prearrival epochs. The loss probabilities of the first-, an arbitrary- and the last-customer in a batch, and other performance measures along with numerical aspects have been discussed. The analysis of actual waiting time of these customers in an accepted batch is also carried out.     

Stationary analysis of a discrete-time GI/D-MSP/1 queue with multiple vacations

This paper analyzes the steady-state behavior of a discrete-time single-server queueing system with correlated service times and server vacations. The vacation times of the server are independent and geometrically distributed, and their durations are integral multiples of slot duration. The customers are served one at a time under discrete-time Markovian service process. The new service process starts with the initial phase distribution independent of the path followed by the previous service process when the server returns from a vacation and finds at least one waiting customer. The matrix-geometric method is used to obtain the probability distribution of system-length at prearrival epoch. With the help of Markov renewal theory approach, we also derive the system-length distribution at an arbitrary epoch. The analysis of actual-waiting-time distribution in the queue measured in slots has also been carried out. In addition, computational experiences with a variety of numerical results are discussed to display the effect of the system parameters on the performance measures.     

Analysis of the BMAP/G/1 retrial system with search of customers from the orbit

We consider a single server retrial queuing model in which customers arrive according to a batch Markovian arrival process. Any arriving batch finding the server busy enters into an orbit. Otherwise one customer from the arriving batch enters into service immediately while the rest join the orbit. The customers from the orbit try to reach the service later and the inter-retrial times are exponentially distributed with intensity depending (generally speaking) on the number of customers on the orbit. Additionally, the search mechanism can be switched-on at the service completion epoch with a known probability (probably depending on the number of customers on the orbit). The duration of the search is random and also probably depending on the number of customers in the orbit. The customer, which is found as the result of the search, enters the service immediately if the server is still idle. Assuming that the service times of the primary and repeated customers are generally distributed (with possibly different distributions), we perform the steady state analysis of the queueing model.