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.     

Analysis of Queueing Systems with Synchronous Single Vacation for Some Servers

We study a multi-server M/M/c type queue with a single vacation policy for some idle servers. In this queueing system, if at a service completion instant, any d (d c) servers become idle, these d servers will take one and only one vacation together. During the vacation of d servers, the other c–d servers do not take vacation even if they are idle. Using a quasi-birth-and-death process and the matrix analytic method, we obtain the stationary distribution of the system. Conditional stochastic decomposition properties have been established for the waiting time and the queue length given that all servers are busy.     

M x /G/1 Queue with Multiple Vacations

Abstract

In this article, we study a queueing system M ^x /G/1 with multiple vacations. The probability generating function (P.G.F.) of stationary queue length and its expectation expression are deduced by using an embedded Markov chain of the queueing process. The P.G.F. of stationary system busy period and the probability of system in service state and vacation state also are obtained by the same method. At last we deduce the LST and mean of stationary waiting time in the service order FCFS and LCFS, respectively.     

Relative priority policies for minimizing the cost of queueing systems with service discrimination

This paper considers several single-server two-class queueing systems with different cost functions. Customers in the two classes are discriminated by service rates and relative priorities. Most attention is focused on the ones with general quadratic bivariable and exponential cost functions that are usually applied in the relatively complicated systems. To the best of the authors’ knowledge, there is no literature analyzing these two kinds of cost functions on the subject of relative priority. We explicitly present the conditions under which relative priority outperforms absolute priority for reducing system cost and further provide the method to find the optimal DPS policy. Moreover, we also discuss variations where service rates of the two classes are decision variables under service equalization and service discrimination disciplines, respectively.     

Equilibrium threshold strategies in observable queueing systems with setup/closedown times

This paper considers two types of setup/closedown policies: interruptible and insusceptible setup/closedown policies. When all customers are served exhaustively in a system under the interruptible setup/closedown policy, the server shuts down (deactivates) by a closedown time. When the server reactivates since shutdown, he needs a setup time before providing service again. If a customer arrives during a closedown time, the service is immediately started without a setup time. However, in a system under the insusceptible setup/closedown policy, customers arriving in a closedown time can not be served until the following setup time finishes. For the systems with interruptible setup/closedown times, we assume both the fully and almost observable cases, then derive equilibrium threshold strategies for the customers and analyze the stationary behavior of the systems. On the other hand, for the systems with insusceptible setup/closedown times, we only consider the fully observable case. We also illustrate the equilibrium thresholds and the social benefits for systems via numerical experiments. As far as we know, there is no work concerning equilibrium behavior of customers in queueing systems with setup/closedown times.     

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.     

供应链中的MQ(t)/Gj/∞排队群

本文研究由一个非时齐Poisson到达过程驱动的多个MQ(t)／Gj／∞型排队系统．当一个K型到达发生时(K(?){1,…,m}),对每一个j∈K,顾客按批量QjK进入排队系统j,(Q1K,…,QmK)是相依的随机向量．我们导出了联合队长过程和联合输出过程的多变量母函数,并给出各排队系统之间的相关性分析．这些结果为非时齐供应链分析提供了理论依据     

具有一般休假的D-BMAP/G/1分解定理

为了直接求出队长向量母函数,省去繁琐的矩阵分析或补充变量过程,给出并证明了具有一般休假的D—BMAP/G/1的分解定理,阐明了任意时刻队长的向量母函数与闲期任意时刻队长向量母函数的关系,离去时刻队长的向量母函数与闲期任意时刻队长向量母函数的关系,闲期任意时刻队长向量母函数与Y_(busy)(θ,z),即忙期任意时刻队长和剩余服务时间的联合向量母函数的关系．     

多级适应性休假的M/G/1排队

在经典M/G/1排队中引入多级适应性休假规则,得到稳态队长、等待时间分布和随机分解,并给出忙期、假期、在线期分布.单重休假和多重休假模型是本文中模型的两个极端情况.     

M/G/1 queue with controllable vacations and optimization of vacation policy

We introduce the control parameterN in a common queue M/G/1 with vacations; the end of a global vacation period is controlled by the parameterN. This extension for a queue with vacations is of significance in certain practical cases. In this paper, we find various transient and steady-state results for the queue size, the delay times and the waiting times for the M/G/1 queue with controllable vacations. Finally, we also discuss optimal selection of the control parameter.