Performance analysis of GI/M/1 queue with working vacations and vacation interruption

In this paper, we analyze a single-server vacation queue with a general arrival process. Two policies, working vacation and vacation interruption, are connected to model some practical problems. The GI/M/1 queue with such two policies is described and by the matrix analysis method, we obtain various performance measures such as mean queue length and waiting time. Finally, using some numerical examples, we present the parameter effect on the performance measures and establish the cost and profit functions to analyze the optimal service rate η during the vacation period.     

The GI/M/1 queue with Bernoulli-schedule-controlled vacation and vacation interruption

Consider a GI/M/1 queue with multiple vacations. As soon as the system becomes empty, the server either begins an ordinary vacation with probability q  (0?q?1)$(0?q?1)$ or takes a working vacation with probability 1-q$1-q$. We assume the vacation interruption is controlled by Bernoulli. If the system is non-empty at a service completion instant in a working vacation period, the server can come back to the normal busy period with probability p  (0?p?1)$(0?p?1)$ or continue the vacation with probability 1-p$1-p$. Using the matrix-analytic method, we obtain the steady-state distributions for the queue length both at arrival and arbitrary epochs. The waiting time and sojourn time are also derived by different methods. Finally, some numerical examples are presented.     

Performance analysis of M/G/1 queue with working vacations and vacation interruption

In this paper, an M/G/1 queue with a working vacations and vacation interruption is analyzed. Using the method of a supplementary variable and the matrix-analytic method, we obtain the queue length distribution and service status at an arbitrary epoch under steady state conditions. Further, we provide the Laplace-Stieltjes transform (LST) of the stationary waiting time. Finally, numerical examples are presented.     

The GI/M/1 queue with start-up period and single working vacation and Bernoulli vacation interruption

Consider a GI/M/1 queue with start-up period and single working vacation. When the system is in a closed state, an arriving customer leading to a start-up period, after the start-up period, the system becomes a normal service state. And during the working vacation period, if there are customers at a service completion instant, the vacation can be interrupted and the server will come back to the normal working level with probability p (0 ? p ? 1) or continue the vacation with probability 1 − p. Meanwhile, if there is no customer when a vacation ends, the system is closed. Using the matrix-analytic method, we obtain the steady-state distributions for the queue length at both arrival epochs and arbitrary epochs, the waiting time and sojourn time.     

Geo/Geo/1 retrial queue with working vacations and vacation interruption

Consider a Geo/Geo/1 retrial queue with working vacations and vacation interruption, and assume requests in the orbit try to get service from the server with a constant retrial rate. During the working vacation period, customers can be served at a lower rate. If there are customers in the system after a service completion instant, the vacation will be interrupted and the server comes back to the normal working level. We use a quasi birth and death process to describe the considered system and derive a condition for the stability of the model. Using the matrix-analytic method, we obtain the stationary probability distribution and some performance measures. Furthermore, we prove the conditional stochastic decomposition for the queue length in the orbit. Finally, some numerical examples are presented.     

Performance analysis of MAP/G/1 queue with working vacations and vacation interruption

We consider the MAP/G/1 queue with working vacations and vacation interruption. We obtain the queue length distribution with the method of supplementary variable, combined with the matrix-analytic method and censoring technique. We also obtain the system size distribution at pre-arrival epoch and the Laplace–Stieltjes transform (LST) of waiting time.     

The GI/M/1 queue with exponential vacations

In this paper, we give a detailed analysis of the GI/M/1 queue with exhaustive service and multiple exponential vacation. We express the transition matrix of the imbedded Markov chain as a block-Jacobi form and give a matrix-geometric solution. The probability distribution of the queue length at arrival epochs is derived and is shown to decompose into the distribution of the sum of two independent random variables. In addition, we discuss the limiting behavior of the continuous time queue length processes and obtain the probability distributions for the waiting time and the busy period.     

Analysis of a GI/M/1 queue with multiple working vacations

Consider a GI/M/1 queue with vacations such that the server works with different rates rather than completely stops during a vacation period. We derive the steady-state distributions for the number of customers in the system both at arrival and arbitrary epochs, and for the sojourn time for an arbitrary customer.     

An M/G/1 queue with single working vacation and vacation interruption under Bernoulli schedule

This paper treats an M/G/1 queue with single working vacation and vacation interruption under Bernoulli schedule. Whenever the system becomes empty at a service completion instant, the server goes for a single working vacation. In the working vacation, a customer is served at a lower speed, and if there are customers in the queue at the instant of a service completion, the server is resumed to a regular busy period with probability p   (i.e., the vacation is interrupted) or continues the vacation with probability 1-p$1-p$. Using the matrix analytic method, we obtain the distribution for the stationary queue length at departure epochs. The joint distribution for the stationary queue length and service status at the arbitrary epoch is also obtained by using supplementary variable technique. We also develop a variety of stationary performance measures for this system and give a conditional stochastic decomposition result. Finally, several numerical examples are presented.     

Performance analysis of a GI/M/1 queue with single working vacation

Consider a GI/M/1 queue with single working vacation. During the vacation period, the server works at a lower rate rather than stopping completely, and only takes one vacation each time. Using the matrix analytic approach, the steady-state distributions of the number of customers in the system at both arrival and arbitrary epochs are obtained. Then the closed property of the conditional probability of gamma distribution is proved and using it the waiting time of an arbitrary customer is analyzed. Finally, Some numerical results and effect of critical model parameters on performance measures have been presented.     

Discrete-time GI/Geo/1 queue with multiple working vacations

Consider the discrete time GI/Geo/1 queue with working vacations under EAS and LAS schemes. The server takes the original work at the lower rate rather than completely stopping during the vacation period. Using the matrix-geometric solution method, we obtain the steady-state distribution of the number of customers in the system and present the stochastic decomposition property of the queue length. Furthermore, we find and verify the closed property of conditional probability for negative binomial distributions. Using such property, we obtain the specific expression for the steady-state distribution of the waiting time and explain its two conditional stochastic decomposition structures. Finally, two special models are presented.      

M/G/1 queue with single working vacation

In this paper, an M/G/1 queue with single working vacation is analyzed. Using the method of supplementary variable and the matrix-analytic method, we obtain the queue length distribution and service status at the arbitrary epoch under steady state conditions. Further, we derive expected busy period and expected busy cycle. Finally, server special cases are presented.     

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.     

Stochastic decompositions in the M/M/1 queue with working vacations

We demonstrate stochastic decomposition structures of the queue length and waiting time in an M/M/1/WV queue, and obtain the distributions of the additional queue length and additional delay. Furthermore, we discuss the relationship between the stochastic decomposition properties of the working vacation queue and those of the standard M/G/1 queue with general vacations.     

Busy period analysis for the GI/M/1 queue with exponential vacations

We consider the busy period in the GI/M/1 queue with multiple exponential vacations. We derive the transform of the joint distribution of the length of a busy period, the number of customers served during the busy period, and the residual interarrival time at the instant the busy period ends.     

TheM/GI/1 Bernoulli feedback queue with vacations

Feedback may be introduced as a mechanism for scheduling customer service (for example in systems in which customers bring work that is divided into a random number of stages). A model is developed that characterizes the queue length distribution as seen following vacations and service stage completions. We demonstrate the relationship that exists between these distributions. The ergodic waiting time distribution is formulated in such a way as to reveal the effects of server vacations when feedback is introduced.This work was supported in part by NSF Grant No. DDM-8913658.     

On stochastic decomposition in the GI/M/1 queue with single exponential vacation

We consider a GI/M/1 queueing system in which the server takes exactly one exponential vacation each time the system empties. We derive the PGF of the stationary queue length and the LST of the stationary FIFO sojourn time. We show that both the queue length and the sojourn time can be stochastically decomposed into meaningful quantities.     

分析M/M/1多重工作休假排队的一种新方法

用一种新方法对经典的M/M/1工作休假排队系统建立模型.对该模型,用无限位相GI/M/1型Markov过程和矩阵解析方法进行分析,不但得到了所讨论排队模型平稳队长分布的具体结果,还给出了平稳状态时服务台具体位于第几次工作休假的概率.这些关于服务台状态更为精确的描述是该排队系统的新结果.     

The discrete-time MAP/PH/1 queue with multiple working vacations

We consider a discrete-time single-server queueing model where arrivals are governed by a discrete Markovian arrival process (DMAP), which captures both burstiness and correlation in the interarrival times, and the service times and the vacation duration times are assumed to have a general phase-type distributions. The vacation policy is that of a working vacation policy where the server serves the customers at a lower rate during the vacation period as compared to the rate during the normal busy period. Various performance measures of this queueing system like the stationary queue length distribution, waiting time distribution and the distribution of regular busy period are derived. Through numerical experiments, certain insights are presented based on a comparison of the considered model with an equivalent model with independent arrivals, and the effect of the parameters on the performance measures of this model are analyzed.