Comparative analysis for the N policy M/G/1 queueing system with a removable and unreliable server

In this paper we analyze a single removable and unreliable server in the N policy M/G/1 queueing system in which the server breaks down according to a Poisson process and the repair time obeys an arbitrary distribution. The method of maximum entropy is used to develop the approximate steady-state probability distributions of the queue length in the M/G(G)/1 queueing system, where the second and the third symbols denote service time and repair time distributions, respectively. A study of the derived approximate results, compared to the exact results for the M/M(M)/1, M/E₂(E₃)/1, M/H₂(H₃)/1 and M/D(D)/1 queueing systems, suggest that the maximum entropy principle provides a useful method for solving complex queueing systems. Based on the simulation results, we demonstrate that the N policy M/G(G)/1 queueing model is sufficiently robust to the variations of service time and repair time distributions.     

Maximum entropy analysis to the N policy M/G/1 queueing system with a removable server

We study a single removable server in an M/G/1 queueing system operating under the N policy in steady-state. The server may be turned on at arrival epochs or off at departure epochs. Using the maximum entropy principle with several well-known constraints, we develop the approximate formulae for the probability distributions of the number of customers and the expected waiting time in the queue. We perform a comparative analysis between the approximate results with exact analytic results for three different service time distributions, exponential, 2-stage Erlang, and 2-stage hyper-exponential. The maximum entropy approximation approach is accurate enough for practical purposes. We demonstrate, through the maximum entropy principle results, that the N policy M/G/1 queueing system is sufficiently robust to the variations of service time distribution functions.     

Age-based Markovian approximation of the G/M/1 queue

We extend the approach of Koole et al. (2012) [15] and Legros et al. (2018) [20] for the G/M/1 queue. The idea is to provide a Markovian approximation where a state represents the oldest customer's wait. This modeling is made possible by creating states with negative wait, representing an estimate of the time at which a new customer would arrive when the system is empty. We apply this method for performance evaluation and routing optimization. Finally, we further extend the model to the G/M/1+G queue.     

Tail asymptotics for M/G/1 type queueing processes with subexponential increments

Bivariate regenerative Markov modulated queueing processes {I _n ,L _n } are described. {I _n } is the phase process, and {L _n } is the level process. Increments in the level process have subexponential distributions. A general boundary behavior at the level 0 is allowed. The asymptotic tail of the cycle maximum, , during a regenerative cycle, , and the asymptotic tail of the stationary random variable L _∞, respectively, of the level process are given and shown to be subexponential with L _∞ having the heavier tail. This revised version was published online in June 2006 with corrections to the Cover Date.     

Interval estimation of mean response time for a G/M/1 queueing system: empirical Laplace function approach

Mean response time is an important performance measure for a queueing system. In this paper, we propose a consistent and asymptotically normal (CAN) estimator of the mean response time for a G/M/1 queueing system, which is based on the fixed point of empirical Laplace function. The confidence interval for the mean response time can be constructed by applying the proposed CAN estimator and its estimated variance. And we carried out a simulation study to perform the accuracy of the constructed confidence interval by calculating the coverage percentage and the relative average length of confidence interval. Detailed discussions of all simulation results for three various models of G/M/1‐type system are presented and some valuable conclusions are provided. Copyright © 2006 John Wiley & Sons, Ltd.     

A recursive method to the optimal control of an M/G/1 queueing system with finite capacity and infinite capacity

We study a single removable server in an infinite and a finite queueing systems with Poisson arrivals and general distribution service times. The server may be turned on at arrival epochs or off at service completion epochs. We present a recursive method, using the supplementary variable technique and treating the supplementary variable as the remaining service time, to obtain the steady state probability distribution of the number of customers in a finite system. The method is illustrated analytically for three different service time distributions: exponential, 3-stage Erlang, and deterministic. Cost models for infinite and finite queueing systems are respectively developed to determine the optimal operating policy at minimum cost.     

Optimal control of the N policy M/G/1 queueing system with server breakdowns and general startup times

This paper deals with an N policy M/G/1 queueing system with a single removable and unreliable server whose arrivals form a Poisson process. Service times, repair times, and startup times are assumed to be generally distributed. When the queue length reaches N(N ? 1), the server is immediately turned on but is temporarily unavailable to serve the waiting customers. The server needs a startup time before providing service until there are no customers in the system. We analyze various system performance measures and investigate some designated known expected cost function per unit time to determine the optimal threshold N at a minimum cost. Sensitivity analysis is also studied.     

An Arrival Time Approach to M/G/1-type Queues with Generalized Vacations

We propose a simple way, called the arrival time approach, of finding the queue length distributions for M/G/1-type queues with generalized server vacations. The proposed approach serves as a useful alternative to understanding complicated queueing processes such as priority queues with server vacations and MAP/G/1 queues with server vacations.     

Wiener-Hopf analysis of an M/G/1 queue with negative customers and of a related class of random walks

Two variants of an M/G/1 queue with negative customers lead to the study of a random walkX _n+1=[X _n + _n ]⁺ where the integer-valued _n are not bounded from below or from above, and are distributed differently in the interior of the state-space and on the boundary. Their generating functions are assumed to be rational. We give a simple closed-form formula for , corresponding to a representation of the data which is suitable for the queueing model. Alternative representations and derivations are discussed. With this formula, we calculate the queue length generating function of an M/G/1 queue with negative customers, in which the negative customers can remove ordinary customers only at the end of a service. If the service is exponential, the arbitrarytime queue length distribution is a mixture of two geometrical distributions.Supported by the European grant BRA-QMIPS of CEC DG XIII.     

M/G/1重试排队系统的渐近稳定性

用算子半群理论研究了带有重试排队的M/G/1系统.通过解算子方程和预解方程,证明了0是系统算子的本征值,且为虚轴上唯一的谱点.从而得出了当时间趋于无穷时系统时间依赖解收敛于稳态解的结论.     

Higher moments of the waiting time distribution in M/G/1 retrial queues

This work analyzes the waiting time distribution in the M/G/1 retrial queue. The first two moments of the waiting time distribution are known from the literature. In this work we obtain all the moments of the waiting time distribution.     

A recursive method for the F-policy G/M/1/K queueing system with an exponential startup time

This paper deals with the optimal control of a finite capacity G/M/1 queueing system combined the F-policy and an exponential startup time before start allowing customers in the system. The F-policy queueing problem investigates the most common issue of controlling arrival to a queueing system. We provide a recursive method, using the supplementary variable technique and treating the supplementary variable as the remaining interarrival time, to develop the steady-state probability distribution of the number of customers in the system. We illustrate a recursive method by presenting three simple examples for exponential, 3-stage Erlang, and deterministic interarrival time distributions, respectively. A cost model is developed to determine the optimal management F-policy at minimum cost. We use an efficient Maple computer program to determine the optimal operating F-policy and some system performance measures. Sensitivity analysis is also studied.     

Markov-modulatedPH/G/1 queueing systems

We study a PH/G/1 queue in which the arrival process and the service times depend on the state of an underlying Markov chain J(t) on a countable state spaceE. We derive the busy period process, waiting time and idle time of this queueing system. We also study the Markov modulated E_K/G/1 queueing system as a special case.     

具有可选服务、反馈、一般重试时间的M/G/1排队系统

本文考虑了一个具有可选服务、反馈的M／G／1重试排队系统。在假定重试区域中只有队首的顾客允许重试的情况下，重试时间具有一般分布时，得到了系统稳态的充分必要条件。求得稳态时系统队长和重试区域中队长分布及相关指标。     

The interdeparture-time distribution for each class in the ∑ i M i /G i /1 queue

A stationary queueing system is described in which a single server handles several competing Poisson arrival streams on a first-come first-served basis. Each class has its own generally distributed service time characteristics. The principal result is the Laplace-Stieltjes transform, for each class, of the interdeparture time distribution function. Examples are given and applications are discussed.     

一类具有到达损失、可选服务、反馈的M/G/1重试排队系统

考虑一个具有到达损失、可选服务、反馈的M/G/1重试排队系统.在假定重试区域中顾客具有相互独立的指数重试时间的情况下,得到了系统的转移概率矩阵和系统稳态的充分必要条件.列出微分方程,求得稳态时系统队长和重试区域中队长分布及相关指标.     

A Markov-modulated M/G/1 queue I: Stationary distribution

We consider an M/G/1 queueing system in which the arrival rate and service time density are functions of a two-state stochastic process. We describe the system by the total unfinished work present and allow the arrival and service rate processes to depend on the current value of the unfinished work. We employ singular perturbation methods to compute asymptotic approximations to the stationary distribution of unfinished work and in particular, compute the stationary probability of an empty queue.This research was supported in part by NSF Grants DMS-84-06110, DMS-85-01535 and DMS-86-20267, and grants from the U.S. Israel Binational Science Foundation and the Israel Academy of Sciences.     

具有随机N—策略的M／G／1排队系统

本文讨论具有随机N-策略的M/G/1排队系统,采用向量Markov过程方法得到该系统有关的排队指标。上述结果可以看作是普通的和N-策略的M/G/1排队系统的推广。     

具有可利用服务员的M/G/1排队模型

本文考虑了具有可利用服务员的M／G／1有有限容量的排队模型．当工作量超过k(k是常数或者随机变量),可利用服务员参与工作,一直到工作量少于或等于k．可利用服务员的速率依赖于目前工作量．应用Level-crossing方法,获得了工作量的平稳分布．应用Kolmogorov向后微分方程方法,构造更新方程以获得忙期的Laplace变换．     

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.