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.     

Optimal Control of an M/G/1/K Queueing System with Combined <Emphasis Type="Italic">F</Emphasis> Policy and Startup Time

We investigate the optimal management problem of an M/G/1/K queueing system with combined F policy and an exponential startup time. The F policy queueing problem investigates the most common issue of controlling the arrival to a queueing system. 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 the system. The method is illustrated analytically for exponential service time distribution. A cost model is established to determine the optimal management F policy at minimum cost. We use an efficient Maple computer program to calculate the optimal value of F and some system performance measures. Sensitivity analysis is also investigated.     

Optimization of the T policy M/G/1 queue with server breakdowns and general startup times

This paper investigates the T$T$ policy M/G/1 queue with server breakdowns, and startup times. Customers arrive at the system according to a Poisson process. Service times, repair times, and startup times are assumed to be generally distributed. The server is turned on after a fixed length of time T$T$ repeatedly until at least one customer is present in the waiting line. The server needs a startup time before starting the service. We analyze various system performance measures and develop the total expected cost function per unit time in which T$T$ is a decision variable. We determine the optimum threshold T^∗$T^{\ast }$ and derive analytical results for sensitivity investigations. The sensitivity analysis is particularly valuable to the system analyst when evaluating future conditions. We also present extensive numerical computation for illustration purpose.     

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.     

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.     

Performance analysis approximation in a queueing system of type M/G/1

In this work, we apply the strong stability method to obtain an estimate for the proximity of the performance measures in the M/G/1 queueing system to the same performance measures in the M/M/1 system under the assumption that the distributions of the service time are close and the arrival flows coincide. In addition to the proof of the stability fact for the perturbed M/M/1 queueing system, we obtain the inequalities of the stability. These results give with precision the error, on the queue size stationary distribution, due to the approximation. For this, we elaborate from the obtained theoretical results, the STR-STAB algorithm which we execute for a determined queueing system: M/Coxian − 2/1. The accuracy of the approach is evaluated by comparison with simulation results.     

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.     

Optimal control of a removable and non-reliable server in an M/M/1 queueing system with exponential startup time

We study a single removable and non-reliable server in the N policy M/M/1 queueing system. The server begins service only when the number of customers in the system reaches N (N1). After each idle period, the startup times of the server follow the negative exponential distribution. While the server is working, it is subject to breakdowns according to a Poisson process. When the server breaks down, it requires repair at a repair facility, where the repair times follow the negative exponential distribution. The steady-state results are derived and it is shown that the probability that the server is busy is equal to the traffic intensity. Cost model is developed to determine the optimal operating N policy at minimum cost.     

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.     

Asymptotic expansions for waiting time probabilities in an M/G/1 queue with long-tailed service time

We consider anM/G/1 queue with FCFS queue discipline. We present asymptotic expansions for tail probabilities of the stationary waiting time when the service time distribution is longtailed and we discuss an extension of our methods to theM ^[x]/G/1 queue with batch arrivals.     

On the convexity of the two-threshold policy for an M/G/1 queue with vacations

In this note, we consider a single server queueing system with server vacations of two types and a two-threshold policy. Under a cost and revenue structure the long-run average cost function is proven to be convex in the lower threshold for a fixed difference between the two thresholds.     

M/G/1/K queues withN-policy and setup times

A steady-state analysis is given for M/G/1/K queues with combinedN-policy and setup times before service periods. The queue length distributions and the mean waiting times are obtained for the exhaustive service system, the gated service system, the E-limited service system, and the G-limited service system. Numerical examples are also provided.     

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.     

Controlling arrivals for a queueing system with an unreliable server: Newton-Quasi method

This paper deals with the control policy of a removable and unreliable server for an M/M/1/K queueing system, where the removable server operates an F-policy. The so-called F-policy means that when the number of customers in the system reaches its capacity K (i.e. the system becomes full), the system will not accept any incoming customers until the queue length decreases to a certain threshold value F. At that time, the server initiates an exponential startup time with parameter γ and starts allowing customers entering the system. It is assumed that the server breaks down according to a Poisson process and the repair time has an exponential distribution. A matrix analytical method is applied to derive the steady-state probabilities through which various system performance measures can be obtained. A cost model is constructed to determine the optimal values, say (F^∗, μ^∗, γ^∗), that yield the minimum cost. Finally, we use the two methods, namely, the direct search method and the Newton-Quasi method to find the global minimum (F^∗, μ^∗, γ^∗). Numerical results are also provided under optimal operating conditions.     

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．     

Asymptotic behavior of the loss probability for an M/G/1/N queue with vacations

In this paper, asymptotic properties of the loss probability are considered for an M/G/1/N queue with server vacations and exhaustive service discipline, denoted by an M/G/1/N-(V, E)-queue. Exact asymptotic rates of the loss probability are obtained for the cases in which the traffic intensity is smaller than, equal to and greater than one, respectively. When the vacation time is zero, the model considered degenerates to the standard M/G/1/N queue. For this standard queueing model, our analysis provides new or extended asymptotic results for the loss probability. In terms of the duality relationship between the M/G/1/N and GI/M/1/N queues, we also provide asymptotic properties for the standard GI/M/1/N model.     

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.     

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.