Statistical inference for G/M/1 queueing system

In this paper, maximum likelihood estimates of the parameters are derived for the G/M/1 queueing model with variable arrival rate. A simulated numerical example is used to illustrate its application for estimating the parameter when the interarrival time distribution is exponential. Problems of hypothesis testing are also investigated.     

On the multi-phase M/G/1 queueing system with random feedback

In this paper we consider an M/G/1 queue with k phases of heterogeneous services and random feedback, where the arrival is Poisson and service times has general distribution. After the completion of the i-th phase, with probability θ _i the (i + 1)-th phase starts, with probability p _i the customer feedback to the tail of the queue and with probability 1 − θ _i − p _i  = q _i departs the system if service be successful, for i = 1, 2 , . . . , k. Finally in kth phase with probability p _k feedback to the tail of the queue and with probability 1 − p _k departs the system. We derive the steady-state equations, and PGF’s of the system is obtained. By using them the mean queue size at departure epoch is obtained.     

On the M/G/1 retrial queueing system with linear control policy

Summary This paper is concerned with the study of a newM/G/1 retrial queueing system in which the delays between retrials are exponentially distributed random variables with linear intensityg(n)=α+nμ, when there aren≥1 customers in the retrial group. This new retrial discipline will be calledlinear control policy. We carry out an extensive analysis of the model, including existence of stationary regime, stationary distribution of the embedded Markov chain at epochs of service completions, joint distribution of the orbit size and the server state in steady state and busy period. The results agree with known results for special cases.     

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.     

Vacation policies in an M/G/1 type queueing system with finite capacity

This paper deals with a queueing system with finite capacity in which the server passes from the active state to the inactive state each time a service terminates withv customers left in the system. During the active (inactive) phases, the arrival process is Poisson with parameter (₀). Denoting byu _n the duration of thenth inactive phase and byx _n the number of customers present at the end of thenth inactive phase, we assume that the bivariate random vectors {(v _n ,x _n ),n 1} are i.i.d. withx _n v+l a.s. The stationary queue length distributions immediately after a departure and at an arbitrary instant are related to the corresponding distributions in the classical model.     

An M/G/1 queueing system with multiple priority classes

We consider anM/G/1 queueing system with multiple priority classes of jobs. Considered preemptive rules are the preemptiveresume, preemptive-repeat-identical, and preemptive-repeat-different policies. These three preemptive rules will be analyzed in parallel. The key idea of analysis is based on the consideration of a busy period as a composite of delay cycle. As results, we present the exact Laplace-Stieltjes (L.S.) transforms of residence time and completion time in the system.     

Continuity theorems for the M/M/1/n queueing system

In this paper continuity theorems are established for the number of losses during a busy period of the M/M/1/n queue. We consider an M/GI/1/n queueing system where the service time probability distribution, slightly different in a certain sense from the exponential distribution, is approximated by that exponential distribution. Continuity theorems are obtained in the form of one or two-sided stochastic inequalities. The paper shows how the bounds of these inequalities are changed if further assumptions, associated with specific properties of the service time distribution (precisely described in the paper), are made. Specifically, some parametric families of service time distributions are discussed, and the paper establishes uniform estimates (given for all possible values of the parameter) and local estimates (where the parameter is fixed and takes only the given value). The analysis of the paper is based on the level crossing approach and some characterization properties of the exponential distribution. Dedicated to Vladimir Mikhailovich Zolotarev, Victor Makarovich Kruglov, and to the memory of Vladimir Vyacheslavovich Kalashnikov.     

Software package for the queueing system Mx/G/1

A user-friendly software package, which should be found useful be researchers, practitioners and students alike, for the bulk-arrival single-server queueing system M^x/G/1 is discussed. It finds numerically the steady-state probabilities and moments for the number in the system at each of the three time epochs (pre-arrival, post-departure and random), as well as moments for waiting time in queue and busy and idle periods.     

Matched queueing system MoPH/G/1

In this paper, we study the matched queueing system, MoPH/G/1, where the type-I input is a Poisson process, the type-II input is a PH renewal process, and the service times are i.i.d. random variables. A necessary and sufficient condition for the stationariness of the system is given. The expectations of the length of the non-idle period and the number of customers served in a non-idle period are obtained.This project is supported by the National Natural Science Foundation of China and partially by the Institute of Mathematics, Academia Sinica.     

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.     

A recursive method for the N policy G/M/1 queueing system with finite capacity

This paper studies a single removable server in a G/M/1 queueing system with finite capacity operating under the N policy. 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 distributions of the number of customers in the system. The method is illustrated analytically for exponential interarrival time distribution. Numerical results for various system performance measures are presented for four different interarrival time distributions such as exponential, 2-stage hyperexponential, 4-stage Erlang, and deterministic.     

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.     

Approximation of departure process from a G/M/1/0 queueing system

The departure (output) process from a (G/M/1/0) queueing system with a stationary counting arrival process, negative exponentially distributed service times, a single server, and no waiting room is approximated. The approximation is based on a two parameter method. Numerical results are presented and concluding remarks are discussed.     

An optimal age maintenance for an M/G/1 queueing system

This paper discusses an optimal age maintenance scheme for a queueing system. Customers arrive at the system according to a Poisson process. They form a single queue and are served by a server with general service distribution. The system fails after a random time and corrective maintenance is performed at the failure. A preventive maintenance is also performed if the system is empty at age T where ‘age’ refers to the elapsed time since the previous maintenance was completed. If the system is not empty at age T, the system is used until it fails. At the failure, the customers in the system are lost and the arriving customers during the maintenance are also lost. By renewal theory, we study the optimal value of T which minimizes the average number of lost customers over an infinite time horizon.     

An M/G/1 queueing model with vacation times

This paper deals with an M/G/1 queueing system with finite capacity for the workload, where the workload at timet is defined as the total amount of work in the system at timet. When the server provides service he will continue servicing until the system becomes empty, after which he leaves the system for a stochastic period of time, which will be called a vacation. When the server, returning from a vacation, finds the system still empty, he leaves for another vacation, otherwise he immediately starts servicing again.Using an embedding approach several characteristics of this system are derived amongst which the joint stationary distribution of the workload and the stage of the server.

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Zusammenfassung Diese Arbeit befaßt sich mit einem M/G/1 Wartesystem, das hinsichtlich der anstehenden Arbeit eine endliche Kapazität hat. Wenn der Bediener tätig ist, bleibt er es solange, bis das System leer ist. Danach ist er während einer stochastischen Pausenzeit nicht verfügbar. Ist am Ende einer Pausenzeit das System immer noch leer, so schließt sich eine weitere Pausenzeit an; ansonsten wird unverzüglich die Bedienung am Ende der Pausenzeit wieder aufgenommen.Unter Verwendung eines eingebetteten Prozesses werden mehrere Kenngrößen des Systems ermittelt, darunter z.B. die gemeinsame Verteilung von anstehender Arbeit und Zustand des Bedieners.