具有两类失效模式的D-策略M/G/1可修排队系统分析

研究具有两类失效模式的D策略M/G/1可修排队系统,其中第一类失效是服务台在服务顾客期间发生的失效,第二类失效是服务台在空闲期间发生的失效,且两类失效模式的失效率不同.使用全概率分解技术和利用拉普拉斯变换与母函数等工具,从任意初始状态出发,讨论了系统队长的瞬时分布和稳态分布,获得了系统稳态队长分布的递推表达式与稳态队长的随机分解结果.进一步,在建立费用模型的基础上,通过数值计算实例讨论了使得系统在长期单位时间内达到最小值的最优控制策略D^*,并在同一组参数取值下与服务台不发生故障时的最优控制策略进行了比较.     

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.     

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.     

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.     

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.

     

Analysis of the M/Gi/1 →/M/1 queueing model

An M/GI/1 queueing system is in series with a unit with negative exponential service times and infinite waiting room capacity. We determine a closed form expression for the generating function of the joint queue length distribution in steady state. This result is obtained via the solution of a new type of functional equation in two variables.     

A queueing model with bonus service for certain customers

A queueing model is introduced in which the management has a policy, because of economic reasons, of not operating the service counter unless a certain number, R + 1, of customers are available during each busy period. Thus, the first R customers who arrive must wait until the service counter is opened. Such a policy may cause the management to provide or render additional services to the first R customers. Assuming Poisson arrivals and that both regular and additional services follow exponential distributions, explicit expressions are derived for the stationary queue length and busy period distributions and their expected values. In the special case where R = 1, an explicit expression is presented for the stationary distribution of the waiting time.     

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.     

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.     

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 periodic phenomena and stationary distribution of queueing system M/G/1 with group arrivals and batch service

In this paper several models of queueing system M/G/1 with group arrivals and batch service are considered, and the following fundamental questions are considered: (1) what is the structure of the phase space of the imbedded Markov chain? (2) what are the necessary and sufficient conditions causing the imbedded Markov chain to be reducible or irreducible, and periodic or aperiodic? (3) what are the necessary and sufficient conditions of the existence of stationary distribution? The generating function of stationary distribution is obtained.     

A queueing model for a nonreliable multiterminal system with polling scheduling

This paper deals with a nonhomogeneous finite-source queueing model to describe the performance of a multiterminal system subject to random breakdowns under the polling service discipline. The model studied here is a closed queueing network which has three service stations. a CPU (single server), terminals (infinite server), a repairman (single server), and a finite number of customers (jobs) that have distinct service rates at the service stations. The CPU's repair has preemptive priority over the terminal repairs, and failure of the CPU stops the service of the other stations, thus the nodes are not independent. It can be viewed as a continuation of papers by the authors (see references), which discussed a FIFO (first-in, first-out) and a PPS (priority processor sharing) serviced queueing model subject to random breakdowns. All random variables are assumed to be independent and exponentially distributed. The system behavior can be described by a Markov chain, but the number of states is very large. The purpose of this paper is to give a recursive computational approach to solve steady-state equations and to illustrate the problem in question using some numerical results. Supported by the Hungarian National Foundation for Scientific Research (grant Nos. OTKA T014974/95 and T016933/95) Proceedings of the Seminar on Stability Problems for Stochastic Models. Hajdúszoboszló, Hungary, 1997, Part, II.     

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.     

A hypercube queueing loss model with customer-dependent service rates

This paper is concerned with the solution of a specific hypercube queueing model. It extends the work that was described in a related paper by Atkinson et al. [Atkinson, J.B., Kovalenko, I.N., Kuznetsov, N., Mykhalevych, K.V., 2006. Heuristic methods for the analysis of a queuing system describing emergency medical services deployed along a highway. Cybernetics & Systems Analysis, 42, 379–391], which investigated a model for deploying emergency services along a highway. The model is based on the servicing of customer demands that arise in a number of distinct geographical zones, or atoms. Service is provided by servers that are positioned at a number of bases, each having a fixed geographical location along the highway. At each base a single server is available. Demands arising in any atom have a first-preference base and a second-preference base. If the first-preference base is busy, service is provided by the second-preference base; and, if both bases are busy, the demand is lost. In practice, because of differences in travel times from the first and second-preference bases to the atom in question, the service rate may be significantly different in the two cases. The model studied here allows for such customer-dependent service rates to occur, and the corresponding hypercube model has 3ⁿ states, where n is the number of bases. The computational intractability of this model means that exact solutions for the long-run proportion of lost demands (p_loss) can be obtained only for small values of n. In this paper, we propose two heuristic methods and a simulation approach for approximating p_loss. The heuristics are shown to produce very accurate estimates of p_loss.     

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.