Shock models under a Markovian arrival process

A device submitted to shocks arriving randomly and causing damage is considered. Every shock can be fatal or not. The shocks follow a Markovian arrival process. When the shock is fatal, the device is instantaneously replaced. The Markov process governing the shocks is constructed, and the stationary probability vector calculated. The probability of the number of replacements during a time is determined. A particular case in which the fatal shock occurs after a fixed number of shocks is introduced, and a numerical application is performed. The expressions are in algorithmic form due to the use of matrix-analytic methods. Computational aspects are introduced. This model extends others previously considered in the literature.     

Reliability of a system under two types of failures using a Markovian arrival process

We consider a system subject to external and internal failures. The operational time has a phase-type distribution (PH-distribution). Failures arrive following a Markovian arrival process (MAP). Some failures require the replacement of the system, and others a minimal repair. This model extends previous papers with arrivals governed by PH-renewal processes.     

Approximations for Markovian multi-class queues with preemptive priorities

We discuss the approximation of performance measures in multi-class M/M/k queues with preemptive priorities for large problem instances (many classes and servers) using class aggregation and server reduction. We compared our approximations to exact and simulation results and found that our approach yields small-to-moderate approximation errors.     

The infinite-buffer single server queue with a variant of multiple vacation policy and batch Markovian arrival process

We consider an infinite-buffer single server queue where arrivals occur according to a batch Markovian arrival process (BMAP). The server serves until system emptied and after that server takes a vacation. The server will take a maximum number H of vacations until either he finds at least one customer in the queue or the server has exhaustively taken all the vacations. We obtain queue length distributions at various epochs such as, service completion/vacation termination, pre-arrival, arbitrary, departure, etc. Some important performance measures, like mean queue lengths and mean waiting times, etc. have been obtained. Several other vacation queueing models like, single and multiple vacation model, queues with exceptional first vacation time, etc. can be considered as special cases of our model.     

Finite buffer vacation models under E-limited with limit variation service and Markovian arrival process

We consider a finite-buffer single server queue with single (multiple) vacation(s) and Markovian arrival process. The service discipline is E-limited with limit variation (ELV). Several other service disciplines like, Bernoulli scheduling, nonexhaustive and E-limited service can be treated as special cases of the ELV service.     

A reliability system under different types of shock governed by a Markovian arrival process and maintenance policy K

A reliability system subject to shocks producing damage and failure is considered. The source of shocks producing failures is governed by a Markovian arrival process. All the shocks produce deterioration and some of them failures, which can be repairable or non-repairable. Repair times are governed by a phase-type distribution. The number of deteriorating shocks that the system can stand is fixed. After a fatal failure the system is replaced by another identical one. For this model the availability, the reliability, and the rate of occurrence of the different types of failures are calculated. It is shown that this model extends other previously published in the literature.     

Discrete NT-policy single server queue with Markovian arrival process and phase type service

We consider a discrete time single server queueing system in which arrivals are governed by the Markovian arrival process. During a service period, all customers are served exhaustively. The server goes on vacation as soon as he/she completes service and the system is empty. Termination of the vacation period is controlled by two threshold parameters N and T, i.e. the server terminates his/her vacation as soon as the number waiting reaches N or the waiting time of the leading customer reaches T units. The steady state probability vector is shown to be of matrix-geometric type. The average queue length and the probability that the server is on vacation (or idle) are obtained. We also derive the steady state distribution of the waiting time at arrivals and show that the vacation period distribution is of phase type.     

Stochastic models of space priority mechanisms with Markovian arrival processes

In this paper, we study a single-server queue with finite capacity in which several space priority mechanisms are implemented. The arrival process is the general Markovian arrival process (MAP) which has been used to model the bursty arrival processes commonly arising in communication applications. The service times are generally distributed. These buffer mechanisms enable the Asynchronous Transfer Mode (ATM) layer to adapt the quality of the cell transfer to the quality of service requirements of the specific broadband ISDN services and to improve the utilization of the network resources. This is done by a selective discarding of cells according to the class they belong to. Computable expressions for various performance parameters are obtained. Numerical results are given for the case of a two-state Markov-modulated Poisson process (MMPP) and deterministic service times. The values derived can be used to evaluate the benefits of using priorities in an ATM network when the traffic is bursty and to make a comparative study of the buffer mechanisms. These results extend the models previously developed, which were limited to Poisson arrivals.     

Analysis of a synchronization station for the performance evaluation of a kanban system with a general arrival process of demands

The kanban system has attracted wide interest in recent years, and a lot of work has been devoted to the modeling of such systems, as well as to methods to evaluate their performance. Especially, in a previous work, we proposed an analytical method to evaluate the performance of a kanban system producing a single part type; we assumed that demands for the finished product arrive in single unit, according to a Poisson process. The present paper proposes an extension of this method to the case where the demands arrive according to a general process. We are more particularly interested in the analysis in isolation of the synchronization station between the finished parts of the system and the external demands. This leads to the resolution of a quasi-birth-and-death process with an infinite number of states and with a very regular structure. We thus propose a matrix-geometric solution of this problem.     

A Markovian single server with upstream job and downstream demand arrival stream

In this paper we consider a Markovian single server system which processes items arriving from an upstream region (as usual in queueing systems) and is controlled by a demand arrival stream for finished items from a downstream area. A finite storage is available at the server to store finished items not immediately needed in the downstream area. The system considered corresponds to an assembly-like queue with two input streams. The system is stable in a strict sense only if all queues are finite, i.e., both random processes are synchronized via blocking. This notion leads to a complementary system with a very similar state space which is a pair of Markovian single servers with synchronous arrivals. In the mathematical analysis the main focus is on the state probabilities and expectation of minimum and maximum of the two input queues. This revised version was published online in June 2006 with corrections to the Cover Date.     

Moment characterization of matrix exponential and Markovian arrival processes

This paper provides a general framework for establishing the relation between various moments of matrix exponential and Markovian processes. Based on this framework we present an algorithm to compute any finite dimensional moments of these processes based on a set of required (low order) moments. This algorithm does not require the computation of any representation of the given process. We present a series of related results and numerical examples to demonstrate the potential use of the obtained moment relations. This work is partially supported by the Italian-Hungarian bilateral R&D programme, by OTKA grant n. T-34972, by MIUR through PRIN project Famous and by EEC project Crutial.     

Analysis of two commodity Markovian inventory system with lead time

A two commodity continuous review inventory system with independent Poisson processes for the demands is considered in this paper. The maximum inventory level for the i-th commodity is fixed asS ⁱ (i = 1,2). The net inventory level at timet for the i-th commodity is denoted byI ⁱ(t),i = 1,2. If the total net inventory levelI(t) =I ¹(t) +I ²(t) drops to a prefixed level s[ leqslant tfrac(S₁ - 2)2ortfrac(S₂ - 2)2]s[ leqslant tfrac{{(S_1 - 2)}}{2}ortfrac{{(S_2 - 2)}}{2}] , an order will be placed for (S ⁱ −s) units of i-th commodity(i=1,2). The probability distribution for inventory level and mean reorders and shortage rates in the steady state are computed. Numerical illustrations of the results are also provided.     

Asymptotic behavior of the first arrival time in Markovian random walks

The article examines the asymptotic behavior of the first arrival time for homogeneous Markov random walks with a fixed stopping boundary. Conditions of convergence to exponential and gamma distributions are established. The results are applied to solve one problem of statistical control.Translated from Statisticheskie Metody, pp. 183–189, 1982.     

Transition phenomena for a Galton-Watson process with immigration in a Markovian environment

The behavior of a Galton-Watson process with state-dependent immigration in a Markovian random environment is investigated. We obtain a limit theorem in the near-critical case. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part II.     

A maintenance model with failures and inspection following Markovian arrival processes and two repair modes

We consider a deteriorating system submitted to external and internal failures, whose deterioration level is known by means of inspections. There are two types of repairs: minimal and perfect, depending on the deterioration level, each one following a different phase-type distribution. The failures and the inspections follow different Markovian arrival processes (MAP). Under these assumptions, the system is governed by a generalized Markov process, whose state space and generator are constructed. This general model includes the phase-type renewal process as a special case. The distribution of the number of minimal and perfect repairs between two inspections are determined. A numerical application optimizing costs is performed, and different particular cases of the model are compared.     

Optimally maintaining a Markovian deteriorating system with limited imperfect repairs

We consider the problem of optimally maintaining a periodically inspected system that deteriorates according to a discrete-time Markov process and has a limit on the number of repairs that can be performed before it must be replaced. After each inspection, a decision maker must decide whether to repair the system, replace it with a new one, or leave it operating until the next inspection, where each repair makes the system more susceptible to future deterioration. If the system is found to be failed at an inspection, then it must be either repaired or replaced with a new one at an additional penalty cost. The objective is to minimize the total expected discounted cost due to operation, inspection, maintenance, replacement and failure. We formulate an infinite-horizon Markov decision process model and derive key structural properties of the resulting optimal cost function that are sufficient to establish the existence of an optimal threshold-type policy with respect to the system’s deterioration level and cumulative number of repairs. We also explore the sensitivity of the optimal policy to inspection, repair and replacement costs. Numerical examples are presented to illustrate the structure and the sensitivity of the optimal policy.     

Stability of a pure random delay system with two-time-scale Markovian switching

This work examines almost sure stability of a pure random delay system whose delay time is modeled by a finite state continuous-time Markov chain with two-time scales. The Markov chain contains a fast-varying part and a slowly-changing part. Using the properties of the weighted occupation measure of the Markov chain, it is shown that the overall system?s almost-sure-asymptotic stability can be obtained by using the “averaged” delay. This feature implies that even if some longer delay times may destabilize the system individually, the system may still be stable if their impact is balanced. In other words, the Markov chain becomes a stabilizing factor. Numerical results are provided to demonstrate our results.