János Sztrik

Finite-source M/M/S retrial queue with search for balking and impatient customers from the orbit

The present paper deals with a generalization of the homogeneous multi-server finite-source retrial queue with search for customers in the orbit. The novelty of the investigation is the introduction of balking and impatience for requests who arrive at the service facility with a limited capacity and FIFO queue. Arriving customers may balk, i.e., they either join the queue or go to the orbit. Moreover, the requests are impatient and abandon the buffer after a random time and enter the orbit, too. In case of an empty buffer, each server searches for a customer in the orbit after finishing service. All random variables involved in the model construction are supposed to be exponentially distributed and independent of each other. The primary aim of this analysis is to show the effect of balking, impatience, and buffer size on the steady-state performance measures. Concentrating on the mean response time, several numerical examples are investigated by the help of the MOSEL-2 tool used for creating the model and calculating the stationary characteristics.

A recursive solution of a queueing model for a multi-terminal system subject to break-downs

Abstract The queueing system to be analysed is a model of a multi-terminal system subject to random breakdowns. All random variables involved here are independent and exponentially distributed. Although the stochastic process describing the systems behaviour is a Markov chain, the number of states becomes very large. The main contribution of this paper is a recursive computational approach to solve the steady-state equations concerning the problem. In equilibrium, the main performance measures of the system, such as the mean number of jobs residing at the CPU, the mean number of functional terminals, the average number of busy servers, the expected response times of jobs, and utilizations are obtained. Finally, some numerical results illustrate the problem in question.

HETEROGENEOUS FINITE-SOURCE RETRIAL QUEUES

In this paper we investigate a single server retrial queue with a finite number of heterogeneous sources of calls. It is assumed when a given source is idle it will generate a primary call after an exponentially distributed time. If the server is free at the time of the request’ s arrival then the call starts to be served. The service time is also exponentially distributed. During the service time the source cannot generate a new primary call. After service the source moves into free state and can generate a new call again. If the server is busy at the time of the arrival of a primary call, then the source starts generating so called repeated calls with exponentially distributed times until it finds the server free. As before, after service the source becomes free and can generate a new primary call again. We assume that the primary calls, repeated attempts and service times are mutually independent. This queueing system and its variants could be used to model magnetic disk memory systems, local area networks with CSMA/CD protocols and collision avoidance local area networks. The novelty of this model is the heterogeneity of the calls, which means that each call is characterized by its own arrival, repeated and service rates. The aim of the paper is to give the usual steady-state performance measures of the system. To do so, an efficient software tool, MOSEL ( Modeling, Specification and Evaluation Language ) developed at the University of Erlangen, Germany, is used to formulate and solve the problem. Several sample numerical results illustrate the power of the tool showing the effect of different parameters on the system measures.

Performance Analysis of a Two-Server Heterogeneous Retrial Queue with Threshold Policy

Abstract In the paper we deal with a Markovian queueing system with two heterogeneous servers and constant retrial rate. The system operates under a threshold policy which prescribes the activation of the faster server whenever it is idle and a customer tries to occupy it. The slower server can be activated only when the number of waiting customers exceeds a threshold level. The dynamic behaviour of the system is described by a two-dimensional Markov process that can be seen as a quasi-birth-and-death process with infinitesimal matrix depending on the threshold. Using a matrix-geometric approach we perform a stationary analysis of the system and derive expressions for the Laplace transforms of the waiting time as well as arbitrary moments. Illustrative numerical results are presented for the threshold policy that minimizes the mean number of customers in the system and are compared with other heuristic control policies.

Machine interference problem with a random environment

Abstract This paper is concerned with a queueing model to analyse the asymptotic behaviour of the machine interference problem with N machines and a single operative. The running and repair times of each machine are supposed to be exponentially distributed random variables with parameter depending on the state of a varying environment. Assuming that the repair rate is much greater than the failure rate (‘fast’ service), it is shown that the time until the number of stopped machines reaches a certain level converges weakly, under appropriate norming, to an exponentially distributed random variable. Furthermore, some numerical examples illustrate the problem in question in the field of textile winding.

Performance analysis of finite-source retrial queues operating in random environments

This paper is concerned with the performance analysis of finite-source retrial queues with heterogeneous sources operating in random environments. All random variables involved in the model construction are assumed to be exponentially distributed with a parameter depending on the source index and on the state of the corresponding random environment. The novelty of the investigation is the involvement of the random environments, which makes the system rather complicated. The MOSEL tool is used to formulate and solve the problem and the main performance measures are derived and graphically displayed.

On the heterogeneous machine interference with limited server's availability

Abstract In this paper three models are developed for the heterogeneous machine interference problem where the occurrence of ancillary duty makes the operator non-available for repair activities. The machines are supposed to be stochastically different, the i -th machine is characterized by exponentially distributed running time with parameter λ −1 i and exponentially distributed required repair time with mean 1/μ i . The ancillary duty which has preemptive priority over the machines arises randomly being governed by exponential law with rate depending on the operators state, i.e., whether he is busy or idle. The time taken to complete an ancillary duty is assumed to be an exponentially distributed random variable. The machines are repaired according to first-in, first-out (FIFO), priority processor sharing (PPS) and preemptive resume priority (PR) service disciplines. The aim of the present paper is to give the main steady-state characteristics of the system, such as operative utilization, mean length of busy period, average number of machines not in working order, machine productivity and duration of time while the machines are failing. In addition some optimization problems are treated. Finally, numerical examples illustrate the problem in question.

Asymptotic analysis of some complex renewable systems operating in random environments

Abstract The present paper is concerned with an asymptotic analysis of some complex renewable systems operating in random environments. Assuming ‘fast’ repair, it is shown that the time to the first system failure, converges in distribution under appropriate norming to an exponentially distributed random variable.

A queueing model for a non-homogeneous terminal system subject to BR992☆

Abstract This paper deals with a non-homogeneous queueing model to describe the performance of a multi-terminal system subject to random breakdowns. All random variables involved here are indepenent and exponentially distributed. Although the stochastic describing the systems behaviour is a Markov chain, the number of states becomes very large. The main contribution of this paper is a recursive computational approach (see(5)) to solve the steady-state equations concerning the problem. It further generalizes the homogeneous model treated in [1]. In equilibrium, the main performance characteristics of the system are obtained. Finally, some numerical results illustrate the problem in question.

Optimization problems on the performance of a nonreliable terminal system

Abstract The aim of this paper is to investigate the effect of the different service disciplines, such as FIFO, PS, Priority Processor Sharing, Polling, on the main performance measures, such as utilizations, response times, throughput, mean queue length. It has been shown by numerical examples that even in the case of homogeneous sources and homogeneous failure and repair times, the CPU utilization depends on the scheduling discipline contrary to the case of reliable terminal systems. All random variables involved in the model construction are supposed to be exponentially distributed and independent of each other.