A transient solution to the M/M/c queuing model equation with balking and catastrophes

In this paper, we consider a Markovian multi-server queuing system with balking and catastrophes. The probability generating function technique along with the Bessel function properites is used to obtain a transient solution to the queuing model. The transient probabilities for the number of customers in the system are obtained explicitly. The expressions for the time-dependent expected number of customers in the system are also obtained. Finally, applications of the model are also discussed

An optional service Markovian queue with working disasters and customer’s impatience

In this paper, we develop a new class of Markov model with working disasters, second optional service, and reneging of customers. The disasters can occur during regular busy period. Whenever a disaster occurs, server continues to serve the customers with a lower service rate instead of completely stopping the service and after the completion of disaster recovery it switches to the regular busy period. Steady-state solution of the model is obtained by using probability generating function technique and stability condition is derived. Further, some important performance measures are presented. A cost model is developed in order to obtain the optimal service rates during first essential service, second optional service and during disaster period using quadratic fit search method. At the end, we provide some numerical examples to visualize the applicability of the model in practical situations.Publisher's Versio

On two modifications of E-r/E-s/1/m queuing system subject to disasters

The paper deals with modelling a finite single-server queuing system with the server subject to disasters. Inter-arrival times and service times are assumed to follow the Erlang distribution defined by the shape parameter r or s and the scale parameter rλ or sμ respectively. We consider two modifications of the model − server failures are supposed to be operate-independent or operate-dependent. Server failures which have the character of so-called disasters cause interruption of customer service, emptying the system and balking incoming customers when the server is down. We assume that random variables relevant to server failures and repairs are exponentially distributed. The constructed mathematical model is solved using Matlab to obtain steady-state probabilities which we need to compute the performance measures. At the conclusion of the paper some results of executed experiments are shown.Web of Science12215814

Transient Analysis of Two- Class Priority Queuing System

A transient solution of Two-class priority queuing system was discussed in this paper, the first class with high priority and the second one with low priority and the capacity of the system is infinite. Various performance measures are examined. Some numerical analyses are discussed and some special cases are deduced and confirmed the results

A double-ended queue with catastrophes and repairs, and a jump-diffusion approximation

Consider a system performing a continuous-time random walk on the integers, subject to catastrophes occurring at constant rate, and followed by exponentially-distributed repair times. After any repair the system starts anew from state zero. We study both the transient and steady-state probability laws of the stochastic process that describes the state of the system. We then derive a heavy-traffic approximation to the model that yields a jump-diffusion process. The latter is equivalent to a Wiener process subject to randomly occurring jumps, whose probability law is obtained. The goodness of the approximation is finally discussed.Comment: 18 pages, 5 figures, paper accepted by "Methodology and Computing in Applied Probability", the final publication is available at http://www.springerlink.co

Transient Analysis of Chemical Queue with Catastrophes and Server Repair

In this paper we deduced explicitly expressions for the transient state distribution for a queueing problem having chemical rules with arbitrary number of customers present initially in the system, in addition to having the possibility of catastrophe and hence system repair. Our calculations based on using generating function and Laplace transformation techniques, the obtained solution of the probabilities distribution enables us to recover the well known formula and other cases such as the transient solution of the standard M/M/1/∞ queue with λ = μ. Finally, the theory is underpinned by numerical results

Analysis of discrete-time queue with two heterogeneous servers subject to catastrophes

This paper studies a discrete-time queueing system with two heterogeneous servers subject to catastrophes. We obtain explicit expressions for the steady-state probabilities at arbitrary epoch using displacement operator method. The waiting time distribution and outside observer’s observation epoch probabilities are deduced. Various performance measures and numerical results have been investigated.Publisher's Versio

Modelování a simulace nespolehlivého E2/E2/1/m systému hromadné obsluhy

This paper is devoted to modelling and simulation of an E2/E2/1/m queueing system with a server subject to breakdowns. The paper introduces a mathematical model of the studied system and a simulation model created by using software CPN Tools, which is intended for modelling and a simulation of coloured Petri nets. At the end of the paper the outcomes which were reached by both approaches are statistically evaluated.Článek je věnován modelování a simulaci E2/E2/1/m systému hromadné obsluhy S obslužnou linkou podléhající poruchám. Příspěvek představuje matematický model studovaného systému a simulační model vytvořený S využitím software CPN Tools, který je určen pro modelování a simulaci barevných Petriho sítí. V závěru článku jsou výsledky dosažené oběma přístupy statisticky vyhodnoceny

Stochastic Processes with Applications

Stochastic processes have wide relevance in mathematics both for theoretical aspects and for their numerous real-world applications in various domains. They represent a very active research field which is attracting the growing interest of scientists from a range of disciplines.This Special Issue aims to present a collection of current contributions concerning various topics related to stochastic processes and their applications. In particular, the focus here is on applications of stochastic processes as models of dynamic phenomena in research areas certain to be of interest, such as economics, statistical physics, queuing theory, biology, theoretical neurobiology, and reliability theory. Various contributions dealing with theoretical issues on stochastic processes are also included