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Mathematics 2018, 6(5), 81; https://doi.org/10.3390/math6050081

A Time-Non-Homogeneous Double-Ended Queue with Failures and Repairs and Its Continuous Approximation

1
Dipartimento di Matematica, Università degli Studi di Salerno, Via Giovanni Paolo II n. 132, 84084 Fisciano (SA), Italy
2
Dipartimento di Informatica, Università degli Studi di Salerno, Via Giovanni Paolo II n. 132, 84084 Fisciano (SA), Italy
3
Department of Mathematics, Anna University, Chennai 600 025, India
*
Author to whom correspondence should be addressed.
Received: 6 April 2018 / Revised: 8 May 2018 / Accepted: 9 May 2018 / Published: 11 May 2018
(This article belongs to the Special Issue Stochastic Processes with Applications)
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Abstract

We consider a time-non-homogeneous double-ended queue subject to catastrophes and repairs. The catastrophes occur according to a non-homogeneous Poisson process and lead the system into a state of failure. Instantaneously, the system is put under repair, such that repair time is governed by a time-varying intensity function. We analyze the transient and the asymptotic behavior of the queueing system. Moreover, we derive a heavy-traffic approximation that allows approximating the state of the systems by a time-non-homogeneous Wiener process subject to jumps to a spurious state (due to catastrophes) and random returns to the zero state (due to repairs). Special attention is devoted to the case of periodic catastrophe and repair intensity functions. The first-passage-time problem through constant levels is also treated both for the queueing model and the approximating diffusion process. Finally, the goodness of the diffusive approximating procedure is discussed. View Full-Text
Keywords: double-ended queues; time-non-homogeneous birth-death processes; catastrophes; repairs; transient probabilities; periodic intensity functions; time-non-homogeneous jump-diffusion processes; transition densities; first-passage-time double-ended queues; time-non-homogeneous birth-death processes; catastrophes; repairs; transient probabilities; periodic intensity functions; time-non-homogeneous jump-diffusion processes; transition densities; first-passage-time
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Di Crescenzo, A.; Giorno, V.; Krishna Kumar, B.; Nobile, A.G. A Time-Non-Homogeneous Double-Ended Queue with Failures and Repairs and Its Continuous Approximation. Mathematics 2018, 6, 81.

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