Abstract
Starting from the definition of fractional M/M/1 queue given in the reference by Cahoy et al. in 2015 and M/M/1 queue with catastrophes given in the reference by Di Crescenzo et al. in 2003, we define and study a fractional M/M/1 queue with catastrophes. In particular, we focus our attention on the transient behaviour, in which the time-change plays a key role. We first specify the conditions for the global uniqueness of solutions of the corresponding linear fractional differential problem. Then, we provide an alternative expression for the transient distribution of the fractional M/M/1 model, the state probabilities for the fractional queue with catastrophes, the distributions of the busy period for fractional queues without and with catastrophes and, finally, the distribution of the time of the first occurrence of a catastrophe.
Highlights
Stochastic models for queueing systems have a wide range of applications in computer systems, sales points, telephone or telematic systems and in several areas of science including biology, medicine and many others
The introduction of the fractional Caputo derivative into the system of differential-difference equations for an M/M/1-type queue was done in [8], where, for a fractional M/M/1 queue, the state probabilities were determined. In this kind of queue model, the inter-arrival times and service times are characterized by Mittag–Leffler distributions [9]; in this case, the model does not have the property of memory loss that is typical of the exponential distributed times of the classical M/M/1 model
Motivated by the mathematical need to enrich the fractional M/M/1 model of [8] with the inclusion of catastrophes, we study in this paper the above model; we determine the transient distribution, the distribution of the busy period and the probability distribution of the time of the first occurrence of the catastrophe
Summary
Stochastic models for queueing systems have a wide range of applications in computer systems, sales points, telephone or telematic systems and in several areas of science including biology, medicine and many others. Motivated by the mathematical need to enrich the fractional M/M/1 model of [8] with the inclusion of catastrophes, we study in this paper the above model; we determine the transient distribution, the distribution of the busy period (including that of the fractional M/M/1 queue of [8]) and the probability distribution of the time of the first occurrence of the catastrophe. Of the number of customers in the system at time t in the fractional M/M/1 queue are characterized by arrivals and services determined by fractional Poisson processes of order ν ∈ They are solutions of the following system of differential-difference equations. Exp {−(α + β)x − xuν cos(νπ)} xr sin (πν − xuν sin(πν)) dx
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