4 research outputs found
Transient behavior of fractional queues and related processes
We propose a generalization of the classical M/M/1 queue process. The
resulting model is derived by applying fractional derivative operators to a
system of difference-differential equations. This generalization includes both
non-Markovian and Markovian properties, which naturally provide greater
flexibility in modeling real queue systems than its classical counterpart.
Algorithms to simulate M/M/1 queue process and the related linear birth-death
process are provided. Closed-form expressions of the point and interval
estimators of the parameters of these fractional stochastic models are also
presented. These methods are necessary to make these models usable in practice.
The proposed fractional M/M/1 queue model and the statistical methods are
illustrated using S&P data
Fractional queues with catastrophes and their transient behaviour
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