Fractional queues with catastrophes and their transient behaviour
Summary: Starting from the definition of fractional \(\mathrm{M}/\mathrm{M}/1\) queue given in the reference by \textit{D. O. Cahoy} et al. in [Methodol. Comput. Appl. Probab. 17, No. 3, 739--759 (2015; Zbl 1322.60189)] and \(\mathrm{M}/\mathrm{M}/1\) queue with catastrophes given in the reference by \textit{A. Di Crescenzo} et al. in [Queueing Syst. 43, No. 4, 329--347 (2003; Zbl 1016.60080)], we define and study a fractional \(\mathrm{M}/\mathrm{M}/1\) 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 \(\mathrm{M}/\mathrm{M}/1\) 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.
fractional differential-difference equations
scientific article; zbMATH DE number 6994723
scientific article