Fractional queues with catastrophes and their transient behaviour
type: Item
instance of: scholarly article
title: Fractional queues with catastrophes and their transient behaviour
zbMATH Open document ID: 1404.60135
author: Giacomo Ascione
author: Enrica Pirozzi
author: Nikolai N. Leonenko
published in: Mathematics
publication date: 2018-12-18T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 60K25
Mathematics Subject Classification ID: 60J80
zbMATH DE Number: 6994723
zbMATH Keywords: fractional differential-difference equations
zbMATH Keywords: fractional queues
zbMATH Keywords: fractional birth-death processes
zbMATH Keywords: busy period
MaRDI profile type: Publication
full work available at URL: math6090159
OpenAlex ID: W2889416606
cites work: Q4070097
cites work: On a new formula for the transient state probabilities for M/M/1 queues and computational implications
cites work: Q4101255
cites work: Introduction to Queueing Systems with Telecommunication Applications
cites work: A transient solution to an M/M/1 queue: a simple approach
cites work: Q5699297
cites work: On Riemann-Liouville and Caputo derivatives
cites work: Transient behavior of fractional queues and related processes
cites work: Mittag-Leffler functions and their applications
cites work: The fractional Poisson process and the inverse stable subordinator
cites work: Fractional Poisson process
cites work: Inverse Stable Subordinators
cites work: Fractional Poisson fields and martingales
cites work: On the M/M/1 queue with catastrophes and its continuous approximation
cites work: Transient solution of an M/M/1 queue with catastrophes
cites work: A state-dependent queueing system with asymptotic logarithmic distribution
cites work: Limit theorems for occupation times of Markov processes
cites work: On densities of the product, quotient and power of independent subordinators
cites work: Transient analysis of the M/M/1 queue
cites work: Basic Theory of Fractional Differential Equations
cites work: Another Note on the Inclusion L p (μ) ⊂L q (μ)
cites work: Q4141653
cites work: Feynman path integrals and asymptotic expansions for transition probability densities of some Lévy driven financial markets
DOI: 10.3390/MATH6090159
description: scientific article; zbMATH DE number 6994723