6533b86dfe1ef96bd12ca07c

RESEARCH PRODUCT

M/M/1 queue in two alternating environments and its heavy traffic approximation

Amelia G. NobileAntonio Di CrescenzoBalasubramanian Krishna KumarVirginia Giorno

subject

Partial differential equationMarkov chainDistribution (number theory)Stochastic processApplied MathematicsProbability (math.PR)010102 general mathematicsMathematical analysisM/M/1 queue60K25 60K37 60J60 60J70Heavy traffic approximation01 natural sciencesSteady-state distribution010104 statistics & probabilityDiffusion approximationFOS: MathematicsAlternating Wiener process0101 mathematicsFirst-hitting-time modelSteady-state distribution; First-passage time; Diffusion approximation; Alternating Wiener processQueueMathematics - ProbabilityAnalysisFirst-passage timeMathematics

description

We investigate an M/M/1 queue operating in two switching environments, where the switch is governed by a two-state time-homogeneous Markov chain. This model allows to describe a system that is subject to regular operating phases alternating with anomalous working phases or random repairing periods. We first obtain the steady-state distribution of the process in terms of a generalized mixture of two geometric distributions. In the special case when only one kind of switch is allowed, we analyze the transient distribution, and investigate the busy period problem. The analysis is also performed by means of a suitable heavy-traffic approximation which leads to a continuous random process. Its distribution satisfies a partial differential equation with randomly alternating infinitesimal moments. For the approximating process we determine the steady-state distribution, the transient distribution and a first-passage-time density.

yearjournalcountryeditionlanguage
2018-09-01Journal of Mathematical Analysis and Applications
https://doi.org/10.1016/j.jmaa.2018.05.043