A c-server Poisson queue with customer impatience due to a slow-phase service
by R. Sivasamy; Peter O. Peter
International Journal of Mathematics in Operational Research (IJMOR), Vol. 20, No. 1, 2021

Abstract: In this paper, we investigate a queue with c-servers in a Markovian environment (ME) under two phases (S - slow and F - fast). We assume that the sojourn times follow an exponential distribution at states S and F with parameters ν and η respectively. Each customer chooses a random relative deadline duration (RDD), which follows an exponential law with parameter α under S-phase but abandons the system as soon as its RDD expires and never returns. When the environment remains under phase j (= S, F), customers arrive according to a Poisson process with rate λ_j, and are served according to an exponential distribution with rate μ_j where μ_S < μ_F. We formulate the queue length process Y_(j,n)(t), representing the number of customers n(t), waiting in the phase j at time t as level dependent quasi birth-death (LDQBD) process in a 2-dimensional space. We apply matrix analytic methods on computational procedures and iterative methods to obtain scalar types of explicit expressions for the stationary probability distributions of Y_(j,n)(t) as t tends to infinity. We present comparable charts to highlight the variations between the steady state measures of an M(λ₀)/M(μ₀ + α)/c queue with customer impatience (CI) and of another M(λ)/M(μ)/c facility without CI.

Online publication date: Fri, 17-Sep-2021

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