A c -server Poisson queue with customer impatience due to a slow-phase service

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 λj, and are served according to an exponential distribution with rate μj where μS < μF. 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.