A c-server Poisson queue with customer impatience due to a slow-phase service. (9th September 2021) - Sivasamy, R.; Peter, Peter O.
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 ( λ 0 )/ M ( μ 0 + α )/ c queue with customer impatience (CI) and of another M ( λ )/ M ( μ )/ c facility without CI.