On Queues with Working Vacation and Interdependence in Arrival and Service Processes

In this paper, we consider two queuing models. Model 1 considers a single-server working vacation queuing system with interdependent arrival and service processes. The arrival and service processes evolve by transitions on the product space of two Markovian chains. The transitions in the two Markov chains in the product space are governed by a semi-Markov rule, with sojourn times in states governed by the exponential distribution. In contrast, in the second model, we consider independent arrival and service processes following phase-type distributions with representation (α,T) of order m and (β,S) of order n, respectively. The service time during normal working is the above indicated phase-type distribution whereas that during working vacation is a phase-type distribution with representation (β,θS), 0<θ<1. The duration of the latter is exponentially distributed. The latter model is already present in the literature and will be briefly described. The main objective is to make a theoretical comparison between the two. Numerical illustrations for the first model are provided.