Algorithmic computation of steady-state probabilities in an almost observable GI/M/c queue with or without vacations under state dependent balking and reneging

We consider an almost observable GI/M/c/N queue with customer impatience with or without multiple synchronous vacations under state-dependent balking. Upon arriving, a customer joins or refuses to join the system based on certain state dependent joining/balking probabilities. Once an arriving customer joins the system and finds all the servers busy, it initiates an exponentially distributed impatience timer with random duration T. An equilibrium balking strategy is discussed for constant balking as a special case of state-dependent balking. In the constant balking case, the waiting time of a customer in a queueing system has been associated with a linear cost-reward structure for estimating the net benefit if a customer chooses to participate in the system. A steady-state system of equations is obtained using the supplementary variable technique. After that, a recursive algorithm is proposed to obtain the stationary system-length distribution at pre-arrival and arbitrary epochs using those steady-state equations, from which the mean system sojourn time, the average reneging rate and the average balking rate are derived. Finally, we produce numerical results relating to system performance and its net benefit when investigated for different model parameters. The proposed model has applications in the modeling of balking and impatient behavior of incoming calls in a call center, multi-core computing, multi-path routing in delay sensitive communications networks.