A multi-server queueing-inventory system with stock-dependent demand

We consider a two-server service system in which idle servers produce and store preliminary services for reducing the sojourn time of incoming customers and increasing customers arrival rate. We model the system as a Markovian process and provide a method, based on matrix geometric (MG) analysis, to obtain closed-form solutions to the steady state probabilities and relevant performance measures. We show the relation of the elements in the rate matrix of the MG to Catalan numbers, and prove that the stability condition remains as is in the traditional M/M/2 queue, although the expected sojourn time of customers has been reduced. We provide an economic analysis for a system in which the PS capacity and the investment to increase customers' arrival rate are decision variables.