A discrete-time Geo/G/1 retrial queue with starting failures and second optional service

We consider a discrete-time Geo/G/1Geo/G/1 retrial queue with starting failures in which all the arriving customers require a first essential service while only some of them ask for a second optional service. We study the Markov chain underlying the considered queueing system and its ergodicity condition. Explicit formulae for the stationary distribution and some performance measures of the system in steady state are obtained. We also obtain two stochastic decomposition laws regarding the probability generating function of the system size. Finally, some numerical examples are presented to illustrate the influence of the parameters on several performance characteristics.