A batch arrival retrial queuing system for essential and optional services with server breakdown and Bernoulli vacation

This paper studies a general retrial M X /G/1 queue with an additional phase of second optional service and Bernoulli vacation where breakdowns occur randomly at any instant while servicing the customers. If an arriving batch finds that the server is busy in providing either first essential service (FES)/second optional service (SOS) or on vacation then arriving batch enters an orbit called retrial queue. Otherwise, one customer from arriving batch starts to be served by the server while the rest join the orbit. The vacation times and service times of both first essential and second optional services are assumed to be general distributed while the retrial times are exponential distributed. Introducing supplementary variables and by employing embedded Markov chain technique, we derive some important performance measures of the system such as average orbit size, average queue size, mean waiting time, expected lengths of busy period, etc. Numerical results have been facilitated to illustrate the effect of different parameters on several performance measures.