An M/G/1 system with startup server and J additional options for service

We consider an M^[x]/G/1 queueing system with a startup time, where all arriving customers demand first the essential service and some of them may further demand one of other optional services: Type 1, Type 2, … , and Type J service. The service times of the essential service and of the Type i  (i=1,2,…,J)$(i=1\text{,}2\text{,}\dots \text{,}J)$ service are assumed to be random variables with arbitrary distributions. The server is turned off each time when the system is empty. As soon as a customer or a batch of customers arrives, the server immediately performs a startup which is needed before starting each busy period. We derive the steady-state results, including system size distribution at a random epoch and at a departure epoch, the distributions of idle and busy periods, and waiting time distribution in the queue. Some special cases are also presented.