On an M/G/1 Queue Providing General First Essential Service with Customer's Choice for General or Deterministic Second Optional Service

We study an M/G/1 queuing system with one by one Poisson arrivals and one by one general first essential service followed by one of the two types of second optional heterogeneous services. A customer entering the service channel compulsorily goes through first essential service. After completion of the first service, the customer has the option to take a second optional service with probability pi no second optional service with probability 1-pi. If the customer decides to take the second optional service, he may take a general second optional service with probability alpha(1) or a deterministic second optional service with probability alpha(2),alpha(1) + alpha(2) = 1. We find steady state solution in terms of the generating function of the queue length as well as the explicit steady state probabilities for various states of the system. Results for a few interesting special cases are derived.