Analysis of A Two-queue Discrete-time Model with Random Alternating Service Under High Occupancy in One Queue

We investigate a discrete-time two-queue system with a single server. At the beginning of every time slot, the single server randomly selects either queue 1 or queue 2 to serve. The decision of the server to select either queue 1 or queue 2 in each time slot is independent of the state of the system and is determined by a single parameter. The service times of the customers are deterministically equal to 1 time slot. The numbers of arrivals into both queues during consecutive slots are characterized by a joint probability generating function. By extending prior research, we analyze the behavior of the system as the number of customers in the first queue approaches infinity, thereby obtaining exact asymptotics for the stationary joint probability distribution of customer numbers in both queues.