Single counter queueing models involving multiple servers

In this paper, two single-counter queueing models are considered. In both models, a pool of R servers is available for service. Servers take vacations when they find the system idle, or when some other server is busy. In the first model, the server who initiates the busy period continues to operate until its completion. In the second model, there is a priority ranking system assigned to the servers. Any server with a higher priority, after returning from a vacation, immediately replaces any busy server with a lower priority. In the first model, in addition to the queue length distribution, several other results of interest are obtained, including the distribution of time between any two consecutive service activations for each server. For the second model, we have developed an algorithm to obtain the state probabilities and the mean queue length. The arrival process is assumed to be Poisson lengths of services and vacations are independent exponentially distributed random variables.