The M/G/1 queue with permanent customers

The authors examine an M/G/1 FCFS (first come, first served) queue with two types of customers: ordinary customers, who arrive according to a Poisson process, and permanent customers, who immediately return to the end of the queue after having received a service. The influence of the permanent customers on queue length and sojourn times of the Poisson customers is studied using results from queuing theory and from the theory of branching processes. In particular, it is shown that, when the service time distributions of the Poisson customers and all K permanent customers are negative exponential with identical means, the queue length and sojourn time distributions of the Poisson customers are the (K+1)-fold convolution of those for the case without permanent customers