On closed ring queueing networks

This paper first studies ring structured closed queueing networks with distinguishable jobs. Under assumptions of periodicity and ergodicity of the service times, essentially the most general, it is shown that the limits defining the average flows of the jobs exist almost surely, and methods for their estimation by simulation are given. However, it turns out that the values of the flows depend on the initial positions of the jobs, due to the emergence of distinct persistent blocking modes. The effect of these modes on the behavior of general networks with queueing loops is examined. For independent and identically distributed service times, conditions are specified for the network to asymptotically approach a steady state at large times. Finally, the paper studies the special case of ring networks with indistinguishable items and stationary and ergodic service times. It is shown that as the number of jobs in the network increases towards infinity, the average circulation time converges to the maximum of the expectations of the service times.