Assume that $k$ jobs circulate clockwise through a cyclic network of $n$ single-server queues, where at each integer time instant the job at the head of each queue moves with probability $p$ to the next queue, independent of the other jobs. The equilibrium distribution for the associated Markov chain is determined, and an exact expression for the expected number of busy servers is obtained. If $n$ and $k$ are large, a simple approximation for the proportion of busy servers is derived. In a second model, where the queues have no waiting room and where movement of a job occurs with probability $p$ only if the next queue was empty, a similar, simple asymptotic representation for the proportion of busy servers is deduced. This representation readily yields a simple expression for the asymptotic cycle time for a single job.
"Asymptotic Behavior of Large Discrete-Time Cyclic Queueing Networks." Ann. Appl. Probab. 4 (2) 591 - 606, May, 1994. https://doi.org/10.1214/aoap/1177005074