Consider a many-server queueing system in which the servers are numbered. If a customer arrives when two or more servers are idle he selects the server with lowest index (this is called the ordered entry selection rule). An explicit expression for the traffic handled by the various servers in a GI/M/s queueing system with ordered entry is derived. For the M/M/s queueing system the probability distribution of the number of busy servers among the first k(k = 1,2, - - -, s) servers will be given. Finally, a formula for the traffic handled by the first server in an MID/s system will be derived. All results are derived under steady-state conditions. As an application some numerical data for the server utilizations will be given and compared to data obtained from simulation studies of a closed-loop continuous belt-conveyor.
Nawijn, W. M. (1983). A note on many-server queueing systems with ordered entry, with an application to conveyor theory. Journal of applied probability, 20(1), 144-152. https://doi.org/10.2307/3213728