Cafeteria process - tandem queues with 0-1 dependent service times and the bowl shape phenomenon

Customers move through a series of M service stations. Each customer, independent of all others, requires service from only one of the stations, for a duration of 1 time unit, this being station i with probability p_i. The customer has zero service at all the other stations, but there is no overtaking between the customers, and so queueing occurs. In the case where there is unlimited waiting room between the servers, we show that the system is interchangeable - permuting the order of the stations has no effect on the distribution of the output stream. When there is no waiting room between the stations we investigate optimal loads of the servers in terms of optimal p_i's for up to 10 stations, and observe that optimal loads exhibit the bowl phenomenon. We also obtain some bounds on the throughput for equal loads as a function of M.