Equilibrium Delay Distribution for One Channel With Constant Holding Time, Poisson Input and Random Service

The equilibrium delay distribution is found for a single-server queueing system with Poisson input, random service and constant holding time. Curves are presented for various occupancy levels, and these are compared with their queued-service constant-holding-time and random-service exponential-holding-time counterparts. In many situations involving waiting lines -- for example, when customers are being served at a bargain counter in a crowded store -- the ideal queue discipline (service order) of first-come first-served is not achieved. Instead, the service order tends to be at least somewhat random, and the probability of long delays is thereby increased over what it would be for the strict first-come first-served discipline. Unfortunately for the analyst, when the queue discipline is somewhere between order-of-arrival and random -- as is often the case in practice -- the problem of calculating the delay distribution seems to be intractable. If the service order is assumed to be actually random, however, then this problem can sometimes be solved, and the delay distribution thus found is useful as a kind of bound on the distributions to be expected in those cases where the queue discipline deviates from order-of-arrival service toward randomness. The term "bound" as used here does not mean a bounding function in the strict sense. Actually, the delay distributions for models which differ only in queue discipline will cross each other, and hence no individual distribution can be a true bound for a family of such distributions.