Use of a gate to reduce the variance of delays in queues with random service

We consider an N-server queuing system with Poisson arrivals and exponential service, in which arriving customers must pass through a gate into a waiting room before becoming eligible for service. Customers who find the gate closed wait outside until the gate opens; customers inside the waiting room are served at random. When the last customer inside acquires a server, the gate admits all those outside and then closes again. If no customer is waiting outside when the gate opens, the gate remains open until there is a queue of k waiting customers. Service offered by this system is intermediary between random service and order-of-arrival service. As long as the gate is open and fewer than N + k customers are in the system, service is purely random. The parameter k can be regarded as a threshold at which the queue is judged too long to permit random service to continue. Our main results are (i) the Laplace-Stieltjes transform of the equilibrium distribution of the waiting time of an arbitrary customer and (ii) a comparison of the second moments of the waiting time for different values of k with those of the waiting time under random service and order-of-arrival service. The service is shown to be “nearly random” at low loads and “not quite order-of-arrival” at high loads; for higher values of k this transition occurs at higher traffic intensities.