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.
[1]
Marcel F. Neuts,et al.
The queue with Poisson input and general service times, treated as a branching process
,
1969
.
[2]
J. F. C. Kingman.
The effect of queue discipline on waiting time variance
,
1962
.
[3]
Roger I. Wilkinson.
Working curves for delayed exponential calls served in random order
,
1953
.
[4]
Marcel F. Neuts,et al.
An Exact Comparison of the Waiting Times Under Three Priority Rules
,
1971,
Oper. Res..
[5]
Marcel F. Neuts,et al.
A Priority Rule Based on the Ranking of the Service Times for the M/G/1 Queue
,
1969,
Oper. Res..
[6]
D. Kendall.
Some Problems in the Theory of Queues
,
1951
.