The M/M/∞ Queue with Varying Arrival and Departure Rates

This paper derives the queue size distribution for infinite server queues with Poisson arrivals and exponential service times when the parameters of both distributions are allowed to vary with time. Both continuous and discrete variation are examined. Infinite server queues realistically describe those queues in which sufficient service capacity exists to prevent virtually any customer waiting time. Such queues are not uncommon in the health services, when a delay in service can sometimes mean death. Specifically, we discuss the queuing problem of an intensive care unit and show that it is unlikely that the hourly variation in the arrival rate of patients to the unit will significantly affect the number of beds occupied.