Double Queues and Impatient Customers with an Application to Inventory Theory

This paper starts by observing that any queuing system can be viewed as two symmetric queues, one of customers in line for service, and one of idle clerks, awaiting customers. Usually at least one of the queues (the idle clerks) has a finite limit, but examples, such as passengers and taxis at a taxi stand, exist where both queues are unlimited. In such cases, if arrival and service rates are independent of queue length, at least one of the queues is unstable and grows indefinitely as time passes. In practice there has to be some factor that limits queue length, and a possible mechanism is to assume that both passengers and taxis become impatient and will leave if, after some fixed amount of time, they have not been “serviced.” The equations governing such a system are developed and steady-state solutions are found. It is shown that steady-state solutions always exist for both queues, the means and variances (which are finite) are given. In the last part of the paper an application to inventory theory is suggested. Here the backlog of orders corresponds to the queue of customers, and items in stock to the idle clerks. It is assumed that orders not filled in a certain time are cancelled and that items are perishable and cannot be sold if stocked too long.