Comparisons of ticket and standard queues

Upon arrival to a ticket queue, a customer is offered a slip of paper with a number on it—indicating the order of arrival to the system—and is told the number of the customer currently in service. The arriving customer then chooses whether to take the slip or balk, a decision based on the perceived queue length and associated waiting time. Even after taking a ticket, a customer may abandon the queue, an event that will be unobservable until the abandoning customer would have begun service. In contrast, a standard queue has a physical waiting area so that abandonment is apparent immediately when it takes place and balking is based on the actual queue length at the time of arrival. We prove heavy traffic limit theorems for the generalized ticket and standard queueing processes, discovering that the processes converge together to the same limit, a regulated Ornstein–Uhlenbeck process. One conclusion is that for a highly utilized service system with a relatively patient customer population, the ticket and standard queue performances are asymptotically indistinguishable on the scale typically uncovered under heavy traffic approaches. Next, we heuristically estimate several performance metrics of the ticket queue, some of which are of a sensitivity typically undetectable under diffusion scaling. The estimates are tested using simulation and are shown to be quite accurate under a general collection of parameter settings.