Optimal staffing for ticket queues

Ticket queues are popular in many service systems. Upon arrival, each customer is issued a numbered ticket and receives service on a first-come-first-served basis according to the ticket number. There is no physical queue; customers may choose to walk away and return later (before their numbers are called) to receive service. We study the problem of optimal staffing in such a system with two capacity levels, where the staffing decision can only be based on ticket numbers, as opposed to the physical queue length in a traditional system. Using renewal reward theorem, we first derive the long-run average total cost (including customer delay and abandonment costs, operating cost and cost for changing staffing levels) and then obtain the optimal solution using fractional programming. In addition, we pursue a random-walk analysis, which leads to some highly accurate approximations.