Ticket queues with regular and strategic customers

We study a Markovian single-server ticket queue where, upon arrival, each customer can draw a number from a take-a-number machine, while the number of the customer currently being served is displayed on a panel. The difference between the above two numbers is called the “virtual queue length.” We consider a nonhomogeneous population of customers comprised of two types: “regular” and “strategic.” Upon arrival, a regular customer, regardless of the value of the virtual queue length, draws a number from the machine, joins the queue and waits in the system until being served. A strategic customer, depending on the virtual queue length, may either join, leave, or go to “orbit” for a random duration. If, upon return from orbit, a strategic customer realizes that s/he missed her/his turn, s/he balks. Otherwise, s/he joins the queue and waits to be served. We analyze this intricate stochastic system, calculate its steady-state probabilities, derive the sojourn time’s Laplace–Stieltjes transform of a regular and of a strategic customer and calculate the system’s performance measures. Finally, an economic analysis is performed to determine the optimal mean orbiting time of strategic customers for two types of objective functions. Numerical examples are presented.