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.

中文翻译：

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与常规和战略客户的票务队列

我们研究了马尔可夫单服务器票务队列，其中，每个客户到达时可以从取号机中抽取一个号码，而当前正在服务的客户的号码显示在面板上。上述两个数字之间的差值称为“虚拟队列长度”。我们考虑由两种类型的非同质客户群组成：“常规”和“战略”。到达后，无论虚拟队列长度的值如何，普通客户都会从机器中抽取一个号码，加入队列并在系统中等待直到被服务。根据虚拟队列长度，战略客户可以加入、离开或进入“轨道”随机持续时间。如果从轨道返回后，战略客户意识到他/她错过了她/他的轮到她/他，他/她会犹豫。否则，他/她加入队列并等待服务。我们分析这个复杂的随机系统，计算其稳态概率，推导出常规客户和战略客户的逗留时间的 Laplace-Stieltjes 变换，并计算系统的性能指标。最后，进行经济分析，以确定两种目标函数的战略客户的最佳平均轨道时间。给出了数值例子。