Abstract This paper studies a Markovian two-station tandem queueing network with impatient customers. Queueing networks with abandonment are common in many industries, e.g., call centers and healthcare. Therefore, their management has received much attention. The resulting model is a level-dependent quasi-birth-and-death (LDQBD) process. Such models are considered analytically intractable and require numerical methods for their solution. We study a specific type of LDQBD process, where the total abandonment rate increases with the number of waiting customers leading to the level-dependent feature. We analyze an equivalent last-come-first-serve system to develop a recursive relation in our LDQBD process, reducing the problem to solving quadratic matrix equations, for which efficient and exact numerical methods exist. We further simplify the analysis by combining the recursive renewal reward theorem with Queueing and Markov chain decomposition (QMCD), so that we only need to solve one quadratic matrix equation instead of infinite ones caused by the system’s level-dependent feature. We develop an exact numerical method to evaluate various performance measures of a tandem queueing network with abandonment. Our method is applicable to the analysis of queueing networks with abandonment under settings with diverse features and in various service disciplines.

中文翻译：

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与不耐烦的客户串联队列

摘要 本文研究了具有不耐烦顾客的马尔可夫两站串联排队网络。具有放弃的排队网络在许多行业中很常见，例如呼叫中心和医疗保健。因此，他们的管理备受关注。由此产生的模型是一个水平相关的准生死 (LDQBD) 过程。这种模型被认为是难以解析的，需要数值方法来求解。我们研究了一种特定类型的 LDQBD 过程，其中总放弃率随着等待客户数量的增加而增加，从而导致级别相关的特征。我们分析了一个等效的后到先服务系统，以在我们的 LDQBD 过程中建立递归关系，将问题简化为求解二次矩阵方程，为此存在有效和精确的数值方法。我们通过将递归更新奖励定理与排队和马尔可夫链分解（QMCD）相结合来进一步简化分析，这样我们只需要求解一个二次矩阵方程，而不是由系统的水平相关特征引起的无穷大方程。我们开发了一种精确的数值方法来评估具有放弃的串联排队网络的各种性能指标。我们的方法适用于分析具有不同特征和不同服务学科的设置下的废弃排队网络。我们开发了一种精确的数值方法来评估具有放弃的串联排队网络的各种性能指标。我们的方法适用于分析具有不同特征和不同服务学科的设置下的废弃排队网络。我们开发了一种精确的数值方法来评估具有放弃的串联排队网络的各种性能指标。我们的方法适用于分析具有不同特征和不同服务学科的设置下的废弃排队网络。