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A queueing system with decomposed service and inventoried preliminary services

机译：具有分解服务和库存初步服务的排队系统

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摘要

We study a single-server queue in which the service consists of two independent stages. The first stage is generic and can be performed even in the absence of customers, whereas the second requires the customer to be present. When the system is empty of customers, the server produces an inventory of first-stage (‘preliminary’) services (denoted PSs), which is used to reduce customers’ overall sojourn times. We formulate and analyze the queueing-inventory system and derive its steady-state probabilities by using the matrix geometric method, which is based on calculating the so called rate matrix R. It is shown that the system's stability is not affected by the production rate of PSs, and that there are cases in which utilizing the server's idle time to produce PSs actually increases the fraction of time during which the server is dormant. A significant contribution is the derivation of an explicit expression of R, whose entries are written in terms of Catalan numbers. This type of result is rare in the literature and enables large-scale problems to be solved with low computational effort. Furthermore, by utilizing Laplace-Stieltjes transform and its inverse, we obtain the distribution function of customers’ sojourn time. Finally, based on the probabilistic study, we carry out an economic analysis using a practical example from the fast food industry.
机译：我们研究了一个单服务器队列，其中服务包含两个独立的阶段。第一阶段是通用的，即使在没有客户的情况下也可以执行，而第二阶段则需要客户在场。当系统中没有客户时，服务器会生成第一阶段（“初步”）服务（称为PS）的清单，用于减少客户的总体停留时间。我们基于计算所谓的速率矩阵R的矩阵几何方法，对排队库存系统进行公式化和分析，得出其稳态概率。结果表明，系统的稳定性不受排队率的影响。 PS，并且在某些情况下，利用服务器的空闲时间来产生PS实际上会增加服务器处于休眠状态的时间比例。一个重要的贡献是R的显式表达式的派生，R的条目以加泰罗尼亚数字表示。这种类型的结果在文献中很少见，并且可以用较少的计算量来解决大规模问题。此外，通过使用Laplace-Stieltjes变换及其逆变换，我们可以获得客户停留时间的分布函数。最后，基于概率研究，我们以快餐业的实际案例为基础进行了经济分析。

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