Stochastic networks with complex structures are key modelling tools for many important applications. In this paper, we consider a specific type of network: retrial queueing systems with priority. This type of queueing system is important in various applications, including telecommunication and computer management networks with big data. The system considered here receives two types of customers, of which Type-1 customers (in a queue) have non-pre-emptive priority to receive service over Type-2 customers (in an orbit). For this type of system, we propose an exhaustive version of the stochastic decomposition approach, which is one of the main contributions made in this paper, for the purpose of studying asymptotic behaviour of the tail probability of the number of customers in the steady state for this retrial queue with two types of customers. Under the assumption that the service times of Type-1 customers have a regularly varying tail and the service times of Type-2 customers have a tail lighter than Type-1 customers, we obtain tail asymptotic properties for the numbers of customers in the queue and in the orbit, respectively, conditioning on the server’s status, in terms of the exhaustive stochastic decomposition results. These tail asymptotic results are new, which is another main contribution made in this paper. Tail asymptotic properties are very important, not only on their own merits but also often as key tools for approximating performance metrics and constructing numerical algorithms.

中文翻译：

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具有非抢占优先级的 $$M_1,M_2/G_1,G_2/1$$ 重试队列的尾渐近

具有复杂结构的随机网络是许多重要应用的关键建模工具。在本文中，我们考虑一种特定类型的网络：具有优先权的重试排队系统。这种类型的排队系统在各种应用中都很重要，包括具有大数据的电信和计算机管理网络。这里考虑的系统接收两种类型的客户，其中类型 1 客户（在队列中）比类型 2 客户（在轨道中）具有非抢占优先权来接收服务。对于这种类型的系统，我们提出了一种详尽版本的随机分解方法，这是本文的主要贡献之一，目的是研究稳态下客户数量的尾概率的渐近行为这个重试队列有两种类型的客户。在假设 1 类客户的服务时间有规律变化的尾部并且 2 类客户的服务时间的尾部比 1 类客户的尾部轻的情况下，我们获得队列中客户数量的尾部渐近特性，并在轨道中，分别根据详尽的随机分解结果调节服务器的状态。这些尾部渐近结果是新的，这是本文的另一个主要贡献。尾渐近特性非常重要，不仅是因为它们本身的优点，而且通常作为逼近性能指标和构建数值算法的关键工具。根据详尽的随机分解结果，我们分别获得队列中和轨道中客户数量的尾部渐近特性，以服务器的状态为条件。这些尾部渐近结果是新的，这是本文的另一个主要贡献。尾渐近特性非常重要，不仅是因为它们本身的优点，而且通常作为逼近性能指标和构建数值算法的关键工具。根据详尽的随机分解结果，我们分别获得队列中和轨道中客户数量的尾部渐近特性，以服务器的状态为条件。这些尾部渐近结果是新的，这是本文的另一个主要贡献。尾渐近特性非常重要，不仅是因为它们本身的优点，而且通常作为逼近性能指标和构建数值算法的关键工具。