Polling systems are systems consisting of multiple queues served by a single server. In this paper, we analyze polling systems with a server that is self-ruling, i.e., the server can decide to leave a queue, independent of the queue length and the number of served customers, or stay longer at a queue even if there is no customer waiting in the queue. The server decides during a service whether this is the last service of the visit and to leave the queue afterward, or it is a regular service followed, possibly, by other services. The characteristics of the last service may be different from the other services. For these polling systems, we derive a relation between the joint probability generating functions of the number of customers at the start of a server visit and, respectively, at the end of a server visit. We use these key relations to derive the joint probability generating function of the number of customers and the Laplace transform of the workload in the queues at an arbitrary time. Our analysis in this paper is a generalization of several models including the exponential time-limited model with preemptive-repeat-random service, the exponential time-limited model with non-preemptive service, the gated time-limited model, the Bernoulli time-limited model, the 1-limited discipline, the binomial gated discipline, and the binomial exhaustive discipline. Finally, we apply our results on an example of a new polling discipline, called the 1 + 1 self-ruling server, with Poisson batch arrivals. For this example, we compute numerically the expected sojourn time of an arbitrary customer in the queues.

中文翻译：

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具有自治服务器的轮询模型分析

轮询系统是由单个服务器服务的多个队列组成的系统。在本文中，我们分析了具有自治服务器的轮询系统，即服务器可以决定离开队列，独立于队列长度和服务客户​​的数量，或者在队列中停留更长时间，即使有没有顾客在排队等候。服务器在服务期间决定这是访问的最后一个服务并在之后离开队列，或者它是常规服务，可能由其他服务跟随。最后一个服务的特征可能与其他服务不同。对于这些轮询系统，我们分别推导出服务器访问开始时和服务器访问结束时客户数量的联合概率生成函数之间的关系。我们使用这些关键关系来推导出任意时间的客户数量和队列中工作负载的拉普拉斯变换的联合概率生成函数。我们在本文中的分析是几种模型的推广，包括具有抢占式重复随机服务的指数限时模型、具有非抢占式服务的指数限时模型、门控限时模型、伯努利限时模型。模型、1 限制学科、二项门控学科和二项穷举学科。最后，我们将我们的结果应用于一个新的轮询规则示例，称为 1 + 1 自治服务器，具有泊松批次到达。在这个例子中，我们以数值方式计算队列中任意客户的预期逗留时间。