Abstract

A system with heterogeneous servers, Poisson stream with Markov modulation, and instant feedback is studied. Primary requests are serviced in a high-speed server, and after servicing, each request, according to the Bernoulli scheme, either leaves the system or requires re-servicing. Repeated requests are serviced in a slow server; at the same time, after servicing, these requests also either leave the system or require repeated servicing in a slow server, according to the Bernoulli scheme. If at the moment of the arrival of the primary request the length of the queue of such requests exceeds a certain threshold value and the slow server is free, then the incoming request, according to the Bernoulli scheme, is either sent to the slow server or joins it when its turn to do so. It is considered that the lengths of the queues in front of each server are infinite. An adequate mathematical model of the system under study is constructed in the form of a three-dimensional Markov chain with an infinite state space. The ergodicity condition for this circuit is obtained, an approximate algorithm for calculating the stationary probabilities of the states is proposed, and its high level of accuracy is shown. The results of the numerical experiments are presented.

中文翻译：

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具有异构服务器和即时反馈的系统的数值分析

摘要

研究了具有异构服务器、具有马尔可夫调制的泊松流和即时反馈的系统。主要请求在高速服务器中提供服务，在服务之后，根据伯努利方案，每个请求要么离开系统，要么需要重新提供服务。重复请求在慢速服务器中提供服务；同时，根据伯努利方案，在服务之后，这些请求也要么离开系统，要么需要在慢速服务器中重复服务。如果在主请求到达的那一刻，此类请求的队列长度超过某个阈值并且慢速服务器空闲，则根据伯努利方案，传入请求要么被发送到慢速服务器，要么被发送到慢速服务器。轮到它时加入它。认为每个服务器前面的队列长度是无限的。所研究系统的适当数学模型以具有无限状态空间的三维马尔可夫链的形式构建。获得了该电路的遍历条件，提出了一种计算状态平稳概率的近似算法，并显示了其高精度。给出了数值实验的结果。