A Queuing Model for Single Phase Server Breakdown Using Markov Chains with Random Transition

Abstract

A novel multi-server vacation queuing approach based on Markov chains with a random transition is being investigated for single phase server failure. The mathematical study of lines of people waiting for something is known as “queuing theory”. A simple queuing system has three main parts: the arrival procedure, the queue, and the service procedure. The model’s self-sufficiency of servers sets it apart from conventional queues. A server can shut down and take a vacation without the permission of the system management or the overall system state. The system administrator’s discretion is whether a server can resume processing clients once vacation is over. One way to think about the arrival process is as a generalized batch Markov one. The question of how many servers to have and the thresholds at which the management makes choices becomes an issue. The system’s behavior might be explained using a three-dimensional Markov chain with a unique generator block structure. The ergodicity of this chain is established, and the issue of computing the steady-state distribution is examined in detail. The distribution of chain states is used to create expressions for performance measures. A representation of a numerical result demonstrates that as N is the number of servers and the average number of consumers in the N buffer grows concerning rising in the parameter J1 and the loss probability P loss increases when the parameter J1 is increased and decreases as the number of servers N is increased.