Analysis of M/M/1 Model with Complete Breakdown during Busy Period, Optional Server with Limited Service, Complete Vacation and Delay in Repair

Abstract

In this paper we have analyzed a single server Markovian queueing model with an optional server for limited-service time, complete breakdown during busy period, complete vacation with some delay in repair. Customers arrival follow Poisson’s distribution with rate λ. Service time during busy period is exponentially distributed with rate. The server goes under complete breakdown during busy period and hence sent for repairing. During breakdown an optional server with limited-service time is available for serving customers rather than stopping service. This optional server after completing busy period moves to working vacation for limited time period, where service time during this period is slower than busy period service time as server has some prior commitments or work to finish. As the limited-service time finishes, the server goes for a complete vacation and will not serve any customer during this period. If main server gets repaired, it immediately resumes busy period but if some delay occurs in repairing of main server, then optional server after completing vacation, will act as main server and resumes busy period until main server get repaired. The closed form expression of various system probabilities is derived. Various system performance measures like waiting time, queue length have been evaluated. Finally, some numerical and graphical results have been shown to model the impact of some parameters on different performance measures.