Performance Analysis of M/M/1 Queueing Model with Working Vacation, Balking, and with Standby Server

The performance analysis of a single server queueing model operating on working vacation, balking, and with a standby server was examined in this paper. The FCFS was used in serving the consumer. The distribution of the services is exponential, and the arrival follows the Poisson Process. The server is on vacation, where the server serves at a lower service rate is working vacation. When a customer balks, it means they do not want to wait in line because they feel the line is too long or they don't like the way they are currently waiting. The customer receives the minimum level of service until the completion of repairs for regular service when the server experiences an unplanned failure and needs to be fixed before continuing with regular service with the help of a standby server. The probability vectors were obtained using the matrix geometric method by considering the stochastic processes {(S(t),N(t)):t≥0}. Performance measures are obtained by adjusting values for specific parameters, such as the mean number of customers present in the system, the probability that the server is busy, working vacation, or on breakdown period. Additionally, certain graphical representations and numerical tables are obtained.