SINGLE SERVER BATCH SERVICE MARKOVIAN(M/M(a,b)/1)QUEUE WITH VARIANT WORKING VACATION

Abstract

This research investigates a single server batch service queuing system with Poisson arrivals, exponential service times and a server that takes variant working vacations. During working vacation, the server continues to serve a customer at a reduced rate. The system’s steady state behavior is analyzed using a Markov chain approach. Mathematical equations governing the system’s dynamics are derived, considering the server’s different states: idle, busy, on vacation, and serving during vacations. The inter-arrival times, service times and the duration of working vacation period are taken to be mutually independent and exponentially distributed. Steady state probabilities are obtained by solving linear equations (forward shift operator). Based on these probabilities, we calculate the average of queue length. Finally, numerical illustrations in the form of tables and graphs were presented using MATLAB to show how various parameters of the model influence the behavior of the system. The research findings demonstrate that the variant working vacation policy can significantly influence the system performance. The specific effect depend on the relative magnitudes of the arrival, service, and vacation rates.