Optimization of the Modified T Vacation Policy for a Discrete-Time $$\mathrm {Geom}^{[X]}/\mathrm {G}/1$$ Queueing System with Startup

In this paper, we discuss a discrete-time Geom\({^{[X]}}\)/G/1 queueing system with modified T vacation policy and startup time. We derive the generating functions and the mean values for the steady state system size and the waiting time, and also get those of the busy period, the vacation period and the vacation cycle by using embedded Markov chain. Finally, we determine the optimal \({(T^*,J^*)}\) to minimize the cost function with fixed cost elements by constructing a cost function.