- - - Department of Mathematics, College of Education for Pure Sciences, University of Babylon, Iraq
- - - Department of Mathematics, College of Education for Pure Sciences, University of Babylon, Iraq

In this article, we consider queueing model with bilevel hysteretic control and multiple vacations. This system's mechanism depends on the queue size and the facility where the server goes to the series of vacation trips when the line is empty and returns but waiting in the system when the number of units is more than level \(M\). In this case, the server doesn't start a new busy period unless this size is more than another level \(N > M\). Furthermore, we employ N-policy and first excess level analysis to derive the probability generating function of queue size. Additionally, we assume that the vacation times are exponentially distributed random variables, and arrival batches are type ۱ geometrically distributed random variables.

hysteretic control, multiple vacations, N-policy, first excess level theory, input batches, bulk input, marked delayed renewal process, delayed renewal process, point process, marked point process, random walk