Abstract:

Based on the take-out operation model of the store, this paper constructed and analyzed a fluid model driven by the M/M/1/N queueing system with set-up time, optional service and two types of mixed vacations. Firstly, the driving system is described and the infinite small generators of the two-dimensional Markov process are decomposed into a blocky Jacobian matrix form. Then, the stationary probability distribution of the driving system is obtained by using matrix-geometric method. Secondly, based on the net input rate structure of the fluid model, the differential and difference equations satisfied by the fluid level in steady-state conditions are obtained using the probability analysis method. Then, the expected buffer content and the probability of the empty buffer under steady-state conditions are obtained by using Laplace transform(LT) and Laplace-Stieltjes transform(LST) methods. Finally, the influence of parameters changing on the performance indicators and cost function are illustrated in numerical analysis.

Key words: Fluid model, Two-stage vacation, Set-up time, Finite capacity, Buffer content