Stationary analysis for the fluid model driven by the working vacation queue

We first analyze a multi-server M / M / c queue with working vacations. Using a quasi birth-and-death (QBD) process and a matrix-geometric solution method, the steady state distribution of the queue length is derived. Furthermore, a fluid flow model driven by this M / M / c working vacation queue is discussed, we obtain the sets of differential equations satisfied by the stationary joint distribution of the buffer content, by which we gain the simple structure of the Laplace transform (LT) of the stationary distribution of the buffer content. Finally, we give the probability of empty buffer content and the mean of the buffer content based on the relationship between the LT and the Laplace–Stieltjes transform (LST) of the stationary distribution.