ASIP tandem queues with Lévy input and consumption

We consider an ASIP (asymmetric inclusion process) tandem queue, in which the first queue receives a fluid input according to a nondecreasing Lévy process. Each queue has a gate that opens after independent, exponentially distributed periods for an infinitesimal amount of time, allowing the queue content to move to the next queue. In addition, again at independent exponentially distributed instants, a fixed fraction of a queue content is removed from the system. For this model, restricting ourselves to steady state, we obtain the following results. (i) We derive the buffer content distribution of the first queue. (ii) For the 2-queue model, we obtain relatively simple explicit expressions for the Laplace transform of the joint buffer content in several special cases. (iii) Asymptotic results are obtained for the 2-queue model when the above-mentioned buffer content removal process approaches a shot-noise process. (iv) For the general n-queue case, we show how all moments of the buffer contents at all queues can be obtained. (v) For the general n-queue case, we sketch an approximation method that allows one in principle to derive tractable expressions for the Laplace transform of the buffer content at each queue, with exact mean buffer contents at all queues.