Markov modulated periodic arrival process offered to an ATM multiplexer

When a superposition of on/off sources is offered to a deterministic server, a particular queueing system arises whose analysis has a significant role in ATM based networks. Periodic cell generation during active times is a major feature of these sources. In this paper a new analytical method is provided to solve for this queueing system via an approximation to the transient behavior of the nD/D/1 queue. The solution to the queue length distribution is given in terms of a solution to a linear differential equation with variable coefficients. The technique proposed here has close similarities with the fluid flow approximations and is amenable to extension for more complicated queueing systems with such correlated arrival processes. A numerical example for a packetized voice multiplexer is finally given to demonstrate our results

A Numerically Efficient Method for the MAP/D/1/K Queue via Rational Approximations

The Markovian Arrival Process (MAP), which contains the Markov Modulated Poisson Process (MMPP) and the Phase-Type (PH) renewal processes as special cases, is a convenient traffic model for use in the performance analysis of Asynchronous Transfer Mode (ATM) networks. In ATM networks, packets are of fixed length and the buffering memory in switching nodes is limited to a finite number K of cells. These motivate us to study the MAP/D/1/K queue. We present an algorithm to compute the stationary virtual waiting time distribution for the MAP/D/1/K queue via rational approximations for the deterministic service time distribution in transform domain. These approximations include the well-known Erlang distributions and the Pad'e approximations that we propose. Using these approximations, the solution for the queueing system is shown to reduce to the solution of a linear differential equation with suitable boundary conditions. The proposed algorithm has a computational complexity independent of the queue storage capacity K. We show through numerical examples that, the idea of using Pad'e approximations for the MAP/D/1/K queue can yield very high accuracy with tractable computational load even in the case of large queue capacities