Discrete Time Single Server Queueing Model with a Multimodal Packet Size Distribution

The importance of IP network will further increase and it will serve as a platform for more and more services, requiring different types and degrees of service quality. In this paper, we investigate the queueing behaviour found in input buffers in IP switch or router. To analyze these types of behaviour, we study the discrete-time version of the “classical” queue model M/M/1/k called Geo/MM/1/k and P/MM/1/k. We use a geometric (Geo) and Pareto (P) distributed inter-arrival time and a discrete multimodal packet length distribution (MM), defined on a range between a minimum and maximum value. In this model, the time unit is reduced from a packet service time to the time to transmit a byte. We develop balance equations for the state of the system, from which we derive packet delay and loss results. The finding that Geometric and Pareto distributions are adequate to model Internet packet interarrival times has motivated the proposal of methods to evaluate steady-state performance measures of Geo/MM/1/k queue. We propose a real time trace simulation model for estimating the steady-state probability showing the loss probability and delay of the P/MM/1/k queue. This model can be used to study the long-tailed queueing systems.