Diffusion Approximation Models for Transient States and their Application to Priority Queues

The article presents a diffusion approximation model applied to investigate the behavior of priority queues. We discusses the use of the diffusion approximation in transient analysis of queueing models in the case of a single station and of a queueing network presenting the solutions. We emphasize the numerical aspect of the solution and analyze the errors. In classical queuing theory, the analysis of transient states is complex and practically does not go far beyond M/M/1 queue and its modifications. However, the time dependent flows in computer networks and especially in Internet focus our interest on transient-state analysis, which is necessary to investigate the dynamics of TCP flows cooperating with active queue management or to see the changes of priority queues which assure the differentiated QoS. With the use of G/G/1/N and G/G/1/N/PRIOR models, we present the potentials of the diffusion approximation and in conclusions we compare it with alternative methods: Markovian queues solved numerically, fluid-flow approximation and simulation. Diffusion approximation allows us to include fairly general assumptions in queueing models. Besides the transient state analysis, it gives us a tool to consider input streams with general interarrival time distributions and servers with general service time distributions. Single server models can be easily incorporated into the network of queues. Here we apply the diffusion approximation formalism to study transient and steadystate behavior of G/G/1 and G/G/1/N priority preemptive models. The models can be easily converted to non-preemptive queueing discipline. The introduction of self-similar traffic is possible as well. The models can be useful in performance evaluation of mechanisms to differentiate the quality of service e.g. in IP routers, WiMAX, metro networks, etc. Index terms — diffusion approximation, transient states, priority queues.