Polling Systems with Periodic Server Routing in Heavy-Tra±c: Distribution of the Delay

We consider polling systems with mixtures of exhaustive and gated service in which the server visits the queues periodically according to a general polling table. We derive exact expressions for the steady-state delay incurred at each of the queues, under standard heavy-tra±c scalings. The expressions require the solution of a set of only M ¡ N linear equations, where M is the length of the polling table and N is the number of queues, but are otherwise explicit. The equations can even be expressed in closed form for several routing schemes commonly used in industry, such as the star and elevator visit order, in a general parameter setting. We also provide a hypothesis for extending our results to non-Poisson arrivals. The results are strikingly simple and reveal a number of asymptotic properties providing new and fundamental insights in the behavior of polling systems. In addition, the results lead to simple and fast approximations for the distributions and the moments of the delay in stable polling systems with periodic server routing. Numerical results demonstrate that the approximations are highly accurate for medium and heavily loaded systems.