Towards a unifying theory on branching-type polling systems in heavy traffic

Abstract

For a broad class of polling models the evolution of the system at specific embedded polling instants is known to constitute a multi-type branching process (MTBP) with immigration. In this paper it is shown that for this class of polling models the vector that describes the state of the system at these polling instants, say X = (X 1 , . . . , X M ), satisfies the following heavy-traffic behavior (under mild assumptions):where γ is a known M-dimensional vector, (α, μ) has a gamma-distribution with known parameters α and μ, and where ρ is the load of the system. This general and powerful result is shown to lead to exact-and in many cases even closed-form-expressions for the Laplace-Stieltjes Transform (LST) of the complete asymptotic queue-length and waiting-time distributions for a broad class of branchingtype polling models that includes many well-studied polling models policies as special cases. The results generalize and unify many known results on the waiting times in polling systems in heavy traffic, and moreover, lead to new exact results for classical polling models that have not been Part of this research has been funded by the Dutch BSIK/BRICKS project.observed before. To demonstrate the usefulness of the results, we derive closed-form expressions for the LST of the waiting-time distributions for models with cyclic globallygated polling regimes, and for cyclic polling models with general branching-type service policies. As a by-product, our results lead to a number of asymptotic insensitivity properties, providing new fundamental insights in the behavior of polling models.

Iterative approximation of k-limited polling systems

“…However, in the present paragraph the system reaches saturation due to an increase in the traffic intensity, whereas in the previous subsection the system got saturated mainly due to the magnitude of the setup times. The difference between these two regimes is enormous, which can be observed by comparing the rigorously proven results in [33] and [49] for the complete class of branching-type policies under Poisson arrival processes. That is, [33] studies the system under an increase of the traffic intensity, which shows that a diffusion limit applies and that the gamma distribution is prevalent, for example, in the scaled cycle lengths and the marginal queue lengths at polling instants.…”

“…The difference between these two regimes is enormous, which can be observed by comparing the rigorously proven results in [33] and [49] for the complete class of branching-type policies under Poisson arrival processes. That is, [33] studies the system under an increase of the traffic intensity, which shows that a diffusion limit applies and that the gamma distribution is prevalent, for example, in the scaled cycle lengths and the marginal queue lengths at polling instants. In contrast, [49] analyzes the effect of an increase of the setup times obtaining a fluid limit with a central role for the deterministic distribution revealing itself again, e.g., in the scaled cycle lengths and the marginal queue lengths at polling instants.…”

Cyclic-type Polling Models with Preparation Times

“…For example, heavy traffic approximations for polling systems seem to be mainly suitable for the study of characteristics of the customers, as is typical in the polling literature, rather than the server, as is our case here; cf. (van der Mei, 2007). Here, we assume that there are always waiting customers in front of each station.…”

Multitype branching processes with immigration in random environment, and polling systems

“…Branching processes with and without immigration are powerful tools in studying various models of queueing systems (see, for instance, [25], [26], [27], [28], [35], [36], [41] and [43]). In this paper we use multitype branching processes with accumulation of a final product and immigration which evolve in random environment (MBPFPIRE) to study the tail distribution of busy periods of a class of polling systems in which input parameters, service disciplines and the distributions of switch-over times vary in a random manner.…”