Self-averaging property of queuing systems

We establish the averaging property for a queuing process with one server, M(t)/GI/1. It is a new relation between the output flow rate and the input flow rate, crucial in the study of the Poisson Hypothesis. Its implications include the statement that the output flow always possesses more regularity than the input flow.Comment: 18 pages, one typo remove

Poisson Hypothesis for Information Networks (A study in non-linear Markov processes) I. Domain of Validity

In this paper we study the Poisson Hypothesis, which is a device to analyze approximately the behavior of large queueing networks. We prove it in some simple limiting cases. We show in particular that the corresponding dynamical system, defined by the non-linear Markov process, has a line of fixed points which are global attractors. To do this we derive the corresponding non-linear equation and we explore its self-averaging properties. We also argue that in cases of havy-tail service times the PH can be violated.Comment: 77 page

Poisson Hypothesis for information networks (A study in non-linear Markov processes)

In this paper we prove the Poisson Hypothesis for the limiting behavior of the large queueing systems in some simple ("mean-field") cases. We show in particular that the corresponding dynamical systems, defined by the non-linear Markov processes, have a line of fixed points which are global attractors. To do this we derive the corresponding non-linear integral equation and we explore its self-averaging properties. Our derivation relies on a solution of a combinatorial problem of rode placements.Comment: 70 page

On the rod placement theorem of Rybko and Shlosman 1

Given n − 1 points x1 ≤ x2 ≤... ≤ xn−1 on the real line and a set of n rods of strictly positive lengths λ1 ≤ λ2 ≤... ≤ λn, we get to choose an n-th point xn anywhere on the real line and to assign the rods to the points according to an arbitrary permutation π. The rod λπ(k) is thought of as the workload brought in by a customer arriving at time xk into a first in-first out queue which starts empty at −∞. If any xi equals xj for i < j, service is provided to the rod assigned to xi before the rod assigned to xj. Let Yπ(xn) denote the set of departure times of the customers (rods). Let Nπ(x1,...,xn−1; λ1,...,λn) denote the number of choices for the location of xn for which 0 ∈ Yπ(xn). Rybko and Shlosman proved that Nπ(x1,...,xn−1; λ1,...,λn) = n! π for Lebesgue almost all (x1,...,xn−1; λ1,...,λn). Let yπ,k(xn) denote the departure point of the rod λk. Let Nπ,k(y) de-note the number of choices for the location of xn for which yπ,k(xn) = y and let Nk(y) = � π Nπ,k(y). In this paper we prove that for every (x1,...,xn−1; λ1,...,λn) and every k we have Nk(y) = (n − 1)! for all but finitely many y. This implies (and strengthens) the rod placement theore