[Paper Review] Queuing Networks with Varying Topology -- A Mean-Field Approach

François Baccelli, Alexandre Rybko+1 more|arXiv (Cornell University)|Nov 15, 2013

Advanced Queuing Theory Analysis2 references3 citations

TL;DR

This paper develops a mean-field approach for queuing networks with moving servers, where nodes exchange positions dynamically, causing customer queue positions to shift relative to destinations. It establishes the convergence of the N-component mean-field network to a Non-Linear Markov Process (NLMP), proving existence and convergence of the limiting process under uniform boundedness of server swap rates.

ABSTRACT

We consider the queuing networks, which are made from servers, exchanging their positions. The customers, using the network, try to reach their destinations, which is complicated by the movements of the servers, taking their customers with them, while they wait for the service. We develop the general theory of such networks, and we establish the convergence of the symmetrized version of the network to the Non-Linear Markov Process.

Motivation & Objective

• To develop a general qualitative theory for queuing networks where servers move on a graph, altering customer queue positions relative to destinations.
• To analyze instability phenomena arising from server mobility, particularly in networks with fast servers and low load that become unstable when nodes move.
• To establish the convergence of the symmetrized N-component mean-field version of the network to a Non-Linear Markov Process (NLMP) as N → ∞.
• To lay the theoretical groundwork for analyzing ergodic properties of the NLMP, which determine network stability or instability.
• To provide a rigorous mathematical framework using Frechet differentiability and semigroup theory for the NLMP limit.

Proposed method

• Model the network as a graph G with nodes hosting single servers, each with a queue of customers destined for specific nodes.
• Introduce stochastic server swaps between neighboring nodes at bounded rates βvv′, with queues moving with servers.
• Define the N-component mean-field network by coupling N identical copies of the network to study macroscopic behavior.
• Formulate the evolution of the system using a measure-valued process μN(t), representing the empirical distribution of queue states.
• Prove convergence of μN(t) to a deterministic limit μ(t) as N → ∞, where μ(t) satisfies a non-linear Fokker-Planck-type equation.
• Apply Frechet differentiation and semigroup theory to show the generator of the limiting process is well-defined and the solution is unique via Gronwall-type estimates.

Experimental results

Research questions

• RQ1 How does server mobility induce instability in queuing networks that are stable under static node conditions?
• RQ2 What is the limiting behavior of a large-scale network of interacting queuing systems with moving servers?
• RQ3 Under what conditions does the mean-field approximation converge to a Non-Linear Markov Process (NLMP) in the limit of large N?
• RQ4 How can the ergodic properties of the NLMP be characterized to determine long-term stability of the network?
• RQ5 What mathematical tools are required to rigorously analyze the convergence and differentiability of the limiting process in such non-linear, spatially extended systems?

Key findings

• The N-component mean-field network converges weakly to a Non-Linear Markov Process (NLMP) as N → ∞, establishing a functional law of large numbers for the system.
• The limiting process μ(t) is shown to exist and be unique, with the solution satisfying a non-linear evolution equation derived from the generator of the semigroup.
• The Frechet differential of the evolution map is well-defined, and the set of uniformly differentiable functions forms a core for the generator, ensuring regularity of the limit.
• The remainder term ζ(t) in the perturbation expansion is bounded by ||h||² and decays uniformly for small t, confirming the validity of the mean-field approximation.
• The convergence is established via Gronwall-type estimates and the application of the Peano theorem for existence and uniqueness of solutions to the limiting ODE system.
• The framework generalizes to sequences of finite graphs Hn converging to an infinite graph G, allowing extension to infinite network topologies.