Queues on a dynamically evolving graph

TL;DR

This paper models a queueing network on a dynamically evolving graph with unreliable links, deriving differential equations, algorithms for moments, and diffusion limits to analyze performance under link outages.

Contribution

It introduces a mathematical framework with differential equations and algorithms for analyzing queueing networks with unreliable, evolving links, including explicit results for special cases.

Findings

Derived coupled PDEs for joint queue distributions

Developed algorithms for moments and performance measures

Established diffusion limits for the queue length process

Abstract

This paper considers a population process on a dynamically evolving graph, which can be alternatively interpreted as a queueing network. The queues are of infinite-server type, entailing that at each node all customers present are served in parallel. The links that connect the queues have the special feature that they are unreliable, in the sense that their status alternates between 'up' and 'down'. If a link between two nodes is down, with a fixed probability each of the clients attempting to use that link is lost; otherwise the client remains at the origin node and reattempts using the link (and jumps to the destination node when it finds the link restored). For these networks we present the following results: (a) a system of coupled partial differential equations that describes the joint probability generating function corresponding to the queues' time-dependent behavior (and a system of ordinary differential equations for its stationary counterpart), (b) an algorithm to evaluate the (time-dependent and stationary) moments, and procedures to compute user-perceived performance measures which facilitate the quantification of the impact of the links' outages, (c) a diffusion limit for the joint queue length process.We include explicit results for a series relevant special cases, such as tandem networks and symmetric fully connected networks.