Queues on a Dynamically Evolving Graph

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.

Modelling complex stochastic systems: approaches to management and stability

“…In recent contributions, results on dynamic random graphs have been reported; see e.g. [47,48,49,50,51]. This chapter can be regarded as being among the first pieces of work to facilitate describing queueing processes on a randomly evolving graph (but it is noticed that the model we propose is substantially more general, as the multiplicative transitions are not restricted to node failures and repairs).…”