Heavy traffic analysis of roving server networks

TL;DR

This paper analyzes the heavy-traffic behavior of queueing networks with a single roving server, providing the first asymptotic results for path-time distribution under general renewal arrivals and proposing an accurate approximation for mean path-time.

Contribution

It introduces the first heavy-traffic asymptotic analysis for path-time distribution in roving server networks with renewal arrivals, extending existing conjectures to this setting.

Findings

First heavy-traffic asymptotic for path-time distribution in such networks.

Proposed highly accurate approximation for mean path-time across various parameters.

Results are exact in certain limiting cases.

Abstract

This paper studies the heavy-traffic (HT) behaviour of queueing networks with a single roving server. External customers arrive at the queues according to independent renewal processes and after completing service, a customer either leaves the system or is routed to another queue. This type of customer routing in queueing networks arises very naturally in many application areas (in production systems, computer- and communication networks, maintenance, etc.). In these networks, the single most important characteristic of the system performance is oftentimes the path time, i.e. the total time spent in the system by an arbitrary customer traversing a specific path. The current paper presents the first HT asymptotic for the path-time distribution in queueing networks with a roving server under general renewal arrivals. In particular, we provide a strong conjecture for the system's behaviour under HT extending the conjecture of Coffman et al. [E.G. Coffman Jr., A.A. Puhalskii, M.I. Reiman 1995 and 1998] to the roving server setting of the current paper. By combining this result with novel light-traffic asymptotics we derive an approximation of the mean path-time for arbitrary values of the load and renewal arrivals. This approximation is not only highly accurate for a wide range of parameter settings, but is also exact in various limiting cases.