Continuity of large closed queueing networks with bottlenecks

TL;DR

This paper analyzes the behavior of large closed queueing networks with a bottleneck station, demonstrating the continuity of queue lengths in non-bottleneck stations under certain service time conditions.

Contribution

It establishes the continuity of queue-length processes in non-bottleneck stations when the hub's service times are close to exponential, extending understanding of network stability.

Findings

Queue-length processes are continuous in non-bottleneck stations.

Continuity holds when hub service times are close to exponential.

Results apply to large systems with a bottleneck station.

Abstract

This paper studies a closed queueing network containing a hub (a state dependent queueing system with service depending on the number of units residing here) and $k$ satellite stations, which are $GI/M/1$ queueing systems. The number of units in the system, $N$, is assumed to be large. After service completion in the hub, a unit visits a satellite station $j$, $1\leq j\leq k$, with probability $p_j$, and, after the service completion there, returns to the hub. The parameters of service times in the satellite stations and in the hub are proportional to $\frac{1}{N}$. One of the satellite stations is assumed to be a bottleneck station, while others are non-bottleneck. The paper establishes the continuity of the queue-length processes in non-bottleneck satellite stations of the network when the service times in the hub are close in certain sense (exactly defined in the paper) to the exponential distribution.