Steady-State Convergence of the Continuous-Time Routing System with General Distributions in Heavy Traffic

TL;DR

This paper proves that in a continuous-time routing system with general distributions operating under specific load balancing policies, the scaled steady-state queue lengths converge to an exponential distribution in heavy traffic, under weaker assumptions than usual.

Contribution

It establishes convergence of steady-state queue lengths to an exponential distribution under minimal moment conditions using the Palm BAR technique.

Findings

Steady-state queue lengths converge to exponential distribution in heavy traffic.

Results hold under the $(2 + oldsymbol{ ext{ extdelta}}_0)$th moment assumption.

The proof introduces a novel use of the Palm version of the basic adjoint relationship.

Abstract

This paper examines a continuous-time routing system with general interarrival and service time distributions, operating under the join-the-shortest-queue and power-of-two-choices policies. Under a weaker set of assumptions than those commonly found in the literature, we prove that the scaled steady-state queue length at each station converges weakly to an identical exponential random variable in heavy traffic. Specifically, our results hold under the assumption of the $(2 + \delta_0)$th moment for the interarrival and service distributions with some $\delta_0 > 0$. The proof leverages the Palm version of the basic adjoint relationship (BAR) as a key technique.