Insensitivity of the mean-field Limit of Loss Systems Under Power-of-d Routing

TL;DR

This paper proves that the mean-field limit of large loss systems under power-of-d routing is insensitive to the service time distribution, extending previous results from exponential to general distributions.

Contribution

It establishes the insensitivity of the mean-field fixed point to general service time distributions using a measure-valued Markov process approach.

Findings

The mean-field fixed point is unique and matches the exponential case.

The approach applies martingale problem techniques to prove insensitivity.

Results extend to other queueing models.

Abstract

In this paper, we study large multi-server loss models under power-of-$d$ routing scheme when service time distributions are general with finite mean. Previous works have addressed the exponential service time case when the number of servers goes to infinity giving rise to a mean field model. The fixed point of limiting mean field equations (MFE) was shown to be insensitive to the service time distribution through simulation. Showing insensitivity to general service time distributions has remained an open problem. Obtaining the MFE in this case poses a challenge due to the resulting Markov description of the system being in positive orthant as opposed to a finite chain in the exponential case. In this paper, we first obtain the MFE and then show that the MFE has a unique fixed point that coincides with the fixed point in the exponential case thus establishing insensitivity. The approach is via a measure-valued Markov process representation and the martingale problem to establish the mean-field limit. The techniques can be applied to other queueing models.