On Pollaczek-Khinchine Formula for Peer-to-Peer Networks

TL;DR

This paper develops a new analytical approach for estimating mean queue length in P2P networks with variable service rates, extending the Pollaczek-Khinchine formula through bounds and simulations.

Contribution

It introduces a limiting analysis method for P2P queueing models, deriving an approximate P-K formula and validating it with extensive simulations.

Findings

Derived bounds for mean queue length in P2P networks.

Proposed an approximate P-K formula based on bounds.

Validated the approximation through simulation studies.

Abstract

The performance analysis of peer-to-peer (P2P) networks calls for a new kind of queueing model, in which jobs and service stations arrive randomly. Except in some simple special cases, in general, the queueing model with varying service rate is mathematically intractable. Motivated by the P-K formula for M/G/1 queue, we developed a limiting analysis approach based on the connection between the fluctuation of service rate and the mean queue length. Considering the two extreme service rates, we proved the conjecture on the lower bound and upper bound of mean queue length previously postulated. Furthermore, an approximate P-K formula to estimate the mean queue length is derived from the convex combination of these two bounds and the conditional mean queue length under the overload condition. We confirmed the accuracy of our approximation by extensive simulation studies with different system parameters. We also verified that all limiting cases of the system behavior are consistent with the predictions of our formula.