Bounds on the Lambert function and their application to the outage analysis of user cooperation

TL;DR

This paper derives bounds on the Lambert W function's branch W_{-1} and demonstrates their application in analyzing outage probabilities in user cooperation scenarios.

Contribution

It provides new tractable bounds on the W_{-1} branch of the Lambert W function and applies them to outage analysis in cooperative communication systems.

Findings

Bounds help identify conditions for improved user cooperation

Bounds simplify outage probability calculations

Application demonstrates practical utility of mathematical bounds

Abstract

Problems formulated in terms of logarithmic or exponential equations often use the Lambert $W$ function in their solutions. Expansions, approximations and bounds on $W$ have been derived in an effort to gain a better understanding of the relationship between equation parameters. In this paper, we focus on one of the branches of $W$, denoted as $W_{-1}$, we derive tractable upper and lower bounds and we illustrate their usefulness in identifying conditions under which user cooperation can yield a lower outage probability than non-cooperative transmission.