An Approximation of the First Order Marcum $Q$-Function with Application to Network Connectivity Analysis

TL;DR

This paper introduces an exponential approximation of the first order Marcum Q-function, facilitating network connectivity analysis in Rician fading environments with high accuracy for small and moderate arguments.

Contribution

It presents a novel exponential-type approximation of the Marcum Q-function that is robust, easily integrable, and applicable to network connectivity analysis, with parameters optimized via analytical and regression methods.

Findings

Approximation is exact for small first argument values.

Numerical results show high accuracy for moderate arguments.

Application to ad hoc network connectivity analysis demonstrates practical utility.

Abstract

An exponential-type approximation of the first order Marcum $Q$-function is presented, which is robust to changes in its first argument and can easily be integrated with respect to the second argument. Such characteristics are particularly useful in network connectivity analysis. The proposed approximation is exact in the limit of small first argument of the Marcum $Q$-function, in which case the optimal parameters can be obtained analytically. For larger values of the first argument, an optimization problem is solved, and the parameters can be accurately represented using regression analysis. Numerical results indicate that the proposed methods result in approximations very close to the actual Marcum $Q$-function for small and moderate values of the first argument. We demonstrate the accuracy of the approximation by using it to analyze the connectivity properties of random ad hoc networks operating in a Rician fading environment.