The lifetime of a network can be increased by increasing the network energy. The network energy can be increased either increasing the number of sensors or increasing the initial energy of some sensors without increasing their numbers. Increasing network energy by deploying extra sensors is about ten times costlier than that using some sensors of high energy. Increasing the initial energy of some sensors leads to heterogeneous nodes in the network. In this paper, we propose a multilevel heterogeneous network model that is characterized by two types of parameters: primary parameter and secondary parameters. The primary parameter decides the level of heterogeneity by assuming the values of secondary parameters. This model can describe a network up to nth level of heterogeneity (n is a finite number). We evaluate the network performance by applying the HEED, a clustering protocol, on this model, naming it as MLHEED (Multi Level HEED) protocol. For n level of heterogeneity, this protocol is denoted by MLHEED-n. The numbers of nodes of each type in any level of heterogeneity are determined by the secondary model parameter. The MLHEED protocol (for all level heterogeneity) considers two variables, i.e., residual energy and node density, for deciding the cluster heads. We also consider fuzzy implementation of the MLHEED in which four variables are used to decide the cluster heads: residual energy, node density, average energy, and distance between base station and the sensor nodes. In this work, we illustrate the network model up to seven levels (1≤n≤7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$1\le n\le 7$$\end{document}). Experimentally, as the level of heterogeneity increases, the rate of energy dissipation decreases and hence the nodes stay alive for longer time. The MLHEED-m, m=2,3,4,5,6,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$m=2,3,4,5,6,7$$\end{document}, increase the network lifetime by 73.05,143.40,213.17,267.90,348.60,419.10%\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$73.05, 143.40, 213.17, 267.90, 348.60, 419.10\,\%$$\end{document}, respectively, by increasing the network energy as 40,57,68.5,78,84,92.5%\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$40, 57, 68.5, 78, 84, 92.5\,\%$$\end{document} with respect to the original HEED protocol. In case of fuzzy implementation, the MLHEEDFL-m, m=2,3,4,5,6,7,\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$m=2,3,4,5,6,7,$$\end{document} increases the network lifetime by 282.7,378.5,435.78,498.50,582.63,629.79%\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$282.7, 378.5, 435.78, 498.50, 582.63, 629.79\,\%$$\end{document}, respectively, corresponding to the same increase in the network energy as that of the MLHEED (all levels) with respect to the original HEED. The fuzzy implementation of the HEED, MLHEEDFL-1, increases the network lifetime by 176.6%\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$176.6\,\%$$\end{document} with respect to the original HEED with no increase in the network energy.
