Technical Note—Analysis of a Layered Defense Model

Abstract

We consider a defense system model in which a cluster of attackers tries to penetrate several layers of defense. The layers of defense are independent and produce attrition in accordance with a binomial distribution—each layer having its own pk. The model is simplistic, but provides a useful approximation in various military settings, e.g., to an ICBM defense system. Questions of interest might pertain to distributions of surviving attackers at each stage, to the ordering of the defense layers, or to the sizing of the total defense. A complete analysis is displayed, based on a Markov chain formulation. It is shown that the distribution of survivors at each stage is binomial if the initial distribution is binomial. The class of transition matrices is shown to commute and to be closed under matrix multiplication. The eigenvectors of these transition matrices can be packaged into a triangular matrix whose nonzero rows are those of Pascal's triangle. The model illustrates a nonstationary Markov chain which admits a closed form analysis. Possible nonmilitary applications of such population modeling are also pointed out.