Monotonicity and comparability in ballistic missile defense systems

A general framework is provided for modeling Ballistic Missile Defense (BMD) systems that are driven by stochastically monotone decreasing discrete time Markov chains, not necessarily temporally homogeneous. Based on a lexicographical ordering of defense vectors, the notions of stochastic monotonicity and comparability for such systems are investigated. Then conditions under which these systems are monotone and comparable are established. For any such BMD system, it is shown that the number of missiles penetrating the defense is decreasing in the likelihood ratio, and the elements of the associated one-step nonhomogeneous transition probability matrix are totally positive of order 2. In addition, the number of missiles penetrating the defense under uniform firing doctrine is shown to be smaller in the reversed hazard rate order than the number of missiles penetrating the defense under random firing doctrine. Particular performance measures are also shown to be monotone and comparable.