Expected Coverage of a Randomly Located Target by Multiple Independent Salvos

In models of the engagement of targets by salvos of weapons, so far mainly two sources of the variation of the relative positions of the weapon impact points and the target have been considered: the random deviation of the mean point of impact from the target center, and the dispersion of the individual rounds about the mean point of impact. New weapon systems, such as multiple-rocket launchers, add a third source of variation which generally cannot be neglected or lumped into one of the two mentioned before: the mean impact points of the individual launcher salvos scatter randomly about a common center. For the assessment of the effectiveness of such weapon systems, a measure is required that would include all three sources of variation. We derive expressions (improper integrals) for the expected destroyed value of a randomly located target, when all underlying distributions are circularly symmetric. Similar to the well-known coverage results for two-stage models, these expressions can be evaluated easily by numerical quadrature. Our coverage result is generalized to even more aggregated systems, where each rocket contains submunitions. Numerical results show that, with regard to a battery of multiple-rocket launchers, two-stage models generally are inadequate; such a model can produce an error whose size makes the result useless.