Estimation of derivatives of nonsmooth performance measures in regenerative systems

We investigate the problem of estimating derivatives of expected steady-state performance measures in parametric systems. Unlike most of the existing work in the area, we allow those functions to be nonsmooth and study the estimation of directional derivatives. For the class of regenerative Markovian systems we provide conditions under which we can obtain consistent estimators of those directional derivatives. An example illustrates that the conditions imposed must be different from those in the differentiable case. The result also allows us to derive necessary and sufficient conditions for differentiability of the expected steady-state function. We then analyze the process formed by the subdifferentials of the original process, and show that the subdifferential set of the expected steady-state function can be expressed as an average of integrals of multifunctions, which is the approach commonly found in the literature for integrals of sets. The latter result can also be viewed as a limit theorem for more general compact-convex multivalued processes.