Control limit policies in a replacement model with additive phase-type distributed damage and linear restoration

We consider a replacement model for an additive damage process with linear restoration. The replacement policy is of the control limit type. Between replacements the damage level process is modeled as a compound Poisson process with positive jumps and negative drift, reflected at 0. We introduce meaningful cost functionals for this system and determine them explicitly for three special cases when the jumps have a phase-type distribution.