## Inspection Policies

The decision maker inspects the system according to a policy n. The policy is a list of inspection epochs n = {ri, T2,...,Tn} and we assume for the moment that inspection is perfect. The decision chooses an action determined by the system state (Xt,Yt) which may change the system state. The actions are assumed to be instantaneous. We consider two cases, a renewal approach where the system is returned to its original state at each intervention, and one where the intervention returns to system to an arbitrary state [SZ91, SZ92]. In the renewal approach if the revealed state on inspection, (Xt,Yt), falls in an interval Ai x Bj, the system is completely restored to the original state with cost Cj,^. In the context of this chapter the renewal approach corresponds to changing, starting, or stopping a treatment and restoring the system to its initial state; the restoration to an arbitrary state corresponds to bringing the system to some state between the current state and the initial state. The system found with state (Xt-,Yt-) = (x,y) is restored to the state (Xt+ ,Yt+) = (x',y') with (x',y') = D(x,y). The function D describes the decision maker's action. Usually the new state lies somewhere between the original state and the present state so that 0 < \x'\ < |x|. In most cases y' = 0 because the decision variable will be reset; in the case when Yt is the maximum y' = x'. Clearly x' = 0 corresponds to restoration to the initial state and x' = x implies no change. The planned inspection period ends normally with the planned inspection or is terminated by the entry into a critical set.

The hitting time of the critical set, starting from (Xt, Yt) = (x, y), is Tx,y. Candidates for the hitting times are:

T^r =inf {t \ (Xt, Yt) G A* x B*}; TX,y = inf {t \ Yt G B*}; TX,y = inf {t \ Xt G A*}.

where A* and B* are critical sets. We shall write the hitting distribution starting from (x,y) as Gx'y (t) and assume it possesses a density gx'y (t). The state probabilities are pxj = vecP [(Xt,Yt) G Ai x Bj ] = vecE [veclKxt,Yt)eAiXBj}] = J J fxy (u,w)dudv

Ai Bj and the hitting time distribution pFy = Gx'y (t) .