## The Structure of the Model

The bivariate process (Xt, Yt) is defined on a product space ^ x R, the state transitions are described by a transition density fX'y (u,w) where fX,y (u, w)dudw = vecP [Xt G (u,u + du), Yt G (w, w + dw) |X0 = x,Y0 = y ] .

The development of this transition density is the key to adapting the general approach to particular cases.

The system stops when the basic process Xt enters a critical set, that is the system continues if Xt gG and stops at time t, the time of first entry into Gc; t = inf {t > 0 | Xt G Gc}. By using the bivariate model more possibilities are available. The times at which the system stops are defined by the excursions of the bivariate process(Xt,Yt) G 0 x R. Decisions can be made by partitioning the state space. For example define the partitions ^ = A0 U {Ui=1..mAj}UA* and R = B0U{Uj=inBi}UB* where A0 and B0 indicate perfect condition and A* and B* are critical sets. Inspection reveals the system state as (Xt,Yt) G Ai x Bj and this determines the action. With each action there is an associated cost. The cost can represent a true cost or another measure of the benefit or harm incurred as a result of the chosen action.