## The Inspection Cycle

The inspection and actions are assumed to occur at the beginning of each interval. This choice allows linking of the chain of decisions required in the dynamic programming solutions later.

### 4.1 System Renewal

The policies are determined by the intervals between inspections. We consider first the the simplest case in which after inspection or entry into a critical set the system is restored to the original state. Consider a single cycle where the policy is the time to the next inspection, t. The decision makers actions are:

1. do nothing if the system is in a "good" state, (Xt,Yt) G A0 x B0;

2. the system state is Ai x Bj, the probability of this state is pi,j, restoration to the initial state costs Ci,j;

3. the system enters the critical state with probability pp-y and is restored with cost Cp.

The expected cost of the planned and unplanned actions is cT(x, y) = y Ci,jpx'y + Cppxpy . i, j

Considering the whole cycle, if the total cost starting in state (x, y) is VTX'y then vecE [VX'y] = vecE [Vf'yvec1{xt,yt)eA0,B0}] + <T(x,y)

where the term vecE \VTX'yvecl{xt,Yi)£A0xB0}] arises because the system state is left unchanged when the system is found in the "good" state A0 x B0. Writing vT (x,y) = vecE [Vf'11 ] it is clear that vT (x,y) = cT (x,y)+ J j vT (u,w)fX'y (u,w)dudw .

A0 B0

### 4.2 Arbitrary Restoration

The state space is subdivided more simply, there are now only two states, "non-critical" and "critical" and the decision maker acts to change the state to D(x,y) on finding the system in state (x,y), D(x,y) ^ (x',y'). The cost of this repair is c(D(x, y)).

Using a similar argument to above, it is clear that

Vt(x, y) =c(D(x, y)) + {vt(0, 0) + Cp}pD(x'y) ■ ■ ■

A0 B0

where v (0, 0) arises from the restoration to the initial state.