Parameter Estimation for Repairable Systems

Suppose that the number of systems under study is K and the qth system is observed continuously from time Sq to time Tq, q = 1, 2, ..., K. During the period [Sq, Tq], let Nq be the number of failures experienced by the qth system and let Xi,q be the age of this system at the ith occurrence of failure, i = 1, 2, ..., Nq. It is also possible that the times Sq and Tq may be observed failure times for the qth system. If then the data on the qth system is said to be failure terminated and Tq is a random variable with Nq fixed. If then the data on the qth system is said to be time terminated with Nq a random variable. The maximum likelihood (ML) estimates of and are values satisfying the Eqns. 4 and 5.

(4)

(5)

where 0 ln 0 is defined to be 0. In general, these equations cannot be solved explicitly for and but must be solved by iterative procedures. Once and have been estimated, the ML estimate of the intensity function is given by:

If S1 = S2 = ... = Sq = 0 and T1 = T2 = ... = Tq (q = 1, 2, ... K) then the ML estimates and are in closed form.

(6)

(7)

The following examples illustrate these estimation procedures.

Fielded Systems Example 1

For the data in Table 10.1, the starting time for each system is equal to 0 and the ending time for each system is 2000 hours. Calculate the ML estimates and .

Table 10.1 - Repairable system failure data

System 1 (Xi1)	System 2 (Xi2)	System 3 (Xi3)
1.2	1.4	0.3
55.6	35.0	32.6
72.7	46.8	33.4
111.9	65.9	241.7
121.9	181.1	396.2
303.6	712.6	444.4
326.9	1005.7	480.8
1568.4	1029.9	588.9
1913.5	1675.7	1043.9
	1787.5	1136.1
	1867.0	1288.1
		1408.1
		1439.4
		1604.8
N1 = 9	N2 = 11	N3 = 14

Solution to Fielded Systems Example 1

Since the starting time for each system is equal to zero and each system has an equivalent ending time, the general Eqns. 4 and 5 reduce to the closed form Eqns. 6 and 7. The ML estimates of and are then calculated as follows:

The system failure intensity function is then estimated by

Figure 10.2 is a plot of over the period (0, 3000). Clearly, the estimated failure intensity function is most representative over the range of the data and any extrapolation should be viewed with the usual caution.

Figure 10.2: Instantaneous Failure Intensity vs. Time plot