Statistics for Maintenance Modelling: Renewal Processes

# Question on Discrete Renewal Processes¤

A long-life light bulb has the following properties:

- probabilities of failure in years 1 through 3 of use are p1 = 0.1, p2 = 0.4, p3 = 0.5
- cost cb of block (scheduled) replacement is £2
- cost cf of failure replacement is £3.

Solve the discrete renewal equation for years n = 1 to n = 10 and find the optimal cost block replacement period.

| Optimal time to replace at n =

Yes, that's right. F1 = p1 = 0.1, F2 = p1 + p2 = 0.5, F3 = p1 + p2 + p3 = 1. Since by year 3 the bulb is certain to fail, F3 = F4 = F5 etc.

The recursive calculation of the renewal function gives:

M0 = 0, M1 = F1 = 0.1, M2 = F2 + p1M1 = 0.5 + 0.1 0.1 = 0.51, M3 = F3 + p1M2 + p2M1 = 1 + 0.1 0.51 + 0.4 0.1 = 1.091,
M4 = F4 + p1M3 + p2M2 + p3M1 = 1 + 0.1 1.091 + 0.4 0.51 + 0.5 0.1 = 1.3631 etc.

The costs per unit time are then:

Year 1 = (cb + cf M0) = 3 0 + 2 = 2, Year 2 = (cb + cf M1)/2 = (3 0.1 + 2)/2 = 1.15, Year 3 = (cb + cf M2)/3 = (3 0.51 + 2)/3 = 1.1767, Year 4 = 1.3182, Year 5 = 1.2179, Year 6 = 1.2472, Year 7 = 1.2601, Year 8 = 1.2400, Year 9 = 1.2513 and Year 10 = 1.2504. Hence you can see that year 2 produces the minimum replacement cost figure.

No, the optimal replacement time is 2. The distribution functions are F1 = p1 = 0.1, F2 = p1 + p2 = 0.5, F3 = p1 + p2 + p3 = 1. Since by year 3 the bulb is certain to fail, F3 = F4 = F5 etc.
The recursive calculation of the renewal function gives:
M0 = 0, M1 = F1 = 0.1, M2 = F2 + p1M1 = 0.5 + 0.1 0.1 = 0.51, M3 = F3 + p1M2 + p2M1 = 1 + 0.1 0.51 + 0.4 0.1 = 1.091,
M4 = F4 + p1M3 + p2M2 + p3M1 = 1 + 0.1 1.091 + 0.4 0.51 + 0.5 0.1 = 1.3631 etc.
The costs per unit time are then:>
Year 1 = (cb + cf M0) = 3 0 + 2 = 2, Year 2 = (cb + cf M1)/2 = (3 0.1 + 2)/2 = 1.15, Year 3 = (cb + cf M2)/3 = (3 0.51 + 2)/3 = 1.1767, Year 4 = 1.3182, Year 5 = 1.2179, Year 6 = 1.2472, Year 7 = 1.2601, Year 8 = 1.2400, Year 9 = 1.2513 and Year 10 = 1.2504. Hence you can see that year 2 produces the minimum replacement cost figure.