ENGG961 Systems Reliability Engineering

Assignment 3: Reliability Modelling for Repairable System


To learn knowledge and approaches for reliability modelling of repairable systems, design for maintainability, and availability.

Learning objectives covered:

  1. Understand and apply reliability concepts and terminology.
  2. Understand and make use of the relationships amongst the different reliability functions.
  3. Collect and analyse reliability data (times to failure and times to repair) using empirical and parametric methods (exponential, Weibull, normal and lognormal are in syllabus); collect and analyse failure times of repairable systems to determine the intensity function (power law model only in-syllabus).
  4. Analyse maintainability, logistic support and the costs of reliability.
  5. Understand design techniques to achieve high reliability and analyse these for optimal reliability.
  6. Have a working knowledge of human factors in reliability and be able to apply prediction models.


  1. A machine has a time-to-first-failure distribution that is Weibull with  = 2.25 and  = 600 operating hours. Assume an average usage of the machine of 120 hours per year. (30 marks) 
  • The manufacturer of the machine offers a 10-year warranty if the customer purchases their 10year service plan. The service plan will replace all worn parts each year, thus restoring the machine to “as-good-as-new” condition. Compare the machine reliability at 10 years with and without the service plan. (7 marks)
  • The repair time distribution of the machine is lognormal with a shape parameter of 0.60 and a median repair time of 8 hours. The service centre advertises that if you bring them a broken machine, it will be repaired by the same the following workday. Find the MTTR and the percent of repairs that are completed within an 8-hour workday. (8 marks)
  • Failure of a broken machine results in minimal repair that can be described by the following intensity function (a nonhomogeneous process): (t) = 5.0  10-6 t2 with t measured in operating hours. Assume that the customer does not accept the service plan in (a).
    • What is the expected number of failures that will occur over 10 years? (5 marks)
    • What is probability of at least one failure during the tenth year? (5 marks)
    • What is the MTBF (instantaneous) at the end of the 10th year? (5 marks)
  1. An essential part of a manufacturing process is the factory assemble and integration linking (FAIL) system. Unfortunately this system has experienced numerous failures over the years under a minimal repair concept that is a nonhomogeneous Poisson process having an intensity function of (t) = 1.20 

10-6 t1.25 with t measured in operating hours. The system averages 4300 operating hours a year and is currently five years old. When the system fails, the repair time is lognormal with tmed = 6.50 hours and s = 1.25. (40 marks)

  • When the system was new, its manufacturer offered a 600-operating-hour warranty. What is the system reliability during the warranty period? (5 marks)
  • What are the expected numbers of failures through the first five years of the system? (5 marks) (c) How many failures are expected in the next two year (year 6 and year 7 of the FAIL system)? (6 marks) 
  • When the system fails, what percentage of repairs will be completed within a single 8-hour shift? (6 marks)
  • When the system fails, it takes a crew of two to repair it. If the labour rate is $65 per hour and based upon the MTTR, what is the maintenance cost of the FAIL system during its 6th year of operation? (6 marks) 
  • If the cost of the system is $25,000, what is its optimal (minimum cost) replacement time? (6 marks)
  • Management group believes that a preventive maintenance (PM) program may be cost effective. A PM consists primarily of replacing certain parts having a cost of $700 and thus restoring it to as-good-as-new condition. If it takes one technician 4 hours to do this, what is the optimum (minimum cost) PM interval? (6 marks)
  1. A company operates an equipment that is essential in the manufacture of their products. This machine has been in operation for four years (1200 operating days). In order to plan for next year’s production levels, the availability of the machine must be determined. The company works a single 8-hour shift, 300 days out of one year. Relevant failure and maintenance data collected over this period is:

Failures: Minimal repair with a power law intensity function having a =

0.000125 and b = 1.650 with time measured in operating days Repairs: Repair time is lognormal with tmed = 2.5 days and s = 0.750

Scheduled maintenance: A preventive maintenance of downtime of one day every five weeks

(30 operating days).

  • What is the mean system downtime over the first four years? (8 marks)
  • Based upon both schedule and unscheduled maintenance, determine the expected availability of the equipment over the coming year (i.e., the next 300 operating days). (12 marks) 
  • If the machine costs $29,500 to replace and a failure costs $200, when should the machine be replaced? (10 marks)

Learning Guides

In order to complete the assignment tasks, you need to read Chapter 9, 10 and 11. While you are reading, you need to understand or answer the following questions:

  • Understand system maintainability: What is MTTR, MTBF, MPMT, MTR, MDT, SDT, MTBM?
  • What is availability? How to calculate availability?
  • How to calculate MTTR given a repair time distribution?
  • What is renewal process, Homogeneous Poisson process, superimposed renewal process?
  • What is minimal repair process? What is Nonhomogeneous Poisson process? What is power law process?
  • Understand intensity functions. Be familiar with Table 9.1 on page 233 in the textbook.
  • Understand system repair time, cycle time.
  • Have a good understanding of reliability under preventive maintenance (Figure 9.3 and Figure 9.4).
  • Know the approach of applying Markov model to repairable systems (pages 241 to 246).
  • Know maintenance concepts and procedures (Figure 10.2).
  • Understand cost model: Repair versus replacement.
  • What is the difference between preventive maintenance and predictive maintenance?
  • Understand maintainability prediction and demonstration.