Dear friends, we are happy to release this 75th video of our technical channel ! In this video, Hemant Urdhwareshe explains the concepts of HPP and NHPP for repairable systems. The NHPP is foundation for Reliability Growth! Hemant is a Fellow of ASQ and is certified by ASQ as Six Sigma Master Black Belt (CMBB), Black Belt (CSSBB), Reliability Engineer (CRE), Quality Engineer (CQE) and Quality Manager (CMQ/OE).
Link to Poisson Distribution Video
This video from the Institute of Quality and Reliability clarifies the concepts of Homogeneous Poisson Process (HPP)and Non-Homogeneous Poisson Process (NHPP) in the context of reliability engineering.
Homogeneous Poisson Process (HPP)
HPP applies when a system is considered “as good as new” after repairs or replacements, and the failure rate (λ) is constant over time. This is also known as a renewal process.
- Expected Failures: The expected number of failures during time T is simply λT.
- Actual Failures: While λT is the expected number, the actual number of failures in an HPP follows a Poisson distribution. The probability of K failures in time T is given by: $- P\left(K}\right)=\frac{e^{-\lambda T}\left(\lambda T\right)^{K}}{K!} -$
- Application: HPP is widely used for modeling repairable equipment, especially during the flat, constant repair rate portion of the bathtub curve. It’s also foundational for many military standards related to system reliability and test planning.
Example: If a machine has an MTBF of 500 hours (λ=0.002 failures/hour), the expected failures in 1,000 hours are 2. The probability of exactly 4 failures in 1,000 hours for a single machine is approximately 0.09022.
Non-Homogeneous Poisson Process (NHPP)
NHPP is used when the failure rate varies with time, meaning the rate at which failures or events occur is not constant. This is common in situations where a system’s reliability is changing, such as during design and development when improvements are made after observing failures.
- Cumulative Intensity Function (M(T)): This represents the expected number of failures up to time T. It’s defined as the integral of the instantaneous intensity function, m(t), from 0 to T: $- M\left(T\right)=\intop_{0}^{T}m(t)dt. -$
- Instantaneous Intensity Function (m(t)): This is the derivative of the cumulative intensity function, m(t)=dM(T)/dT.
- Modeling NHPP: Two common models for characterizing the failure intensity function in NHPP are:
- Power Law Model: M(T)=aTb. If b<1, the system is improving (failure rate is reducing). If b>1, the system is deteriorating. The instantaneous intensity function is m(t)=abTb−1.
- Exponential Model: M(T)=aebT. The instantaneous intensity function is m(t)=abebT.
- Constants ‘a’ and ‘b’ are estimated from data.
- Inter-Arrival Times: In repairable systems, while the time to first failure might follow an exponential or Weibull distribution, the times to subsequent failures (inter-arrival times) typically do not follow a single distribution.
- Expected Failures in an Interval: The expected number of failures between time T1 and T2 is M(T2)−M(T1).
- Probability of Failures in an Interval: For relatively small intervals, if you know the expected number of failures (M(T2)−M(T1)), the probability of K failures in that interval can still be estimated using the Poisson distribution.
Example: A repairable system has a cumulative failure intensity function M(T)=0.007T1.3.
- The instantaneous failure intensity function is m(t)=0.0091T0.3.
- The expected number of failures between 50 and 100 hours is approximately 1.65.
- The probability of no failures (reliability) during the 50 to 100-hour interval is e−1.65≈0.191.
In essence, HPP deals with constant failure rates and “as good as new” repairs, while NHPP accounts for changing failure rates, often seen during product development and improvement cycles.
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