The Moranda Geometric Model
Also called the De-Eutrophication model, Moranda's Geometric model relaxes one of Jelinski-Moranda's most restrictive assumptions: instead of requiring a fixed, known total fault count, it lets the hazard rate decrease by a constant ratio after each failure — allowing for an effectively open-ended fault population.
A geometric relaxation of the Jelinski-Moranda model
Jelinski-Moranda requires assuming a fixed total number of faults N, decreasing the hazard rate by an equal arithmetic step after each fix. Moranda's Geometric model instead assumes each fault has a hazard-rate contribution that's a constant fraction of the previous one — a geometric, not arithmetic, decline — which sidesteps the need to know or assume a fixed inherent defects up front.
Times-Between-Failures, Geometric Decline
A close relative of Jelinski-Moranda that replaces the fixed-N, arithmetic-decline assumption with a geometric decline.
1975 — Paul Moranda
Developed as a direct relaxation of the Jelinski-Moranda model's most restrictive assumption.
Not one of the recommended models
A standard alternative to Jelinski-Moranda when a fixed total fault count can't be reasonably assumed.
The IEEE 1633 Working Group members were surveyed with regards to their experiences with all software reliability growth models. No one model works for every situation, but this one had no clear success stories.
Requs Trend does not include this model as it focuses on the IEEE 1633 recommended models that have been shown to have value.
How the Moranda Geometric Model works
Rather than stepping down by a fixed arithmetic amount after each fix, the hazard rate here decreases by a constant multiplicative ratio, producing a geometric sequence of failure rates.
Hazard Function Between Failures
Each successive hazard rate is φ times the previous one, so the sequence of rates forms a geometric progression rather than the arithmetic (equal-step) progression used by Jelinski-Moranda.
- Z(tᵢ) — the (constant) hazard rate during the interval before the ith failure
- D — the initial hazard rate at the start of testing
- φ — the geometric decay ratio (0 < φ < 1); each fix multiplies the hazard rate by φ
- i — the index of the current failure interval
Mean Value Function
Because there's no fixed N, the summation runs conceptually over an unbounded set of possible faults, each with its own small geometric detection probability p_a.
- μ(t) — expected cumulative number of failures by time t
- The sum runs over the (conceptually unbounded) population of individual faults
- p_a — the per-fault detection probability for fault a, following the geometric sequence set by D and φ
Key Assumptions
The model relaxes the fixed-N assumption but keeps the rest of the Jelinski-Moranda structure intact.
- The hazard rate decreases by a constant geometric ratio φ after each fix, rather than a fixed arithmetic step
- No fixed, pre-specified total fault count N is required
- Failures are independent, and inter-failure times are exponentially distributed within each interval
- Fault repair is instantaneous and perfect — no new faults introduced
Parameter Estimation
D and φ are estimated using maximum likelihood estimation over the sequence of observed inter-failure times, similar in approach to Jelinski-Moranda but fitting a geometric rather than arithmetic decline pattern.
- D — the initial hazard rate, from the earliest observed inter-failure times
- φ — the geometric decay ratio, from how quickly successive intervals lengthen relative to each other
Strengths, limitations, and best fit for the Moranda Geometric Model
Doesn't require assuming a fixed fault count
Useful when there's genuinely no reasonable estimate of total remaining faults, unlike Jelinski-Moranda which requires exactly that assumption.
Geometric decline can be too aggressive
If the true hazard rate doesn't actually decay geometrically, the model can produce overly optimistic reliability projections late in testing.
Needs individual failure timestamps
Like Jelinski-Moranda, it needs the actual time of each individual failure rather than grouped interval counts.
See some more realistic models
Start with the Requs Trend demo or a discussion of your current reliability growth modeling approach.