Musa-Okumoto Logarithmic Poisson Model
Unlike finite-failure models, the Musa-Okumoto Logarithmic Poisson Execution Time Model assumes the total number of faults is effectively unbounded, and that earlier-discovered faults have a disproportionately larger impact on failure intensity than later ones. It's the go-to model when failure data doesn't level off the way a finite-failure model expects.
A model for software that never quite runs out of faults
The Logarithmic Poisson model drops the finite-fault assumption entirely. Instead, it assumes the software has an effectively infinite population of faults with unequal severity — early faults are found and fixed quickly and have an outsized effect on reducing failure intensity, while later faults matter progressively less, producing a logarithmic (rather than exponential) growth curve.
Infinite-Failure NHPP
Total fault count is not assumed finite — the mean value function grows without bound, just very slowly over time.
1984 — Musa & Okumoto
Developed as a companion to the Basic Execution Time model for projects where the finite-failure assumption doesn't hold.
Covered — Logarithmic
This model is one of the five named model families in Requs Trend's ensemble averaging.
How the Logarithmic Poisson Model works
The defining trait of this model is that failure intensity decreases as a function of the number of failures already observed, not directly as a function of time — producing a logarithmic mean value function.
Mean Value Function
Unlike the Basic model, μ(τ) here has no upper asymptote — it keeps growing, just increasingly slowly, reflecting the unbounded-fault assumption.
- μ(τ) — expected cumulative number of failures by execution time τ
- λ₀ — initial failure intensity at τ = 0
- θ — the failure intensity decay parameter
Failure Intensity Function
Intensity decreases as a reciprocal function of τ rather than exponentially, so it falls off more slowly in absolute terms as testing continues.
- λ(τ) — instantaneous failure intensity at execution time τ
- λ₀ and θ — same parameters as the mean value function
- Failure intensity decreases by a constant factor each time a fixed number of additional failures occurs
Key Assumptions
This model relaxes the finite-fault assumption but adds an assumption about how fault severity is distributed.
- The total number of faults is effectively unbounded (infinite-failure category)
- Faults discovered earlier have a proportionally larger impact on failure intensity than faults discovered later
- Failures occur according to a Poisson process with logarithmically decreasing intensity
- Fault repair is instantaneous and does not introduce new faults
Parameter Estimation
λ₀ and θ are estimated with maximum likelihood estimation against observed failure times or grouped counts. Because the model has no upper asymptote, fitting focuses on matching the curvature of the logarithmic growth rather than an eventual plateau.
- λ₀ — fit from the observed initial failure intensity
- θ — fit from how quickly that intensity decays as failures accumulate
- Model fit is typically compared against finite-failure alternatives (like Musa Basic or Goel-Okumoto) using AIC or PSSE
Strengths, limitations, and best fit for the Logarithmic Poisson Model
Fits data that never plateaus
Handles large, long-lived, or continuously evolving codebases where a finite-failure model would systematically underestimate remaining risk.
No natural stopping point
Because the mean value function is unbounded, the model can't directly answer "how many total faults remain" the way a finite-failure model can.
Large or continuously growing systems
Best suited for waterfall development.
Included in the ensemble
Listed directly among Requs Trend's five named model families, alongside Exponential, NHPP, Rayleigh, and Weibull.
See this model fit against real project data.
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