Musa-Okumoto Logarithmic Poisson Execution Time Model

Introduction

• Predicts expected failures.

• Incorporates both execution and calendar times.

• Model Innovation Intention: Simple model with high predictive validity.

• Failure process in a non homogeneous Poisson process.

Introduction Contd.

• It has been observed that reductions in failure rate resulting from repair action following early failures are often greater because they tend to be the most frequently-occurring ones, and this property has been incorporated in the model.

Execution Time Component

• Model defined in terms of a random process {M(τ), τ ≥ 0} – representing number of failures experienced by execution time τ.

• Counting process – characterized by distribution of M(τ), mean value function μ(τ) and failure intensity function λ(τ).

Model Assumptions

• Assumption 1: There is no failure observed at time τ = 0, i.e., M(0) = 0 with probability one.

• Assumption 2: The failure intensity will decrease exponentially with the expected number of failures experienced, i.e.

Model Assumptions Contd.

• Many other models postulate equal reduction in failure intensity as each failure is experienced and fixed (e.g. Jelinski-Moranda).

• In this model, the repair of early failures reduces the failure intensity more than later ones, as expressed by Assumption 2.

Model Assumptions Contd.

• Assumption 3: For a small time interval Δτ, the probabilities of one and more than one failure during (τ, τ + Δτ] respectively are : λ(τ)Δτ + o(Δτ) and

o(Δτ) respectively.

where o(Δτ) / Δτ -> 0 as Δτ -> 0

• Probability of no failure during (τ, τ + Δτ] is given by 1 – (λ(τ)Δτ + o(Δτ)).

Derivation - Mean Value Function and Failure Intensity Function

Failures Experienced

• The failures experienced by time τ, M(τ), is a random quantity. Using assumptions 1 & 3,

– This is a Poisson distribution with a mean and variance of μ(τ).

Failures Experienced Contd.

• Suppose that me failures have been observed during (0, τe]. Since the Poisson process {M(τ), τ ≥ 0} has independent increments, the conditional distribution of M(τ) given M(τe)=me for τ > τe is the distribution of number of failures during (τe , τ] i.e. for m ≥ me

Failure Time, Time Between Failures

• Failure Time, Time Between Failures – Such quantities help a project manager to predict the time it takes to experience a certain number of failures and probability of failure free operation for a certain amount of time.

Failure Time, Time Between Failures

T2’T1

’ T3’

T1

T2

T3

Failure Time, Time Between Failures

• Let Ti’(i=1,2,…) be a random variable

representing the ith failure interval and define Ti (i=1,2,…) as a random variable representing time to ith failure.

Failure Time, Time Between Failures

Event E1: There are atleast i failures experienced by time τ.

Event E2: Time to the ith failure is atleast τ.

E1 and E2 are equivalent.

**Note: This equivalence is defined so as to use previously derived quantities in current derivation.

Failure Time, Time Between Failures

• Hence, the c.d.f of Ti can be obtained as …

Failure Time, Time Between Failures

• Further, the conditional c.d.f of Ti given that M(τe) = me , where i > me is derived as,

Reliability

• The conditional reliability of Ti’ on the last

failure time Ti-1 = τi-1 can be obtained as:

Reliability

• The second term in previous expression, is the sum of Poisson probabilities, except for one term. Thus :

Calendar Time Component

• To relate execution time τ to calendar time t.

• Mostly applied during system test phase of a project.

Assumptions

• Assumption 4: The pace of testing at any time is constrained by one of three limiting resources: failure-identification (test team)personnel (I), failure-correction (original designer) personnel (F), or computer time (C).

• In most projects during test, there will be from one to three periods, each characterized by a different limiting resource.

Assumptions

• Let dtI / dτ, dtF / dτ, and dtC / dτ represent the instantaneous calendar time to execution time ratios that result from the effects of each of the resource constraints taken alone, respectively. Then, the assumption can be written as:

Assumptions

• Assumption 5: The rate of resource expenditures with respect to execution time dχk/dτ can be approximated by:

• Θk is an execution time coefficient of resource expenditure and μk is a failure coefficient of resource expenditure.

• Resource Expenditure: E.g. Person-hours

Assumptions

• Assumption 6: The quantities of the available resources are constant for the remainder of the test period. The maximum utilization of each of the available resources is also constant. Therefore, if we denote by Pk and ρk(k = I,F,C) the fixed available quantity and the utilization factor, respectively, of resource k, then the effective available quantity of resource k is (Pk ρk)

Derivation

• To derive a relationship between calendar time and execution time.

Derivation Contd.

• Since effective available resource is Pk ρk(Assumption 6), the rate of calendar time with respect to execution time associated with each resource is given by:

Derivation Contd.

• From assumption 4, we can obtain the instantaneous calendar time to execution time ratio as:

References

• J. D. Musa , K. Okumoto, A logarithmic poisson execution time model for software reliability measurement, Proceedings of the 7th international conference on Software engineering, p.230-238, March 26-29, 1984, Orlando, Florida, United States.