Software Engineering Littlewood and Verall’s Model

Littlewood and Verrall’s Model is a software reliability model that was proposed by Littlewood and Verrall in the 1990s. It is an extension of the Jelinski-Moranda (J-M) model and is also known as the J-M/L-V model. It estimates the reliability of software systems.

Assumptions:

The assumptions in this model include the following:

1. Failures are independent and identically distributed
2. Failure rates over time due to changes in patterns and software environment
3. The failure rate function λ(t) is non-decreasing and non-negative over time.
4. The time intervals between failures are distributed exponentially.
5. The size of the software system under test is known and has remained constant over time.

The formula for the software reliability function in Littlewood and Verall’s Model is given by,

F(t) = λ(t) * S(t)

where:

F(t) = number of failures up to time t.
S(t) = size of the software system being tested up to time t.
λ(t) = time-varying failure rate.

Partial distribution function (pdf) of ti: f(ti) = λ1 * (1 + (i-1) * λ2)^(-1) * e^(-λ1 * (1 + (i-1) * λ2)^(-1) * ti)

Cumulative distribution function (CDF) of ti: F(ti) = 1 – e^(-λ1 * (1 + (i-1) * λ2)^(-1) * ti)

Where:

1. λ1 is the initial failure rate
2. λ2 is the failure rate increase factor
3. ti is the time between the (i-1)th and the (i)th failures

Note that these formulas are based on the assumptions of the Littlewood and Verall Model, which include that the program contains a fixed number of faults, the failure rate increases with each failure detected, and the time intervals between failures follow an exponential distribution.

Comparison between Littlewood and Verall’s Model and the Jelinski-Moranda (J-M) Model

Littlewood and Verall’s Model	Jelinski-Moranda (J-M) Model
To predict the expected number of faults remaining  in a software system at any given time	To predict the program failure rate at each interval of time
The faults in the software are corrected at a constant rate and  the rate of correction is proportional to the number of faults present	There are N initial faults in the program and each fault is  independent and equally likely to cause failure. The time intervals between failures are independent of each  other and whenever a failure occurs, a corresponding fault is removed
R(t) = R(0) * e^(-lambda * t)	λ(ti) = φ [N-(i-1)]

It can be seen that Littlewood and Verall’s Model is focused on the number of faults remaining in the system, while the J-M model is focused on the program failure rate at each interval of time.

In general, Littlewood and Verrall’s Model is a more sophisticated and realistic approach to modeling software reliability than the J-M model. It is a useful tool for predicting software reliability in real-world applications because it accounts for variable failure rates.