Mixture models for reliability of software with imperfect debugging: identifiability of parameters

A class of software-reliability mixture-type models is introduced in which individual bugs come with i.i.d. random failure-causation rates λ, and have conditional hazard function φ(t|λ) for software failure times. The models allow the possibility of imperfect debugging, in that at each failure a new bug (possibly with another rate-parameter λ) is introduced, statistically independently of the past, with probability p. For φ(t|λ)=λ, it is shown that the unknown parameters p, n₀ (the initial number of bugs), and G (the Cdf for λ) are uniquely determined from the probability law of the failure-count function (N(t), 0⩽t⩽δ), for arbitrary δ>0. The parameters (n₀,G) are also uniquely determined by the mean failure-count function E{N(t)} when p is known (e.g., is assumed to be 0), but not when p is unknown. For special parametric classes of G, the parameters (n₀,p,G) are uniquely determined by (E{N(t)}, 0⩽t⩽δ)