An analysis of estimate bias of steady-state availability for various test plans

Any process of technical product development may involve dependability testing. If in the course of operation, the recovery of an entity after a failure is the norm, then test plans of types NRect, NRecR, NNoRect и NNoRecR are normally used, where N is the number of tested same-type entities; t is the testing time of each of the N entities; R is the number of failures; Rec (NoRec) is the characteristic of the plan that indicates that the entity’s operability after each failure within the testing time is recovered (not recovered). Normally, NRect and NRecR indicate that, in the process of testing, failures are recovered immediately. In order not to confuse plans NRect, NRecR, NNoRect and NNoRecR with test plans with long recovery times, let us denote the latter as NRec!t, NRec!R, NNoRec!t and NNoRec!R respectively. Let us simplify the problem description and require, for test plans of types NRec!t, NRec!R, NNoRec!t and NNoRec!R, the fulfilment of condition D = R, where D is the number of recoveries, i.e. after the conclusion of testing, at the moment of time t, the recovery of entities continues until the last of R failed entities is recovered. We will denote such test plans NRec!t(D=R), NRec!R(D=R), NNoRec!t(D=R) and NNoRec!R(D=R). As the dependability model, an exponential distribution is adopted. Steady-state availability is normally defined as the composite dependability indicator of recoverable entities. Finding efficient estimates is one of the primary goals of the dependability theory. Since the 1960s, Russian scientific literature has featured next to no research dedicated to the properties of steady-state availability estimates. The best known work in the steady-state availability estimates for a NRecR test plan is in the book: Beletsky B.R. [Dependability theory of radio engineering systems (mathematical foundations). Study guide for colleges]. Moscow: Sovetskoye radio; 1978. This paper makes up for this deficiency. In order to identify the efficient steady-state availability estimate out of infinite many, first, an estimate efficiency comparison criterion is to be constructed. The paper

Aims to construct a simple criterion of steady-state availability estimation for test plans with long recovery times and identify the efficient estimate out of the available ones using the constructed criterion.

Methods of research. The efficient estimate was found using integral numerical characteristics of the accuracy of estimate, i.e., the sum square of the displacement of the expected realization of an estimate from the considered parameters of the distribution laws.

Conclusions. The authors constructed simple criteria of efficiency of steady-state availability estimation for test plans with long recovery time (case of N≥1). Estimate G₃=(1+VR/S(R1))^-1 is bias-efficient out of those available for test plans of types NRec!t(D=R) and NNoRec!t(D=R). Conventional estimate G₁=(1+ V/S)^-1 is bias-efficient out of those available for test plans of types NRec!R(D=R) and NNoRec!R(D=R).

1. Barzilovich E.Yu., Beliaev Yu.K., Kashtanov V.A. et al. Gnedenko B.V., editor. [Matters of mathematical dependability theory]. Moscow: Radio i sviaz; 1983. (in Russ.)

2. Gnedenko B.V., Beliaev Yu.K., Soloviev A.D. [Mathematical methods in the dependability theory. Primary dependability characteristics and their statistical analysis. Second edition, corrected and enlarged]. Moscow: Knizhny dom LIBROKOM; 2013. (in Russ.)

3. Beletsky B.R. [Dependability theory of radio engineering systems (mathematical foundations). Study guide for colleges]. Moscow: Sovetskoye radio; 1978. (in Russ.)

4. Mikhailov V.S., Yurkov N.K. [Integral estimation in the dependability theory. Introduction and main findings]. TEKHNOSFERA; 2020. (in Russ.)

5. Cox D.R., Smith W.L. Renewal theory. Moscow: Sovetskoye radio; 1967.

6. Borovkov A.A. [Probability theory]. Moscow: Editorial URSS; 1999. (in Russ.)

7. Shulenin V.P. [Mathematical statistics. Part 1. Parametric statistics: a textbook]. Tomsk: Izdatelstvo NTL; 2012. (in Russ.)