Arithmetic and Geometric Processes

Abstract

arithmetic process (AP)geometric process (GP) Section 49.1 introduces two special monotone processes. A stochastic process is an AP (or a GP) if there exists some real number (or some positive real number) such that after some additions (or multiplications) it becomes a renewal process (RP). renewalprocess (RP) Either is a stochastically monotonic process and can be used to model a point process, i.e. point events occurring in a haphazard way in time or space, especially with a trend. For example, the events may be failures arising from a deteriorating machine, and such a series of failures is distributed haphazardly along a time continuum. Sections. 49.2–49.5 discuss estimation procedures for a number K of independent, homogeneous APs (or GPs). More specifically; in Sect. 49.2, Laplaceʼs statistics are recommended for testing whether a process has a trend or K processes have a common trend, and a graphical technique is suggested for testing whether K processes come from a common AP (or GP) as well as having a common trend; in Sect. 49.3, three parameters – the common difference (or ratio), the intercept and the variance of errors – are estimated using simple linear regression techniques; in Sect. 49.4, a statistic is introduced for testing whether K processes come from a common AP (or GP); in Sect. 49.5, the mean and variance of the first average random variable of the AP (or GP) are estimated based on the results derived in Sect. 49.3. Section 49.6 mentions some simulation studies performed to evaluate various nonparametric estimators and to compare the estimates, obtained from various estimators, of the parameters. Some suggestions for selecting the best estimators under three non-overlapping ranges of the common difference (or ratio) values are made based on the results of the simulation studies. In Sect. 49.7, ten real data sets are treated as examples to illustrate the fitting of AP, GP, homogeneous Poisson process (HPP) and nonhomogeneous Poisson process (NHPP) models. In Sect. 49.8, new repair–replacement models are proposed for a deteriorating system, in which the successive operating times of the system form an arithmetico-geometric process (AGP) arithmetico-geometric process (AGP) and are stochastically decreasing, while the successive repair times after failure also constitute an AGP but are stochastically increasing. Two kinds of replacement policy are considered, one based on the working age (a continuous decision variable) of the system and the other determined by the number of failures (a discrete decision variable) of the system. These policies are considered together with the performance measures, namely loss (or its negation, profit), cost, and downtime (or its complement, availability). Applying the well-known results of renewal reward processes, expressions are derived for the long-run expected performance measure per unit total time, and for the long-run expected performance measure per unit operation time, under the two kinds of policy proposed. In Sect. 49.9, some conclusions of the applicability of an AP and/or a GP based on partial findings of four real case studies are drawn. Section 49.10 gives five concluding remarks. Finally, the derivations of some key results are outlined in the Appendix, followed by the results of both the APs and GPs summarized in Table 49.6 for easy reference. Most of the content of this chapter is based on the authorʼs own original works that appeared in Leung et al. [49.1,2,3,4,5,6,7,8,9,10,11,12,13], while some is extracted from Lam et al. [49.14,15,16]. In this chapter, the procedures are, for the most part, discussed in reliability terminology. Of course, the methods are valid in any area of application (see Examples 1, 5, 6 and 9 in Sect. 49.7), in which case they should be interpreted accordingly.