On Gaver's parallel system

Abstract

Gaver’s parallel system (and its variants) has received considerable attention in the literature. The Laplace-transform (LT) of the survival function, corresponding to the underlying systems, has been derived by various (alternative) methods. Unfortunately, little-to-no attention has been paid to invert the corresponding transform. First, we present a general reliability analysis of Gaver’s basic parallel system (valid for an arbitrary repair time distribution). Then, we formulate some general hints to obtain the numerical inverse. Finally, we propose a tangible methodology to derive the exact inverse in some particular but important cases of non-rotational transforms.

Gaver se parallelsisteem (en variante daarvan) is reeds dikwels behandel in die literatuur. Die Laplacetransformasie van die oorlewingsfunksie van die onderliggende sisteme is reeds afgelei op verskillende uiteenlopende wyses. Ongelukkigerwys is min aandag gegee aan die ooreenstemmende inverse transformasie. Ten aanvang word 'n algemene betroubaarheidsontleding van Gaver se basiese parallelsisteem voorgehou (geldig vir 'n arbitrêre hersteltyd-verdeling). Dit word gevolg deur wenke vir bepaling van die numeriese inverse. Ten slotte word 'n tasbare metodologie vir die afleiding van die presiese inverse vir sekere belangrike nie-rotasionele transformasies voorgestel.