Universal Generating Function Method

Abstract

In recent years a specific approach called the universal generating function (UGF) technique has been widely applied to MSS reliability analysis. The UGF technique allows one to find the entire MSS performance distribution based on the performance distributions of its elements using algebraic procedures. This technique (sometimes also called the method of generalized generating sequences) (Gnedenko and Ushakov 1996) generalizes the technique that is based on a well-known ordinary generating function. The basic ideas of the method were primarily introduced by I. Ushakov in the mid 1980s (Ushakov 1986, 1987). Then the method was described in a book by Reinshke and Ushakov (1988), where one chapter was devoted to UGF. (Unfortunately, this book was published only in German and Russian and so remained unknown for English speakers.) Wide application of the method to MSS reliability analysis began in the mid-1990s, when the first application was reported (Lisnianski et al. 1994) and two corresponding papers (Lisnianski et al. 1996; Levitin et al. 1998) were published. Since then, the method has been considerably expanded in numerous research papers and in the books by Lisnianski and Levitin (2003), and Levitin (2005).

Here we present the mathematical fundamentals of the method and illustrate the theory by corresponding examples in order to provide readers with a basic knowledge that is necessary for understanding the next chapters.