Basic notions of the system multi-state reliability analysis are introduced. The multi-state series system with degrading components is defined and its exact reliability function is determined. The limit reliability function for large multi-state systems is introduced. An auxiliary theorem on limit reliability function of a large multi-state series system, which is necessary for its approximate reliability evaluation, is presented. On the basis of this auxiliary theorem some corollaries are formulated and proved and then applied to approximate reliability and availability determination of a large multi-state series system. Evaluations are given of a multi-state system reliability function, its mean lifetimes in the state subsets and their standard deviations, mean value and variance of the lifetime up to and the number of exceeding the system reliability critical state and its availability coefficient. The cost analysis of a multi-state series system is performed as well. The results are applied to a large telecommunication network reliability, availability and cost analysis.