الفهرس 
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Abstract
The increasing of reliability for any system become more important in many fields of communications technology and industry with the automation of industrial processes and the complex mechanization. The reliability of a system is the probability that the system performs a specified function throughout a specified time under environmental conditions and specified operational. When ”stress-strength reliability” term appears in the statistical literature, typically denotes to the quantity 𝑅=𝑃(𝑌<𝑋), where a random stress Y affects the system with a random strength X a system such that if the stress exceeds the strength. Availability combines aspects of reliability and maintainability. it implies that the system is either able to operate or in active operation if required. Markov models are the most important and popular analytical techniques which can be utilized to calculate either reliability or availability. Markov Models yield results for both the steady-state of the system and the time-dependent evolution of the system. Occasionally, in reliability theory, the analysis of failure times is done by mixture distributions, and This thesis is concerned with a mixture of Rayleigh distribution and Half-Normal distribution. The thesis consists of five chapters, and it is structured as follows: This main objective of Chapter One is to make the thesis as much as possible a self-contained piece of research. It includes most of the main definitions, concepts, and theoretical results. It also contains a brief description of estimation.