Signature reliability of the system using commutative and associative properties

The present paper studied the air conditioner binary system which is the combination of seven elements listed in series parallel arrangement having independent identically distributed units. Coherent system very useful tools for computing nature of working units in any system. The air conditioner has two states either working or failed. We explicitly evaluate the various factors of consider air conditioner system such as signature, tail signature, Barlow Proschan index and expected lifetime with the help of universal generating function with commutative and associative properties. A statistical approach also used for evaluating reliability structure function of the units. For signature and others measures analysis Boland and Owen's methods has been discussed A needed numerical example also discussed for computational result.