Lifetime or Failure Time Lifetime / Survival time / Failure time :
          the time to the occurrence of event (always nonnegative) .


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*Lifetime and Survival time can be synonymous .  But why Failure time  can be used as alternative of lifetime / survival time ?

In the book Statistical Models and Methods for Lifetime Data , it is written :
"Survival Analysis consists of techniques for positive-valued random variables ."
Also it has mentioned in the book that :
"Lifetime is  the time to the occurrence of event (always nonnegative) ."

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*Why haven't they mentioned either nonnegative or only positive in both cases ?

 A: I think this is a linguistic point.
For "failure time" to be synonymous with lifetime / survival time, then you can think of "failure time " as meaning "time to failure", whereas I think you might be interpreting it as "time elapsed while the thing is failing".  The time to failure is the same as the survival time.  
To address your comment under the post:

But i meant this time denotes length of time to occur the interested event and if the occurrence of the event is considered as success, then the length of time to occur the event is failure time.

Whether the event at the end (failure / death / whatever) is considered a success depends on the context entirely - for instance in a protective component such as a fuse, a short time to failure (lower value of failure/survival time) might be success, whereas for a car tyre, the reverse would be true.
A: I think this is probably just an editorial glitch in the writing publishing of the book.
Technically, I suppose it is possible in some cases to have a 0, and the authors may have been thinking of some such cases when they wrote one description and not when they wrote the other and no one caught it. E.g. if you are testing light bulbs and some blow out the instant they are switched on.  Hardly the worst editorial glitch ever.
