@acer I am not sure that I completely follow the numerical approximation you presented.  I am interested in how you found a numerical approximation for the LambertW function.  I updated the question with some more specifics.  Say that the desired reliability R is 90%, the time period of operation t is 500hrs excluding downtime, and the number of allowed failures n is 20 each, if that helps

sorry I added this comment regarding the numerical approximation before I saw your new comment.  I will have to review the new info you provided

@acer thank you for the answer with using Sum to remove the Gamma function. Excel actually does have a built in Gamma function, but it does not have a version for the incomplete Gamma function. I could implement the Sum approach over a number of rows for a finite upper bound, although I am not sure about an infinite upper bound. Maybe there is a way to simplify only using regular gamma functions, GAMMA(z), instead of GAMMA(a,z), that way we wouldn't have to iterate over a number of rows.

also what about the LambertW function? Excel does not have any built in function for LambertW