Mathematicians: please help!

I am still stuck with a problem with the M/G/1 queue: not quite the same as my original problem (discussed here) as I understand that now – but the next stage really – involving some manipulation of Laplace transforms.

I won’t post all the details here, because you can read them here instead and, if this sort of thing matters to you (and why wouldn’t it?) pick up the bounty I am offering on the maths Stack Exchange.