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.
Reading An Introduction to Laplace Transforms and Fourier Series I reach the point where it is stated, rather axiomatically, that: .
This is a beautiful formula and has always suggested to me some sort of mystical inner mathematical harmony (yes, I am a materialist, but I cannot help it).
But these days I also want to see the proof, so here is one:
We know that complex numbers can be described in polar co-ordinates:
So too where and depend on .
Now (and applying the product rule)
So we equate the real and imaginary sides of both sides of this equality we have:
Then, recalling , we have
By inspection we can see that and , giving us:
and multiplying both sides by we have:
Reversing the signs:
But what of and ? Well, we have and .
So is constant with respect to while varies as .
If we set then – a wholly real number, so and . Thus and we can replace with throughout.