Algebra with random variables

I've done something strange, potentially bad, and i just wanted to check how bad.

So i have the p.d.f of waiting time as:

Pr(Q>t)=\frac{\lambda}{\mu}e^{-({\mu} - {\lambda})t}

where lambda is the inverse arrival rate and mu the inverse departure rate from a queue.

What i'm interested in is finding the maximum arrival rate the system can cope with before we would expect over 5% of people to have to wait 15 minutes or longer.

I interpreted this as meaning i want to know when \( C= Pr(Q>t) = 0.05 \)

when t=15

So i rearranged the first equation:


Then i used Lambert's W function to get:

\({\lambda} = \frac{W(C{\mu}e^{\mu}t)}{t} \)

Thus i found an arrival rate up to which we expect queueing system to be able to be able to see 95% of people within 15 minutes. It gave sensible sounding answers so i used it.

But i've never seen random variables jostled about as if they were normal variables in standard algebra, so i'm worried i may have been doing some very bad things (other than it probably would have been better if i had used inequalities).

Any comments?
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Ambassador to the humans
But i've never seen random variables jostled about as if they were normal variables in standard algebra
Where exactly did you do that?

As a side note I think you need to use curly braces when doing your exponents because you're dropping the t from the exponent and it looks to just be another value you're multiplying by.
Well this is the source of my confusion.

I started with \(Pr(Q>t)\)

then i stated i want to find lambda such that:

\(Pr(Q>15) = 0.05\)

and so i solved for lambda from the orginal p.d.f.

I guess at this point it is no longer random variable but a realisation? Doesn't feel right though.


Ambassador to the humans
Q is a random variable but it pretty much only relates to the problem in that it dictates what the distribution is that we're taking probabilities with respect to. Once we have that distribution though... Q is irrelevant to the problem. You aren't dealing with anything random. You aren't even dealing with realizations of this random process.