# Algebra with random variables

#### Prometheus

##### Member
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:

$$C{\mu}e^{{\mu}t}={\lambda}e^{{\lambda}t}$$

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).

Last edited:

#### Dason

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.

#### Prometheus

##### Member
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.