Share
Explore BrainMass

Queueing Systems

This exam covers customer queuing systems including arrival rates, Poisson processes, probability density functions, steady states, operator utilisation, machine efficiency, and exponentially distributed service times.

Attachments

Solution Preview

a) This question can be solved using Bayes' Theorem. The theorem states that, given f(x|y) (the conditional density function of x given y) and the density function f(y),

inf.
f(x) = integral f(x|y)f(y)dy
y=0

Let's see how to use this theorem to solve this problem. We are asked to find the probability function of the number of customers that arrive during an interval t of time, where t is a random variable with density function f(t). Since customers arrive according to a Poisson process, we have that, given t,

P(An = k | t) = (lambda*t)^k * exp(-lambda*t)
-------------------------------------
k!

(that's the Poisson probability function) This would be the f(x|y) in Bayes' Theorem. Since we know that t follows the distribution f(t) then, using the theorem, we arrive at:

inf.
P(An = k) = integral (lambda*t)^k * exp(-lambda*t)
t=0 -------------------------------------- f(t)dt
k!

We can take lambda^k and k! out of the integral (since they don't depend on t) to get:

inf.
P(An = k) = lambda^k * integral t^k * exp(-lambda*t) * f(t)dt
------------ t=0
k!

That's the probability function we were looking for.

The second part of this question can be easily solved using the conditional probability theorem that states that:

E(X) = E[ E(X|Y) ]

Again, given the interval of time t, we have that:

E(An | t) = lambda*t

This is simply the expected value of the Poisson process given t. So, using the theorem:

E(An) = E[ E(An | t) ] = E(lambda*t) = lambda*E(t)

(since lambda is a constant, it can be ...

Solution Summary

This solution set includes complete calculations and explanations. 900 words.

$2.19