# Queuing Theory problem (M/M queue)

1. Jul 3, 2011

### A-fil

1. The problem statement, all variables and given/known data
Given an M/M type queuing model on an first come first save basis, where 2 kinds of customers (type 1 and type 2) arrive as poisson processes with arrival rate λ1 and λ2 respectively, being served by one cashier with timewise exponential distribution with parameter μ. However, if the ammount of customers in the system (type 1 and type 2) are more than K people, then customers of type 1 can skip queing and leave the cashier. Type 2 will always queue no matter what.
(1) Give the balance equation expressed with the balance state probability for n customers in the system pn.
(2) Solve the equation from (1). Find the conditions so that there exists an balance state.
(3) Find the average number of customers in the system.

2. Relevant equations

3. The attempt at a solution
(1)
I think I got this one right, I made a simple graph
Numbers below show states (how many people are in the queue) and the arrows with probabilies indicate change of states (not a beautiful illustration, but hopefully you understand).
1 >-λ12-> 2 >-λ12-> ... >-λ12-> n-1 >-λ12-> n >-λ12-> n+1 >-λ12-> ... >-λ12-> K >-λ2-> ...
1 <-μ-< 2 <-μ-< ... <-μ-n<-μ-< ... <-μ-< K <-μ-< ...

This gives:
μ p1 = (λ12) p0

(μ + λ1 + λ2) pn = (λ12) pn-1 + μ pn+1 for 0 < n < K

(μ + λ2) pK = (λ12) pK-1 + μ pK+1

(μ + λ2) pn = λ2 pn-1 + μ pn+1 for n > K

And maybe also use

$\sum_{n=0}^{\infty}p_{n} = 1$

(2) Solving this is where I got stuck, I was thinking about doing it with Z-transform but it didn't really work out. He seems to just see how it turns out in the other examples in the book, but I have a hard time to see how to solve this. As for the conditions for convergence, I'm a bit clueless.

(3) I think I'm supposed to do this with Little's Formula