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I have a stochastic modelling test tomorrow, I'm stuck on one practice question. We have an open Jackson network which is as follows:

Arrivals in queue 1 are a Poisson process with rate $\lambda$. After being served, they go to queue 2 with probability $p$, with probability $1-p$ they go to queue 3. After being served at queue 2 they go back to queue 1 with probability $q$, with probability $1-q$ they go to queue 3. After being served at queue 3, customers leave the system. The service times at the three queues are exponential with rate 3. All the queues have an infinite amount of servers.

Here are the routing equations I calculated:

$\gamma_1 = \frac{\lambda}{1-pq}$

$\gamma_2 = \frac{p\lambda}{1-pq}$

$\gamma_3 = \frac{1}{1-pq}$

At an individual level these now count as arrival rates for the queues (if I understand correctly).

The question I struggle with is as follows: Determine the expected total number of customers in the system in terms of $\lambda, p$ and $q$.

Should I determine $\mathbb{E}L_1$, $\mathbb{E}L_2$ and $\mathbb{E}L_3$ individually and just sum them? If this is the case I think I can just use $\mathbb{E}L_i = \gamma_i\mathbb{E}S$ because of Little's Law? (L is number of customers in system and S is service time).

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