This question is about exercise 70 of the book 'Queueing Systems' (https://www.win.tue.nl/~iadan/queueing.pdf) from Ivo Adan and Jacques Resing. The task is the following:
Consider a queueing system where on average 3 groups of customers arrive per hour. The mean group size is 10 customers. The service time is exactly 1 minute for each customer. Determine the mean sojourn time of the first customer in a group, the last customer in a group and an arbitrary one in the following two cases:
(i) the group size is Poisson distributed; (ii) the group size is geometrically distributed.
First I computed the mean waiting time for an arbitrary customer with the formula on page 107:
$\mathbb{E}[W] = (\rho * \mathbb{E}[R]) / (1-\rho) + ((\mathbb{E}[G^2]-\mathbb{E}[G])*\mathbb{E}[B]) / (2*\mathbb{E}[G]*(1-\rho))$
I got $\mathbb{E}[W] = 10.5$ for exercise (i) and $\mathbb{E}[W]=20.5$ for exercise (ii).
With $\rho= \lambda * \mathbb{E}[G]*\mathbb{E}[B]=1/20 * 10 * 1 = 1/2$ and $\mathbb{E}[R]=\mathbb{E}[B^2]/(2*\mathbb{E}[B])=1/(2*1)=1/2$
But then I'm stuck. I don't know how to compute the mean sojourn time after having these values for the mean waiting time.
And I also don't know how to compute the mean waiting time and after that the mean sojourn time of the first customer and the last customer in the group for (i) and (ii).
I think I have to use formula (10.6) on page 106 and do something with the last term at the right-hand side of (10.6). But I have no idea what exactly to do. Can you help me with this exercise?