Following task (and sorry for the long text in advance, I want to give all the information I have and my thoughts about it):
"Consider a machine which is turned off when there is no work. It is turned on and restarts work when enough orders, say $N$, arrived to the machine. The setup times are negligible. The processing times are exponentially distributed with mean $1/\mu =2$ and the average number of orders arriving per unit time is $\lambda=1/3$. Suppose that the setup cost is 54 dollar. In operation the machine costs 12 dollar per minute. The waiting cost is 1 dollar per minute per order.
Determine, for given threshold N, the average cost per unit time."
I know that the expected number of waiting customers is $$E[L_q]=\frac{\rho^2}{1-\rho} + \frac{N-1}{2}$$
My thoughts regarding the cost was the following: Average cost/ min= setup cost * setup/min (1) + operating cost *fraction of time operating (2) + waiting cost * orders waiting (3).
So I computed (1): $$54 [$/setup] \cdot (1-\rho) \cdot \frac{\lambda}{N} [setup/min]=\frac{6}{N} [$/min] $$
(2): $$12 [$/min] \cdot \rho = 8 [$/min]$$
and (3): $$1 [$/(min \cdot order)] \cdot E[L_q] [order]=\frac{4}{3} + \frac{N-1}{2} [$/min]$$ and then I would (obviously) some this up to get the total costs/min. Unfortunately, the last expression (3) is wrong and should be instead $$4+\frac{3}{2}(N-1)$$.
What is my mistake? It looks a lot to me as if my expression needs to be multiplied with the mean interarrival time $\frac{1}{\lambda}=3$ but then my units do not make any sense anymore.