# Operations Research – Queue Theory Help M / M / 3

Comtex plc employs three people in the post office to sort and send the letters passing through the internal postal system. Letters arrive on average at 150 hours per hour and each employee can process 60 letters per hour. Assuming you can treat this as an M / M / 3 / ∞ / FIFO system, calculate:

1. The probability that no letters are waiting to be sorted
2. The proportion of time that each sorter is busy
3. The probability of having more than 4 letters in the sorting office

I've tried these questions … please can someone tell me if I'm doing it right

1. Dienstrate is now S$$mu$$ so now is the traffic intensity $$lambda$$/$$mu$$S where S is 3, since there are 3 people. So $$rho$$ = 150/180 = 0.833333333 and 1$$rho$$= 0.166666 = $$p_0$$

2. the proportion of time that each sorter is busy
$$rho$$= 0.833333333 ie 83.3333% of the time? not sure if this is correct

3. One wonders if there are now more than 4 letters in the sorting office $$n> S$$ so
$$p_4$$ = $$2.5 ^ 4$$/ 4! * 0.166666 = 0.27126 …

Is that all right? Many Thanks