1. ## Poisson and Queues

Ok so I'm really struggling with the following question... I'm just not getting it. The maths is pretty basic (well at least for you guys) but the logic is confusing me. I'm sorry the question is really long... I was just wondering if one of you could let me know if I'm completely off track.

The coffee shop has a busy time from 10:30am till 2:30pm on weekdays. They recorded the customers who arrive during this period on a typicla weekday and find 285 of whom 146 just order a coffee and 139 order something in addition (muffin etc). There are two employees. The process is as follows:

Stage 1: The customers order and James writes what type of coffee and customer's name on a cardboard cup and gives this to Peter.
Stage 2: If the customers wants anything in addition to coffee, James fetches the item and gives it to the customer.
Stage 3: James asks customer to pay and operates till.
Stage 4: Peter makes coffees and shouts out names when they are ready. there is a queue of cardboard cups waiting to be filled. These stages are represented below.

Stage 1
Fixed time
12 seconds

Stage 2
Fixed time
16 seconds

Stage 3
Fixed time
21 seconds

Stage 4
Normal distribution
standard deviation: 10 secs
mean= 35 secs

A) Estimate the average number of customers waiting to be served by James.

Ok so av number of ppl in queue is
(CV² + 1)/2 x u² /(1-u)

λ : 285 customers between 10.30-2.30 (4 hours) = 71 per hour = 1.183 per min = 0.01972 per second.
u: λ x p
What I'm not sure of is how do you get the mean?

b) There is a queue of customers waiting for their coffees after having paid. If the arrivals to this queue followed a Poisson process, what would you estimate to be the average number of people who have paid but not yet picked up their coffee?

c) Explain why the arrival process for Peter is not Poisson. State whether the actual number waiting will be more or less than your answer in (b) and give reasons for your answer.