You are the manager of a grocery store Trader M's in downtown Berkeley. Because of COVID- 19, you limit the number of customers who can be in a store at the same time, in order to facilitate social distancing. This limit is m customers. The store has two doors as entrances - one from the parking garage and the other from the street front. Two exits are through the same doors as the two entrances. Customers who finish shopping exit from either of these two doors. To manage the limit of customers, two queues are marked at the two entrances. You appoint two associates to count and direct customers in each entrance. Once the store hits the capacity limit m, customers will be admitted into the store following the policy "one-out-one-in." That is, one customer lined up at the garage entrance will only be admitted if one customer exits to the garage.

The current time is marked as t = 0. Suppose that customers arrive at the garage entrance according to a Poisson process N1=(N1(t):t0) with rate 50 customers per hour and the customers arrival process to the street front entrance is a Poisson process N2=(N2(t):t0) with rate 10 customers per hour. Given the nature of the business, these two arrival streams are independent.

i.) Rigorously describe how a Poisson process is defined through the exponentially distributed inter-arrival times.

ii.) What is the probability distribution of total number of customers who arrive at the store within a range of two hours? (Both of the two entrances count.)

iii.) The associate begins to count customers arriving at the garage entrance from time 0. What is the expected time that the 10-th customer arrives at the garage entrance?

iv.) What is the probability that the first 5 customers that arrive at the store are all from the garage entrance?

v.) Conditional on that 120 customers arrive at the garage entrance in the first hour, in expectation what is the total number of customers that arrive in the last 10 minutes?

vi.) Conditional on that 200 customers arrive at the store in the first hour, in expectation what is the total number of customers that arrive at the street entrance in the last 10 minutes?

vii.) Suppose that each associate holds a thermometer to take the temperature of every customer immediately after the customer arrives at the entrances before joining the queue. Suppose that 1% of all customers have a temperature while 2% of all customers have contracted the virus but with no symptoms at all. Those without symptoms cannot be detected by the associates. Find the distribution of the first time that an infected customer arrives but is not detected by the associate.