5. Customers arrive according to a Poisson process of rate 30 per hour. Each cus

Question

5. Customers arrive according to a Poisson process of rate 30 per hour. Each customer is served and leaves immediately upon arrival. There are two kinds of service, and a customer pays $5 for Service A or S15 for Service B. Customers independently select Service A with probability and Service B with probability (a) What is the distribution of the number of customers who select Service A during the period 9:00am to 9:10am? (Give its name and any parameter(s).) (b) During the period 9:30-10:30am, there were 32 customers in total. What is the probability that none of them arrived during 10:25-10:30am?

Explanation / Answer

a.

During a 10 minute perios, expected number of customers = 30/6 = 5

Out of these 1/3 take service A and 2/3 take service B

SO, distribution of customers choosing service A is also Poisson with parameter = 5-1/3 = 5/3

b.

In 60 mins we have 32 customers.

To find the probability of no customers in 5 mins given that there are 32 customers in 60 mins,

P(32 customers in 55 mins and 0 customers in 5 mins | 32 customers in 60 mins) = P(32 cust in 55 mins).P(0 cust in 5 mins)/P(32 cust in 60 mins)

Arrival rate for 55 mins is 30*55/60 = 27.5 and for 5 mins is 30*5/60 = 2.5

Hence the required probability is P(32 cust in 55 mins).P(0 cust in 5 mins)/P(32 cust in 60 mins) = 0.04959*0.0821/0.0659 = 0.0617

c.

Probability that both customers request server B = (2/3)^2 = 4/9, assuming independent choices of both

d.

During a 10 minute period, expected arrivals = 30*10/60 = 5

Expected amount paid by each customer = 1/3*15 + 2/3*5 = 25/3

So, amount paid by all customers in a 10 minute period = 5*25/3 = $125/3

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