(20 points) The Comet is a very small coffee shop, in the Nichols Arcade, with g

Question

(20 points) The Comet is a very small coffee shop, in the Nichols Arcade, with great coffee and room for at most five customers. That is, if Comet is full, no other customer can even enter. Suppose that customers arrive according to a Poisson Process with rate 30 per hour and that there is only one server with a service time that can be modeled as an independent exponential random variables with a mean of three minutes. Assume that once served, customers leave. (a) (5 points) What is the average number of customers in Comet? Draw the rate diagram clearly defining the states and label all transition rates. (b) (5 points) How many customers enter Comet per hour? (c) (5 points) If the server could work twice as fast, how much more business would be done? That is, on average, how many more customers would be able to enter Comet per hour with respect to (b)?) Draw the rate diagram, clearly defining the states and labelling all transition rates. (d) (5 points) Now, if the server was working at the original rate, but she could hire arn additional worker with the same service rate, how much more business would Comet do. (That is, on average, how many more customers would be able to enter Comet per hour with respect to (b)?) Draw the rate diagram, clearly defining the states and labelling all transition rates.

Explanation / Answer

Arrival rate, A= 30 customers per hour, or a customer every 2 minutes

Service rate, M = 60/3 = 20 customers per hour

p = Avg Utilization = 30/20 = 1.5 or 150%

a) Avg number of customers in system = L = A /(M - A) = 30 / (20 - 30), since arrival rate is greater than service rate, the queue will keep on growing. On an average, there will be 5 customers inside the coffee shop always and other's won't enter as queue would keep on growing.

b) Customer's entering Comet per hour = Service rate = 20 customers per hour, Other customers would leave seeing the queue (5 in line)

c) New Service rate = 1.5 minutes per customer or 60/1.5 = 40 customer per hour

Since service rate > Arrival rate, Comet would be able to serve all arriving customers and hence handle 10 more customers per hour compared to b.

d) Average utilization = p = Arrival Rate / (service rate * number of server (s)) = 30 / 40 = 75%

Comet would be able to serve 10 more customers per hour i.e. 30 customers per hour (arrival rate). Since utilization rate is less, Comet can serve 10 more customers in case arrival rate increases to 40.

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