ISyE 3232 – Exam 1 Summer 2012

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- Place your photo ID on your desk at all times during test. - Show your photo ID when turning in your exam. - Do not share Calculator. - NO cell-phone calculator; NO cell-phone on the desk. - Show intermediate steps to justify your work and to save maximum partial points if mistakes are made. - Grading will be solely based on the work presented in exam. - When wanting to detach the last sheet, write down your name on the top and keep the sheet at the center of your area. Turn in the detached sheet as well.

Followings may or may not be helpful:

#1 /10 #2 /10 #3 /13 #4 /22 #5 /10 #6 /15 #7 /30 (Bonus) Total /110

#1. This problem is about exponential distribution and its memoryless property.(a) Complete the following statement with mathematical expression. [4pts]

A random variable X is said to be memoryless1 if...

(b) Precisely state the CDF (Cumulative distribution function) of exponential distribution with pa-rameter λ. [2pts]

(c) Prove that exponential distribution is memoryless. [4pts]

1Or, A random variable X is said to have memoryless property if...

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#2. In post office, there are two servers, A and B. A’s service time follows exponential distributionwith mean 3 minutes and B’s service time follows exp(1/5) distribution. Their service times areindependent. Alice and Betty came to post office at noon and services started. (Alice by server A,Betty by server B)(a) What is the chance that Betty will leave the post office first? [5 pts](b) What is the chance that they will be still in the service at 12:05 pm? [5 pts]

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#3. A store stocks a particular item. The demand for the product each day is 1 item with proba-bility 1/6, 2 items with probability 3/6, and 3 items with probability 2/6. Assume that the dailydemands are independent and identically distributed. Each evening if the remaining stock is strictlyless than 3 items, the store orders enough to bring the total stock up to 5 items. These items reachthe store before the beginning of the following day. Assume that any demand is lost when the itemis out of stock. Let Xn be the amount in stock at the beginning of day n

(a) Define state space. [3 pts]

(b) Draw transition diagram [5 pts]

(c) What is the transition matrix?2 [5 pts]

2Make sure you specify which row/column corresponds to which state, see the transition matrix in #5 for example.

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#4. A store sells a particular brand of fresh juice. By the end of the day, any unsold juice will besold at a discount price of $2 per gallon. At every morning, the store gets the juice from a localproducer at the cost of $5 per gallon, and it sells the juice at $10 per gallon. Assume the dailydemand for the juice is uniformly distributed between 100 and 200 gallons.(a) What is the optimal order quantity that maximizes the expected daily profit? [7 pts](b) If 150 gallons are ordered, what is the expected profit per day? (answer in number) [15 pts]

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#5. Suppose we have following transition matrix for discrete time Markov chain {Xn : n ≥ 0, n ∈ N}

P =123

0.7 0.3 00.5 0.5 00 0.2 0.8

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Suppose that P(X0 = 3) = 1, then what is P(X2 = 2) [10 pts]?

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#6. A small bank is staffed by a single server. During a normal business day, the inter-arrivaltimes of customers to the bank follow exponential distribution with mean 3 minutes. On the otherhand, the service time for a customer follows normal distribution with mean 2 minutes and stan-dard deviation of 1 minute. Answer following questions in number. It is OK to use previous answer.3

(a) What is the long-run fraction of times that the server is busy? Is this system stable? [5 pts]

(b) What is the long-run average waiting time of each customer in the queue?[5 pts]

(c) What is the long-run average number of customers waiting for service? [5 pts]

3For example, if you didn’t solve problem (a) but know that the answer for (b) is [the answer for (a)] times 5, thenyou may answer question (b) by “5 × [ans in (a)]”.

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#7. You are working for revenue management team in airline company. You need to determine howmany flight tickets to sell for ATL-DCA at Next Thursday 6:00 AM, flight number 3232. The cabinsize is 200 and a ticket is sold for $150. Suppose that if z number of tickets are sold, then the numberof people who show up as passengers is between z− 10 to z− 1, equally likely.4 If a passenger showsup and there is no seat for the passenger, then airline company must provide compensation of 500dollars to the passenger. Assume that the Airline company do not issue a refund to ticket price andhas no liability after compensation $500 is given. Marginal cost of carrying an passenger from ATLto DCA is nominal and can be ignored to be zero.

(a) During the classes, we defined overstock cost as “economic cost due to having overstock”, and wedescribed the situation as “too much preparation for demand”. How would you quantify overstockcost in this example? (answer in dollar) [5 pts]

(b) During the classes, we defined understock cost as “economic cost due to having understock”,and we described the situation as “too less preparation for demand”. How would you quantify un-derstock cost in this example? (answer in dollar) [5 pts]

(c) Suppose you sold 205 tickets and let X be the number of passenger who show up at gate forflying. Then X is distributed between [205-10,205-1]=[195,204], equally likely. What is the totalexpected profit5 for airline company on this flight 3232?6 Express the answer in terms of expectationand evaluate the expectation to present answer in number. [10 pts]

(d) See next page

4That is, there will be 1 ∼ 10 people who will not show up, each case of 10% chance. For example, if you sell200 tickets then the distribution of passenger will be either 190,191,...,199 with equal probability of 0.1. Indeed, it isknown that about 1∼2% of people who buy flight ticket do not show up as passenger for whatever reason.

5Hint1: Profit=Revenue-Cost; revenue comes from ticket sale and cost comes from overbooking compensation.6Hint2: If X is the number of passengers who show up at gate, and the flight has 200 seats, then how many

passengers will not have seats?

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(d) (Bonus Question) Find the optimal number of ticket sale. Show your work. [10 pts]

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