Practice Questions 1

Upon observing the current trends toward commercialization, Valentine realized that he can turn his matchmaking skills into a business, and decided to open a matchmaking agency. Valentines goal is to find a girlfriend/boyfriend for every client. After opening and running the agency for a few weeks, Valentine felt that the agency might not have the most efficient setup and decided to use simulation to analyze it.

He observed that new client requests arrive to his agency with exponentially distributed inter-arrival times with mean 60 minutes. Each customer goes through a two-stage process. The first stage is an initial screening (interview) to identify the clients personality. The interview takes an exponentially- distributed processing time with mean 25 minutes, and is performed by Valentines assistant, Cupid; if Cupid is not available, the clients have to wait. The second stage consists of Valentine searching through his database to find a good match for the client and calling the client to notify him/her about the results. This process takes Valentine a triangularly-distributed amount of time with a minimum of 20 minutes, a maximum of 2 hours, and a most likely time of 1 hour. Assuming the clients leave the system after the second stage, create an Arena simulation model and run it for 10 replications of 7 days each. Moreover, assume Valentines agency is open 24 hours a day, 7 days a week.

Find the average waiting time at stage 1 (interview). State your answer, calculated in minutes, below. Please state the numerical value only.

B)

b) Find the average waiting time at stage 2 (database search). State your answer, calculated in minutes, below. Please state the numerical value only.

After seeing the results of the first simulation, Valentine thought that the results are surprising, so he decided to analyze the data more carefully. He found that out of all the questions that are asked at the interview, there is one that is the best predictor of whether two people will like each other. This question is: Do you like tea more than coffee? After doing some research, Valentine found that in Canada (where his agency is located), approximately 30% of people like tea more than coffee.

Thus, it usually takes longer to find a match for somebody who likes tea. Therefore, the second-stage for tea-drinkers actually takes a triangularly-distributed amount of time with a minimum of 40 minutes, a maximum of 2 hours, and a most likely value of 1.5 hours. The second-stage for coffee-drinkers actually takes a triangularly-distributed amount of time with a minimum of 20 minutes, a maximum of 1 hours, and a most likely value of 45 minutes.

Note that you find out if a person is a tea or coffee drinker only after the interview. To easier see tea-drinkers and coffee-drinkers in the simulation, change the pictures of both (so that they would be different).

Find the proportion of time that Valentine is busy. State your answer below. Please state the numerical value only.

Based on his analysis, Valentine decides that he would like to replace the interview at the first stage of the process by an online questionnaire. This would allow him to free Cupid from the interview stage and instead use Cupid to perform the search at the second stage.

This way, both Cupid and Valentine will be

performing the database search. Moreover, Cupid is expected to be as efficient as Valentine at the second stage. Valentine also estimates that the online questionnaire will take an exponentially- distributed processing time with mean 15 minutes.

What is the average client system time?

a) Find the average system time for tea-drinkers. State your answer, calculated in minutes, below. Please state the numerical value only.

b) Find the average system time for coffee-drinkers. State your answer, calculated in minutes, below. Please state the numerical value only.

Valentine is working crazy hours right now. Thus, he wants to simulate the system assuming he is open 9am to 5pm for seven days a week in order to determine what effect the working hours will have. Closing at 5pm means that after 5pm, no new client requests are allowed, but those already in the system will be processed. Run the simulation for 15 replications (each of which is from 9am to 5pm).

What is the average number of clients served? State your answer below. Please state the numerical value only.

(a) Valentine further realizes that he missed modelling a major feature of the situation. In particular, if a client is not happy with the girlfriend/boyfriend found by Valentine, that client will return to the second- stage. From experience, Valentine estimates that with probability 0.47, a client is not happy (and therefore will return). These clients have a different processing time distribution than first-time clients. In particular, they have a processing time distribution that is x% longer than the clients original processing time, where x is uniformly distributed between 0.1 and 0.3. Moreover, they get priority over first-time clients (i.e., if there is a first-time client and a return client in the queue, the return client will be served first).

It also turns out that Cupid likes to have his lunch at the same time every day: 1pm to 2pm. Moreover, he is pretty selfish when he is hungry, so even if he is performing search for a client, when it is 1pm, he stops and leaves, resuming his search from the point where he left off before his lunch.

Find the average number of returns to the second stage.

(b) Valentine further realizes that he missed modelling a major feature of the situation. In particular, if a client is not happy with the girlfriend/boyfriend found by Valentine, that client will return to the second- stage. From experience, Valentine estimates that with probability 0.47, a client is not happy (and therefore will return). These clients have a different processing time distribution than first-time clients. In particular, they have a processing time distribution that is x% longer than the clients original processing time, where x is uniformly distributed between 0.1 and 0.3. Moreover, they get priority over first-time clients (i.e., if there is a first-time client and a return client in the queue, the return client will be served first).

It turns out that Cupid likes to have his lunch at the same time every day: 1pm to 2pm. However, he is very responsible so if he is in the process of searching at 1pm, he will finish what he is doing before going on break. He will still take a break of exactly 1 hour.

Find the average number of returns to the second stage.

a) Valentine wants to advertise his company by the following policy If you dont find somebody you really like after three matches suggested by Valentine & Cupid, you get your money back. Before putting this policy in place, he wants to find out the number of clients who return after three attempts. State the average number of clients who return after three attempts in the space below.

b) Valentine wants to advertise his company with If you dont find somebody you really like after five matches suggested by Valentine & Cupid, you get your money back. Before putting this policy in place, he wants to find out the number of clients who return after five attempts. State the average number of clients who return after five attempts in the space below.