An NYC taxi service begins the year with 100 active taxis, and 20 taxis held in reserve. Each taxi’s lifespan (the time it lasts in service before requiring repairs) follows a lognormal distribution with ??=-1/2,??=1. Once broken, a reserve taxi is immediately put into the active pool, and the broken taxi is sent to a repair service, that services one taxi at a time; the service time for each taxi follows an exponential distribution with rate ??. Once fixed, the taxi is sent to the reserve pool. The taxi service would like to know how long they can stay in operation until they have no more taxis in reserve.

Assume ??=5/365 (i.e. ??=365/5), and run the simulation for 1000 trials; on average, how many days does it take for there to be no more taxis in reserve? (t = 1 equals one year, or 365 days)

This is question relies on coding

The company owners would like the repair company to work faster, so that the company can be in business longer without the taxi reserve running dry.

Based on the model we complemented in Q1, what is the largest value of ?? that would allow the company to run for at least t = 5 years on average, before the reserve is empty?

[When you try different ??, use step 0.001]

This is question relies on coding

Setting

Customers phone into a customer support line, with an exponential rate ??=1/5 per minute. Once they call, they will need to talk to an automated system. The time consumers takes to speak to the automated system follows a uniform distribution with U[1,5] minutes. There is unlimited compacity for the automated system, meaning system could serve unlimited number of customers at the same time. After finish talking to the automatic system, the consumers will then need to move on to talk to a representitve. Let’s assume that there is only one representive. The representive deals with every customer one at a time; each customer spends an exponential amount of time with the representative with ??(??)=5 minutes. The customer who finishes talking to the automated system first gets the priority when multiple people are lining up.

Question 1 [10 points]

Write down the simulation algorithm. This algorithm should compute the amount time of the 200th arrival spent in the system. Please use the markdown box below. do not attach another file or include a image in your answer.

[There are lots of examples in Chapter 6 of the text book]

For example, the algorithm of a simple system is as follows:

**Step 1: Initialization**

total number of customers ??=0

time ??=0

**Step 2: Simulate the arrival of a customers**

Draw ????? from distribution ????????(1/5)

??=??+?????

??=??+1

**Step 3: Repeat the Step 2 until n==200**

This is question does not rely on coding. The goal is to help the readers (and you) understand what you will be doing in your coding in a simpler way

Implement the system to compute the amount time the 200th arrival spent in the system. Simulate the system 1000 times and give the 95% confidence interval for the average time the 200th arrival spent in the system.

This is question relies on coding

Implement the system to compute the amount time the 200th arrival spent in the system. Simulate the system 1000 times and give the 95% confidence interval for the average time the 200th arrival spent in the system.

This is question relies on coding