Reference no: EM13174647

1) Consider an inventory system in which the sequence of events during each period is as follows. 
(1) We observe the inventory level (call it i) at the beginning of the period. 
(2) If i < 1, then 4- i units are orders. If i > 2, then 0 units are ordered. Delivery of all ordered units is immediate.
(3) With probability 1/3, 0 units are demanded during the period; with probability 1/3, 1 unit is demanded during the period; and with probability 1/3, 2 units are demanded during the period. 
(4) We observe the inventory level at the beginning of the next period.

Define a period's state to be the period's beginning inventory level. Determine the transition matrix that could be used to model this inventory system as a Markov chain.

2) A company has two machines. During any day, each machine that is working at the beginning of the day has a 1/3 chance of breaking down. If a machine breaks down during the day, it is sent to a repair facility and will be working two days after it breaks down. (Thus, if a machine breaks down during day 3, it will be working at the beginning of day 5.) Letting the state of the system be the number of machines working at the beginning of the day, formulate a transition probability matrix for this situation.

3. For each of the following Markov chains, determine the long-run fraction of the time that each state will be occupied.

4. Customers buy cars from three auto companies. Given the company from which a customer last bought a car, the probability that she will buy her next car from each company is as follows: 
Will buy next from
Last bought from Co. 1 Co. 2 Co. 3
Co. 1 .85 .10 .05
Co. 2 .10 .80 .10
Co.3 .15 .10 .75
If someone currently owns a company 1 car, what is the probability that at least one of the next two cars she buys will be a company 1 car?

5. Each airline passenger and his or her luggage must be checked to determine whether he or she is carrying weapons onto the airplane. Suppose that at Gotham City Airport, an average of 10 passengers per minute arrive (interarrival times are exponential). To check passengers for weapons, the airport must have a checkpoint consisting of a metal detector and baggage X-ray machine. Whenever a check-point is in operation, two employees are required. A checkpoint can check an average of 12 passengers per minute (the time to check a passenger is exponential). Under the assumption that the airport has only one checkpoint, answer the following questions:

a. What is the probability that a passenger will have to wait before being checked for weapons?
b. On the average, how many passengers are waiting in line to enter the checkpoint?
c. On the average, how long will a passenger spend at the checkpoint?