MATH268

Operational Research: Probabilistic Models

1. Ahead of Olympic Games, one country faces the problem of whether athletes should be tested for drug use. After a test, the underlying authlete can be accused or not. Suppose you are the decision maker (the chief of the Olympic Association of that country).

Suppose that 5% of all athletes are drug users, and that the test used is 90% reliable. That is, if an athlete uses drugs, there is a 90% chance that the test will detect it, returning a positive result, and if an athlete does not use drugs, there is a 90% chance that the test will return a negative result.

Let C1 be the cost incurred if an athelete is falsely accused of drug use, C2 the cost if a drug user is not identified, and C3 the cost due to invasion of privacy if a nonuser is tested. All the cost is in the unit of £100.

By using a decision tree, answer the following two questions.

(a) Show that if C1 > C2 > C3 > 0, then it is optimal not to test. (You need plot a decision tree corresponding to this problem, where for each chance note, compute and indicate all the probabilities along the branches out of it.) [40 marks]

(b) Suppose C1 = 10, C2 = 5, C3 = 1. What is the expected cost (in unit of £) following the optimal policy? [15 marks]

2. Consider the M/M/1 queueing system with arrival rate λ > 0 and service rate µ > 0. Assume λ < µ. Recall the notation used in the lecture notes: W is the waiting time of a “typical” customer (in the long run/steady state). Compute the variance of W. (You need explain how you reached your conclusion to get full marks. You may use any result obtained in the lecture notes.) [15 marks]

3. Consider a manager of a shop, which sells diamond rings. The manager keeps a safety stock of S ∈ {0, 1, 2, . . . } rings on hand. Customers arrive in a Poisson process with rate λ > 0 per day. Each customer demands one ring. Every time a request for a ring is made (a customer demand), the manager of the shop places an order at the factory to manufacture another ring. The amount of time required to manufacture a ring follows exponential distribution, and on average, it takes 1/µ > 0 days to make one. Suppose the manufacturing times of rings are independent, and are also independent of the Poisson process modeling the customer arrivals. A customer receives a ring immediately when there is at least one ring in the shop at his/her arrival. It is assumed that customers who request a ring but find there is none immediately available in the shop will wait until the stock is replenished by orders due in (this is called backordering or backlogging).

Let Z represent the steady-state inventory level: if Z ≥ 0, then there are Z rings in the shop; if Z < 0, then there are |Z| rings in backorder. Denote the steady-state probabilities by qm = P(Z = m) for all m ∈ {S, S−1, S−2, . . . }.

Compute qm for each m ∈ {S, S − 1, . . . } in terms of S, λ and µ. (You need explain how you reached your conclusion to get full marks.)

(Hint. It is useful to consider the relationship between Z and N, the steady-state number of orders outstanding i.e., the number of orders currently being processed at the factory.) [20 marks]

4. Suppose it is known how to simulate a random variable X with the cumulative distribtuion function F defined by F(x) = P(X ≤ x) for all x ∈ (−∞,∞). Based on this fact, suggest how to simulate a random variable Y whose cumulative distribution function G is given by G(x) = 1−(1−F(x))ⁿ, where n ≥ 2 is an integer. Explain why your method is correct for this purpose. [10 marks]