Question 1.

An investor has just bought shares of a speculative stock at the share price of $48, and has given orders to his broker to sell the stock as soon as the price either rises to $50 or above or falls to $45 or below. From observations about this stock, over the last few weeks, he estimates that the probability of a price rise of two dollars is 0.2, and the probability of a price decline of one dollar is 0.3 for each day; otherwise the price remains unchanged.

(a) Describe this as a Markov chain and set up the transition matrix.

(b) What is the expected number of days until his shares are sold either at the share price of $45 or $50 or more?

(c) What is the expected long-term gain or loss per share if he sells the shares when the selling conditions are met?

Question 2.

Harkness Industries is starting up a new fishery which they intend to stock with immature fish. During its life, each member of the species of fish matures and grows through 6 size categories (1, 2, 3, 4, 5, and 6) starting as an immature fish (or size 1 fish) and reaching full maturity at size category 6 . Each month, size 1 fish have a 0.4 probability of growing and becoming a size 2 fish, a 0.4 probability of remaining size 1, or a 0.2 probability of perishing. Similarly fish, in other size categories, will also either grow into the next size category, remain the same size, or perish. The probabilities of these occurrences are summarised in Table 1.

Table 1 also gives the profit (revenue minus costs) for selling fish in each category size. Note fish of size 1 cannot be sold and fish that reach size 6 category remain in the same category before being sold.

Harness Industries intends to catch and sell only fish that reach the largest size category with other fish of smaller size categories thrown back if caught. Currently the company intend to sell all fish that reach size 6. However the cost of feeding increases the longer the fish are kept and so Harkness Industries is not certain if they should wait until all fish reach size 6 before they are caught and sold.

Help Harkness industries determine if this gives the best expected profit or if it is better to catch and sell all fish once they reach one of the smaller size categories (1{5). The only restriction is that Harkness industries will only sell fish of the same size. Which size category should Harkness catch and sell all fish (of that size) to give the optimal profit per month? What is the average lifetime of a fish (from being stocked as size 1 fish to being caught) under each option?

Question 3.

A biologist is observing the behaviour of a frog in a small lily pond which has four lily pads. The frog jumps from one lily pad to another lily pad but will not enter the water or leave the pond. From observations, the probability the frog will jump from one lily pad to another is inversely proportional to the distance between the lily pads. That is, the frog is more likely to jump to a closer lily pad than a far one i.e. for instance the probability of jumping from lily pad 1 to lily pad 2 is 4=5 and from pad 1 to pad 3 is 1=5.

The distances from one lily pad to another (in centimetres) are given in the table below

Note due to distance the frog cannot jump from lily pad 1 to lily pad 4 or vice a versa.

(a) Formulate the evolution of the position of the frog as a Markov chain, by identifying four possible states and then constructing the (one-step) transition matrix.

(b) Find the expected first passage from state i to state j for all i and j and the steady state for each state.

(c) Suppose the frog is currently on lily pad 4. Determine the probability the frog is on lily pad 2 after 3 jumps.

Question 4.

The Eat and Gas service station has four self-service pumps. A lane leading up to the pumps can house at most one extra car excluding those being serviced. Customers go elsewhere if all pumps are busy and the lane is full.

Prospective customers (vehicles) arrive according to a Poisson process at the mean rate of three per minute. Service time is exponentially distributed with a mean of two minutes.

(a) On average, how many customers are at the service station?
(b) What proportion of prospective customers are turned away?
(c) What proportion of the time is a particular pump free?
(d) What is the probability that an arriving car will not start service immediately but will find an empty space in the lane.

Question 5.

A fast food outlet has one-drive in window. Cars arrive according to a Poisson distribution at a rate of four cars every 15 minutes. Assume that cars are willing to queue in the neighbouring streets of the outlet if the drive-in lane is full. The service time per customer is exponential with a mean of 2.6 minutes.

(a) What is the probability that at least six customers (cars) arrive in the next twelve minutes?

(b) Draw the state transition diagram and write down the steady-state (balance) equations for the model. Solve these equations to find an expression for the probability, that n cars are in the system.

(c) On average, how many cars are waiting in the queue (not being served)?

(d) How long on average would a car needs to wait before being served?

(e) What is the probability that the drive-through window is idle?

Question 6.

A manager of a Hyneman Bank is considering how many clerks are needed to serve customers on the following day which is expected to be busy.

The average interarrival time of customers is 3 minutes which is exponentially distributed. Each clerk on average takes 9 minutes to serve each customer which is also exponentially distributed.

The manager estimates the delay cost per minute per customer he or she waits in line (including in-service) is 20 cents per minute. It also costs the post office $14 per hour to employ each clerk.

How many clerks should the bank manager have on sta the next day to minimise the total cost (wages and delay costs)?

State any assumptions you may make.