In this problem the goal is to model the use of automated teller machines (ATMs) at a bank. After inserting a card into an ATM a customer may perform three types of transactions: deposit, withdrawal, and obtain account information. The bank believes that 50% of all customers start with a withdrawal, 40% start with a deposit, and the remainder start by requesting account information. After completing a transaction, 90% of the customers complete their business (obtain their card and leave); those who do not complete their business are equally likely to select one of the other two types of transactions (for instance, if they just made a withdrawal and they do not complete their business, then they are equally likely next to select a deposit or request account information). This pattern continues until their business finally is completed.
(a) Derive a Markov chain model capable of answering the questions below. Be sure to define your state space, time index, and one-step transition matrix.
(b) Evaluate the Markov and stationarity properties for this situation. Do you think they are appropriate? Why or why not?
(c) Including inserting the card as a transaction, what is the probability distribution of the number of transactions that a customer performs on an ATM? Carry the calculation out to n = 20 transactions.
(d) If customers withdraw $100 on each withdrawal they make, what is the expected amount of money withdrawn by customers each time they use an ATM? See Exercise 23 for some help.
Exercise 23
For a Markov chain with finite state space M and one-step transition matrix P, let T be the set of transient states (assume that there are mT
> 0 transient states). Let M be the mT
× mT
matrix with elements μij
denoting the expected number of times the process is in transient state j given {S0
= i}. Show that
where (i = j) is the indicator function taking the value 1 if i = j and 0 otherwise.