(This exercise is repeated from Exercise 27 in Chapter 6, with an embellishment.) In this problem the goal is to model the use of automated teller machines (ATMs) at a bank. After inserting their card into an ATM, there are three types of transactions that customers may perform: 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 example, 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. Suppose that we have the following additional information about how long a customer spends on each transaction. All times are modeled as exponentially distributed random variables.
(a) Derive a Markov process model capable of answering the questions below. Be sure to define your state space, time index and generator matrix.
(b) Do you think the assumption of exponentially distributed transaction times is appropriate? Why or why not?
(c) What is the probability that a customer takes longer than 4 minutes at the ATM? Calculate a numerical result.
(d) Use the result in Exercise 11 to calculate the expected time a customer spends at the ATM.
Exercise 27 in Chapter 6
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.
Exercise 11
For a Markov process with generator matrix G, let ( be the set of transient states (assume that there are mM
> 0 transient states). Let μij
denote the expected total time the process spends in transient state j given {Y0
= i}. Show that
(where pik
is a one-step transition probability from the embedded Markov chain) by conditioning on the first transition out of state i.