1. Problems Markov Chains 1 1) Given the following one-step transition matrices of a Markov chain, determine the classes of the Markov chain and whether they are recurrent. a. b.

2. Problems Markov Chains 3 2) The leading brewery on the West Coast (A) has hired a TM specialist to analyze its market position. It is particularly concerned about its major competitor (B). The analyst believes that brand switching can be modeled as a Markov chain using 3 states, with states A and B representing customers drinking beer produced from the aforementioned breweries and state C representing all other brands. Data are taken monthly, and the analyst has constructed the following one-step transition probability matrix. A B C What are the steady-state market shares for the two major breweries?

3. Problems Markov Chains 3 3)Louise Ciccone, a dealer in luxury cars, faces the following weekly demand distribution. Demand012 Probability.1.5.4 She adopts the policy of placing an order for three cars whenever the inventory level drops to two or fewer cars at the end of a week. Assume that the order is placed just after taking inventory. If a customer arrives and there is no car available, the sale is lost. Show the transition matrix for the Markov chain that the describes the inventory level at the end of each week if the order takes one week to arrive. Compute the steady- state probabilities and the expected number of lost sales per week.