Please solve attached problem and show all work. Thank you!

Q3. One way to mitigate the effects of disasters that disrupt the power grid is to introduce more power storage in the system through the use of batteries. This can be done at a large scale through \"big battery\" technology, as well as at a small scale, such as at an individual home level. In other words, if you were to invest in a large battery for your home, then you could charge the battery when prices are low and use the battery power when prices are high, as well as during times when there is a disruption in the power grid and battery power is all that you have available. Suppose that you have invested in such a battery and have established an inventory policy for chargmg/using the battery. Now you are interested in the long-run behavior of the inventory system that you have established. You are interested in this since you know that a disaster can occur randomly on any given day, and you'd like to know the probability that you will have enough charge in the battery when disaster strikes Let X, be the number of kWh that you have stored in the battery at the beginning of day t. Based on your household's average usage of electricity each day, the average price of electricity each day, and other factors, you determine that, for all t E {0, 1, 2, . . . }, Xt-l-l = maX{0, mjn{Xt + Dt, 5}}, where lP[Dg = 2] = 0.1, lP'[Dt = 1] = 0.3, IND; = 0] = 0.4, and IND; = 1] = 0.2. (a) (10 points) Formulating this system as a Markov chain, what is the transition probability matrix? [b] (5 points) Determine classes for the states of this Markov chain. For each class, determine if it is recurrent or transient. Is any recurrent state absorbing? (c) (10 points] State the equations dening the steady state probabilities for this Markov chain, then solve them to determine the steady state probabilities for the Markov chain. ((1) (5 points) Suppose that when a disaster strikes, the cost that you will incur depends on how many kWh you have stored in your battery at that time. In particular, suppose that the cost that you will incur is described by the function CU whose values are 0(0) = $2000, 0(1) = $1000, 0(2) = $500, 0(3) = $250, 0(4) = $125, and 0(5) = $0. Since a disaster can strike randomly at any given day in the future, what is your expected cost when a disaster strikes? 5