Consider a roller coaster that lasts for 1 minute and we will discretize time such that one-time slot equals 1 minute. During a ride, one new customer arrives with probability Pl- two new customers arrives with probability p2, and with probability 1 — p1 — p2 there are no new customers. Customers wait in line in a first-come first-served manner (no line cutting). Assume the line can be in?nitely large. A customer that just ?nished hisi’her ride, may choose to repeat the ride. Because the ride is quite intense, customers choose not to repeat the ride with probability q; with probability 1 — q they choose to have one more ride. If a customer chooses not to repeat the ride, the customer departs and the next one rides the roller coaster. Notice that customer arrivals and departures are independent in the various time slots. 1) Draw the state transition diagram for the ?rst 4 states of this DTMC. Note: This is an in?nite state Markov chain. Note: Computing the average time spent on the roller coaster (waiting in the line and going on the ride), W, from the 151’s, the stationary distribution, can be hard. So, another method of solving for W is using the equation: Xn+1 =Xn ‘5n 'Dn +An (1) where Xn denotes the number of customers in the system at the beginning of the nth th time slot, on indicates if there are customers in the system at the beginning of the n th time slot, [JI1 denotes the number of departures at the end of the n th time slot, and An denotes the number of arrivals during the n time slot. 2) Write the distributions of Dn and An- 3) Express P (X 3 O), the probability that there are customers in the system, in terms of p1 , p2 and q. 4) Square both sides of Equation (1), take expectations and then let n —) m to obtain the average number of customers as seen by a departure. 5) Compute W from the average number of customers as seen by a departure. 6) What is the stability condition of this queue?

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