1. (50 points) M/M/1 with blocking. Consider a M/M/1 queue where there is a maximum number of k customers allowed in the system. Assume that customers arrive at the system according to a Poisson process at rate X. Each customer has an independent service requirement - Exponential(). If there are k customers in the system then an arriving customer is blocked and lost, otherwise she joins the queue (if the server is busy) or enters service (if the system is empty). (a) [10 points) Formulate the headcount process {X(t), + > 0} as a continuous-time Markov chain. (Specify the state space, transition probabilities, and holding time parameters.) (b) [20 points) Find the stationary distribution. What if > ? (c) (10 points) What is the long-run proportion of customers who are blocked? (d) [10 points) What is the long-run proportion of customers who join the system but have to wait? 1. (50 points) M/M/1 with blocking. Consider a M/M/1 queue where there is a maximum number of k customers allowed in the system. Assume that customers arrive at the system according to a Poisson process at rate X. Each customer has an independent service requirement - Exponential(). If there are k customers in the system then an arriving customer is blocked and lost, otherwise she joins the queue (if the server is busy) or enters service (if the system is empty). (a) [10 points) Formulate the headcount process {X(t), + > 0} as a continuous-time Markov chain. (Specify the state space, transition probabilities, and holding time parameters.) (b) [20 points) Find the stationary distribution. What if > ? (c) (10 points) What is the long-run proportion of customers who are blocked? (d) [10 points) What is the long-run proportion of customers who join the system but have to wait