3. (a) During lunch hour, arrivals of customers at a pizza hut restaurant follows a Poisson process with the rate of 120 customers per hour. The restaurant has one line, with three workers taking food orders at independent services stations. Each worker takes an exponen- tially distributed amount of time-on average 1 minute-to serve a customer. Let X, denote the number of customers in the restaurant (in line and being serviced) at time t. Then, the process (Xt20) is a continuous-time Markov chain. (i) Show that the process is a birth-and-death process by giving the birth and death rates. [4 marks] (ii) Find the generator matrix of the above process. [5 marks] (iii) For each integer k ≥0, derive the long-term probability that there are k customers in the restaurant. [8 marks] (iv) Calculate the long-term probability that all three workers are busy. [6 marks] (v) Find the average number of customers in the restaurant in the long term. [7 marks] (b) A student support center has 3 tutors who help students with their home work. Students arrive at the center according to a Poisson process at rate A = 3 per hour. Each tutor's service time is exponentially distributed with average of 1/10 hours. Tutors' service times and student arrival times are independent. If all the tutors are busy when a student arrives at the center, the student will leave. Let X, denote the number of tutors who are busy at time t. Determine its generator matrix and stationary distribution. [10 marks]