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Question 2 A take-away food counter has one server. Customers arrive randomly at a rate of A per hour. If there is a queue, some customers go elsewhere 69 so that the probability of a potential customer staying for service is zy when there are n customers in the shop already waiting or being served. Service times are independent, but the server is new to the job and tends to make mistakes under pressure, so the rate of service drops to nti per hour when there are n customers present, where a > A. Model this system as a birth-and-death process. (a) Show that Pa (n = 0, 1,2, ...), the equilibrium probability that there are n customers in the system, is given by: Pn = (n + 1)p" Po, with p = 4 (b) Show that the server will be busy for a proportion p(2 - p) of the time. Hint: You may need to use the identity E(n+ 1)an = (1 - 1)2 for |z|