Stochastic Modelling

Question 2 A takeaway food counter has one server. Customers arrive randomly at a rate of A per hour. If there is a queue, some customers go elsewhere so that the probabilityr of a potential customer staying for service is n+4 when there are 71 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 r, per hour when there are n customers present, where p. r: A. Model this system as a birth anddeath process. {a} Show that p\" {n = U, 1, 2, . . .), the equilibrium probability that there are n customers in the system, is given by: . J. pa = {11+ Uri-\"Pu, \"nth P: p. {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 I36 Z(n+1}:r\" = for I31 :1: 1. nzl] Question 3 A small supermarket has two selfservice scanners where customers can pay for their purchases. Customers arrive at these randomly at an aver age rate of 2'3 per hour and have independent exponentially distributed ser vice times, taking on average ve minutes each to complete their purchases. Currently there is a single queue for both scanners, but a new manager is considering two further options: (13] relocating one of the scanners to the opposite end of the shop so that there will be separate queues for each machine, with customers assumed equally likely to enter either queue [it] selling one scanner, leaving only one to serve all the customers For the three possible options (including the existing setup] calculate, where possible, the proportion of customers who will be served inunediately on arrival [i.e. without joining the queue} and advise the manager appropriately