STOCHASTIC PROCESS

lQuestion 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 2D 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: {1'} 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 [ii] 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 immediately on arrival (i.e. without joining the queue} and advise the manager appropriately