Discuss the below:
Q: In the last few years, many banks have switched from a system having a line at each teller's window to a single waiting line cordoned off by a set of ropes or chains. Examine this change by considering the following two systems. In each system, it is assumed that the number of arrivals is random and has a Poisson distribution. Moreover, the service times are exponentially distributed.
System 1: Customers arrive at an average rate of 48 per hour and wait for service in a single line. There are three tellers and each of them can process an average of 20 customers per hour.
System II: Customers arrive at a teller's window at a rate of 16 per hour (so that with three tellers there are 48 customers per hour on average). There is a separate line at each window and velvet ropes are placed to prevent customers from switching between lines. As in System I, the average service rate is 20 customers per hour.
(a) In System 1, what is the average waiting time before being served? What is the average time that a customer has been spent at the bank?
(b) In System II, what is the average waiting time before being served? What is the average time that a customer has been spent at the bank?
(c) Based on your findings in (1) and (2) above, which of the two systems is more efficient?