College Credit UnionCollege Credit Union (CCU) was created several years ago in order to provide the students andalumni of a local college with convenient banking options complete with school-related financialservices that larger retail banks in the area did not offer. Due to increasing enrollment and agrowing body of alumni who use CCU's services, more and more customers are visiting the bank.Recognizing this trend, Adrien, the CCU's Vice President of Operations, has decided to model thesystem in hopes that he can make informed resource-allocation decisions to support the increasingdemand for CCU's teller services.Every weekday, customers arrive randomly during the 3:00 P.M. to 4:00 P.M. peak hour (rightbefore the bank closes) at an average rate of one person every 40 seconds. Since CCU's inception,Adrien has collected historical service-time data during this one-hour period, and has determinedthat a teller spends an average of 4 minutes with each customer, no matter what weekday it is. Thequeuing model assumptions of Poisson arrivals, exponential service times, and infinite customerpopulationsize were evaluated and found to be appropriate for use in modeling CCU's telleroperations.Adrien is worried that, if the customer base grows too fast, the tellers will frequently becomeoverloaded. This is especially worrisome, since CCU lacks a significant number of ATM's aroundthe city where people tend to make night deposits and withdrawals. While Adrien is not meetingthese two service criteria at present, he is committed to having a high level of customer serviceand wants the tellers to be able to service at least 90% of arriving customers immediately, (i.e.,without any waiting in the queue). He also, wants no more than 1% of the customers to find thewaiting area full because there is no other appropriate space in which to wait.With the current configuration, the bank has enough space to accommodate 5 waiting customersin a single line, which feeds all of the tellers on a first-come-first-served basis. Some tellers aremore senior than others, but on average, the cost of a teller is $17 per hour.Adrien has hired you as a consultant to conduct a separate analysis to complement his own work.Please help Adrien address the congestion issues facing CCU by using your understanding ofqueuing systems to answer the following questions
Part A
Please use your understanding of the Poisson and negative exponential probability distributions toaddress the following
1. What is the probability that no customers enter the teller system during a 3-minute periodbetween 3:00 P.M. and 4:00 P.M.? Five customers? More than five customers?
2. What is the probability that a bank teller spends less than 4 minutes with a customer? Between3 and 5 minutes? More than 6 minutes?
3. What is the probability that a teller spends at least 4 more minutes with a customer who hasalready been in service for 4 minutes?