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Question

First Local Bank would like to improve customer service at its drive-in facility by reducingwaiting and transaction times. On the basis of a pilot study, the bank's process managerestimates the average rate of customer arrivals is 15 cars per hour. All arriving cars line upin front of a window and are served on a first-come, first-served basis. Each transactionrequiresonaverage3minutestocomplete.AssumethatthearrivalofcarsfollowsaPoissonprocess,andthatthetransactiontimesareexponentiallydistributed.

The bank now has one window that serves the drive-in customers. It costs $20 an hour tostaff this window. The customer relations (CR) manager estimates that the cost (loss ofgoodwill and future business, etc.) due to a customer's waiting in the queue has a linearform, i.e., the bank incurs a cost of 10 dollars per hour per customer in queue. On the otherhand,assoonasacustomergoesintoservice,thebankdoesnotincuranygoodwillloss.

1. Whatistheaveragetime(inminutes)acustomerspendsinqueue?

1. Whatistheprobabilitythatacustomerneedstowaitbeforebeingserved?

1. Whatistheaveragenumberofcustomersinthesystem(includingtheoneswaitingin lineandtheonesreceivingserviceat thewindow)?

1. Whatistheprobabilitythatthereare3customersinthesystem(includingtheoneswaiting inline andtheonesreceiving serviceatthewindow)?

1. Whatistheutilizationofthecustomerservicerepresentativeatthewindow?

1. The bank is considering the possibility of leasing high-speed information-retrievalandcommunicationequipmentthatcosts$20anhour.Theequipmentisoperated by the original staff employee. As a result, the average transaction time will bereduced by one minute. To minimize the total cost of staffing, equipment, and customerwaiting, should the bank lease the communications equipment? That is, how does thetotalcostafterleasingcomparetothatbefore?

1. Another way to improve customer service is to open a second windowinstead of leasing high-speed information-retrieval and communication equipment. Itcosts $20 an hour to staff the second window, and both windows have the same averageservice time, 3 minutes. The customers will form two separate lines, each in front of onewindow. Upon arrival, a customer will choose one of the windows to line up with equalprobability(i.e.,50%chance),andjockeyingbetweenqueuesisnotallowed.Theservicedisciplineisfirst-come,first-served.Basedonthetotalcostofstaffingandcustomerwaiting,isthisalternativebetterthanstatusquo?Whatisthetotalcostnow?

The manager realized that letting customers form two separate lines is not an efficientqueue layout. Instead, the customers, upon arrival, are asked to form one single line in frontofthetwowindows.(Thereare stilltwowindows open).

1. Comparing to the two separate lines layout, what is the difference, if any, inthe average number of customers waiting in line when using the single line layout?Explainthedifference (orindifference) withintuition.

1. Comparing to the two separate lines layout, what is the difference, if any, inthe utilization of the service windows when using the single line layout? Explain thedifference(orindifference)withintuition.

1. Now,supposethecustomersformasinglequeue,andtheservicedisciplineisfirst-come,first-served.Basedonthetotalcostofstaffingandcustomerwaiting,isthisalternativebetter thanthestatusquo?Whatisthetotalcostnow?