Assignment 6

• You may work individually, or as a group of 2 students. (If you work as a group, please submit one assignment per group.)

1. The Manager of the American Savings Bank wants to determine the minimum number of tellers needed to handle lunchtime customers. The actual time between two customers arriving follows an exponential distribution. Each teller can handle an average of 12 customers an hour, but the time for each customer varies an according to an exponential distribution.

(a) Suppose there is only one teller, and customers arrive at a rate of one about every 7.5 minutes. Find the values of the average waiting time Wq , average time in the system W , mean queue length Lq , and mean number in the system L.

(b) Suppose there is only one teller. The manager has observed that there seems to be an average of 5 customers in the bank at any given time. Find the values of Wq , W , Lq and L.

(c) Suppose there are two tellers, and customers arrive at a rate of one about every 7.5 minutes. Find the values of Wq , W , Lq and L.

2. The Vice-President of Operations for Texas Airways, you, Ms. Kimberly Brown, have been asked to look into complaints that passengers in Dallas are spending too much time at the check-in counter and that the line is too long. To learn more about this problem, you developed a queuing model to study the existing system, which currently consists of one waiting line and four ticket agents.

After preliminary data analysis, you discovered that the passengers arrive during the busiest hours according to a Poisson process at a rate of 132 per hour. Each ticket agent requires an average of 1.75 minutes to process a passenger, but the time for a customers is random and follows an exponential distribution.

(a) For the busy hours, determine the average number of passengers waiting in line and how much time (in minutes) they spend both in line waiting for service and for the complete check-in process.

(b) In view of the results above, you decided to discuss the problem with the ticket agent to ?nd out what is causing the delays. Some customers, the agents said, need only to check-in, and they move quickly. Other customers take longer, checking ?ight schedules, and so on. On the basis of these discussions, it was suggested that service could be improved by having two separate lines: one line (and one ticket agent) devoted to passengers with time-consuming special problems and the other lines (and three ticket agents) for check-in passengers only. To examine this data, further data analysis was performed to determine that an average of 12 of the 132 passengers arriving each hour require special handling. Their random service time is not exponential but instead takes an average of 3 minutes with a standard deviation of 1 minute. The remaining 120 passengers that arrive each hour requires an average of 80 seconds for check-in, and this time is random and follows an exponential distribution.

For these two queues, report the same statistics as you did in part (a) for passengers requiring special handling and for those needing only to check-in. Assume that arrivals for each queue still follow a Poisson process with appropriate arrival rates. 1 (c) Mr. Carlos Moreno, Manager of the Dallas Airport check-in counter for Texas Airways, suggested that further improvements might be possible because many check-in passengers are daily commuters with no baggage. Perhaps, he suggested, it might be advisable to split the newly proposed check-in lines into two lines: one for passengers with baggage and one for passengers without baggage. Data analysis revealed that the average 120 passengers arriving for check-in per hour, 45% of them had baggage and required an average of 96 seconds each for check-in; this time is random and follows an exponential distribution. The remaining 55% had no baggage and so could be processed in an average of 48 seconds; this time is random and follows an exponential distribution. Assuming that arrivals in both queues follow a Poisson process with appropriate arrival rates, explain why in this scenario, you should not consider having one ticket agent for passengers with baggage and two agents for passengers without baggage if you have two separate lines for check-in passengers. Instead you should consider having two ticket agents for passengers with baggage and one ticket agent for passengers with no baggage.

(d) What are the waiting-time statistics for passengers with and without in the system proposed in (c)? Who waits longer in line, on average: passengers with baggage or passengers without baggage?

(e) Assume that the President of the company wants check-in passengers to spend an average of less than 5 minutes waiting in line for check-in services. Which system would you choose, the one in (b), or the one in (c)? Explain.

3. The PayFast supermarket currently has four cashiers. The company wants to upgrade its service by hiring and additional cashier or by putting in bar-code scanners at each checkout. Historical data indicates that (a) customers arrive according to a Poisson process at the average rate of 45 per hour and (b) the amount of time a cashier needs to handle a customer follows an exponential distribution with an average rate of 15 customers per hour.

Consultants estimate that modernizing with bar-code equipment would increase e?ciency by 20%. The Accounting Department suggests that the "cost" of a waiting customer should be valued at $30 per hour and that the hourly cost of a cashier is $15, including bene?ts.

However to cover the cost of the bar-code equipment, the hourly rate of the casher should be increased to $18. Determine whether the company should hire an additional casher or install the bar-code equipment at the existing check-outs. (Assume that the customers form a single queue when checking out.) (Use EXCEL, if necessary.)

4. Consider a variation of the M/M/1 queue where the maximum number of customers permitted in the system is K. Any customer that arrives when the system has K customers is turned away. Show that the probability that the total number of customers in the system is n is given by: Pn = ρn · for n = 0, 1, . . . , K, where ρ = λ/µ. 2 1-ρ 1 - ρK +1.