1、1 CHAPTER 1 QUEUEING MODELS SOLUTION TO SOLVED PROBLEMS 11.S1 Managing Waiting Lines at First Bank of Seatle Sally Gordon has just completed her MBA degre and is proud to have earned a promotion to Vice President for Customer Services at the First Bank of Seattle. One of her responsibilities is to m
2、anage how telers provide services to customers, so she is taking a hard look at this area of the banks operations. Customers needing teler service arrive randomly at a mean rate of 30 per hour. Customers wait in a single line and are served by the next available teler when they reach the front of th
3、e line. Each service takes a variable amount of time (assume an exponential distribution), but on average can be completed in 3 minutes. The telers earn an average wage of $18 per hour. a. If two telers are used, what wil be the average waiting time for a customer before reaching a teler? On average
4、, how many customers wil be in the bank, including those currently being served? This is an M/M/s queueing system. The mean arrival rate is = 30 customers per hour. The mean service rate is = (60 min/hr) / (3 minutes/customer) = 20 customers per hour. The number of servers is s = 2. The Template for
5、 the M/M/s queueing model is shown below. The average waiting time for a customer before reaching a teler is Wq= 0.064 hours = 3.86 minutes. The average number of customers in the bank, including those currently being served, is L = 3.43. 12345678910111213A B C D E G H ITemplate for the M/M/s Queuei
6、ng ModelData Results =30(mean arrival rate) L =3.428571429 =20(mean service rate) Lq=1.928571429s = 2 (# servers) minutesW =0.1142857146.857142857Pr(W t) =0.50480467Wq=0.0642857143.857142857when t =0.08333333 = 0.75Prob(Wq t) =0.27938456when t =0.08333333n Pn0 0.1428571432 b. Company policy is to ha
7、ve no more than a 10% chance that a customer wil need to wait more than 5 minutes before reaching a teler. How many telers need to be used in order to met this standard? As in part a, this is an M/M/s queueing system with mean arrival rate = 30 customers per hour and mean service rate = 20 customers
8、 per hour. To satisfy the company policy that there be no more than a 10% chance that a customer wil need to wait more than 5 minutes before reaching a teler, we need to asure that Pr(Wq t) in cel C11 is no more than 10% when t = 0.0833 hours (5 minutes). From part a, when s = 2, Pr(Wq t) = 27.9%. A
9、s shown below, when s = 3, Pr(Wq t) = 1.9%. Thus, First Bank wil need at least 3 telers to met this standard. c. Sally fels that a significant cost is incurred by making a customer wait because of potential lost future busines. Sally estimates the cost to be $0.50 for each minute a customer spends i
10、n the bank, counting both waiting time and service time. Given this cost, how many telers should Sally employ? This question can be answered using the template for economic analysis of the M/M/s queueing model. The cost of service is Cs= $18/hour/server. The cost of waiting is Cw= $0.50/minute/serve
11、r = $30/hour/server. The total cost of service and waiting is Cs+ CwL. As sen in the templates below, with 2 servers the total cost is $138.86. With 3 servers, the total cost is $106.11. With 4 servers the total cost is $118.34. Therefore, Saly should employ 3 servers. 12345678910111213A B C D E G H
12、Template for the M/M/s Queueing ModelData Results =30(mean arrival rate) L =1.736842105 =20(mean service rate) Lq=0.236842105s = 3 (# servers)W =0.057894737Pr(W t) =0.23946063 Wq=0.007894737when t =0.08333333 = 0.5Prob(Wq t) =0.01944118when t =0.08333333 n Pn0 0.2105263163 12345678910111213141516171
13、81920A B C D E F GTemplate for Economic Analysis of M/M/s Queueing ModelData Results =30(mean arrival rate) L =3.428571429 =20(mean service rate) Lq=1.928571429s = 2 (# servers)W =0.114285714Pr(W t) =0.50480467 Wq=0.064285714when t =0.08333333 = 0.75Prob(Wq t) =0.27938456when t =0.08333333 n Pn1 0 0
14、.1428571432Economic Analysis: 1 0.2142857140 Cs =$18.00(cost / server / unit time) 2 0.1607142860 Cw =$30.00(waiting cost / unit time) 3 0.1205357140 4 0.0904017860 Cost of Service$36.00 5 0.0678013390 Cost of Waiting$102.86 6 0.0508510040 Total Cost$138.86 7 0.03813825312345678910111213141516171819
15、20A B C D E F GTemplate for Economic Analysis of M/M/s Queueing ModelData Results =30(mean arrival rate) L =1.736842105 =20(mean service rate) Lq=0.236842105s = 3 (# servers)W =0.057894737Pr(W t) =0.23946063 Wq=0.007894737when t =0.08333333 = 0.5Prob(Wq t) =0.01944118when t =0.08333333 n Pn1 0 0.210
16、5263162Economic Analysis: 1 0.3157894741 Cs =$18.00(cost / server / unit time) 2 0.2368421050 Cw =$30.00(waiting cost / unit time) 3 0.1184210530 4 0.0592105260 Cost of Service$54.00 5 0.0296052630 Cost of Waiting$52.11 6 0.0148026320 Total Cost$106.11 7 0.0074013164 1234567891011121314151617181920A
17、 B C D E F GTemplate for Economic Analysis of M/M/s Queueing ModelData Results =30(mean arrival rate) L =1.544751381 =20(mean service rate) Lq=0.044751381s = 4 (# servers)W =0.051491713Pr(W t) =0.1974963 Wq=0.001491713when t =0.08333333 = 0.375Prob(Wq t) =0.00115636when t =0.08333333 n Pn1 0 0.22099
18、44752Economic Analysis: 1 0.3314917131 Cs =$18.00(cost / server / unit time) 2 0.2486187851 Cw =$30.00(waiting cost / unit time) 3 0.1243093920 4 0.0466160220 Cost of Service$72.00 5 0.0174810080 Cost of Waiting$46.34 6 0.0065553780 Total Cost$118.34 7 0.0024582675 d. First Bank has two types of cus
19、tomers: merchant customers and regular customers. The mean arrival rate for each type of customer is 15 per hour. Both types of customers currently wait in the same line and are served by the same telers with the same average service time. However, Sally is considering changing this. The new system
20、she is considering would have two linesone for merchant customers and one for regular customers. There would be a single teler serving each line. What would be the average waiting time for each type of customer before reaching a teler? On average, how many total customers would be in the bank, inclu
21、ding those currently being served? How do these results compare to those from part a. The queueing systems for merchant and regular customers are two separate, but identical, M/M/s queueing systems. For each, the mean arrival rate is = 15 customers per hour. The mean service rate is = 20 customers p
22、er hour. The number of servers is s = 1. The template for the M/s queueing model is shown below. The average waiting time for a customer before reaching a teler is Wq= 0.15 hours = 9 minutes. The average number of customers of each type in the bank, including those currently being served, is L = 3.
23、Thus, the total number of customers (of both types) is 6. These results are significantly worse than those from part a. 00.050.10.150.20.250.301234567891011213141516171819202122232425Number of Customers in SystemProbability6 e. Sally fels that if the telers are specialized into merchant telers and r
24、egular telers, they would be more eficient and could serve customers in an average of 2.5 minutes instead of 3 minutes. Answer the questions for part d again with this new average service time. Like part d, the queueing systems for merchant and regular customers are two separate, but identical, M/M/
25、s queueing systems. For each, the mean arrival rate is = 15 customers per hour. The mean service rate is = (60 min/hr) / (2.5 minutes/customer) = 24 customers per hour. The number of servers is s = 1. The template for the M/M/s queueing model is shown below. The average waiting time for a customer b
26、efore reaching a teler is Wq= 0.069 hours = 4.17 minutes. The average number of customers of each type in the bank, including those currently being served, is L = 1.67. Thus, the total number of customers (of both types) is 3.33. These results are similar to those from part a. The waiting time before reaching a teler is slightly higher (4.17 minutes vs. 3.86 minutes), but the total number of customers in the bank is slightly smaler (3.33 vs. 3.43). 00.050.10.150.20.250.30.350.401234567891011213141516171819202122232425Number of Customers in SystemProbability
