运营管理基础 4E-戴维斯英文IMChap011s

1、SUPPLEMENT 11 WAITING LINE THEORYChapter OverviewOne method of analyzing waiting line is queuing theory. It is used extensively in manufacturing and service environments, and is a standard tool in such areas as service delivery systems design, scheduling, and machine loading. Specifically, this chap

2、ter introduces some of the more common queuing models.Major Points of ChapterThe six major components of a waiting line are source population, arrival characteristics, line(s), selection from line(s), and service facility.To select the appropriate model, the various characteristics of the line must

3、be described.If an appropriate queuing model is not available, computer simulation can be used to analyze the waiting line system.There is a trade-off between capacity utilization and waiting time. Managers need to be aware of this relationship.Review and Discussion QuestionsHow many waiting lines d

4、id you encounter during your last airline flight?Answers will vary, but possibilities include baggage check-in, ticket counter, security check, check-in at gate, at boarding, and baggage pickup.Distinguish between a channel and a phase.A channel is the initial service point of a queuing system. A ph

5、ase refers to the number of stages that the service points provide. It is possible to have single to multiple service channels and single to multiple service phases.Which assumptions are necessary to employ the formulas given for Model 1?Poisson arrival rates, exponential service rates, which imply

6、a purely random process, but with a known mean (and hence known variance). Also assumed is that the process has reached a point of stochastic equilibrium. In other words, steady state conditions prevail.151Supplement 11In what way might the first-come, first served rule be unfair to the customer wai

7、ting for service in a bank or hospital?In a bank, FCFS may be perceived to be unfair by customers who have large accounts, but who must wait while the less important” customers obtain service.In a hospital, especially in an emergency room, FCFS is probably the exception rather than the rule. FCFS wo

8、uld be unfair when a patient with a minor problem is treated before another experiencing severe pain, for example.Identify the various types of waiting lines you encounter in a “normal day.Answers will vary, but typical examples include stores, restaurants, phones, intersections, professor office, c

9、omputer, etc.Compare the queuing systems of McDonalds and Wendy9s?McDonalds typically uses a multiple-line system (one line for each cashier), while Wendys typically uses a single line, single cashier system.Why do you think doctors5 and dentists offices usually have such long waits?There are severa

10、l possible answers, including variability in times required to see patients, the addition of patients at the last minute to the schedule. But, probably the primary reason is to maximize utilization of the doctors or dentists. The concept is that their time is more valuable than the patients and, the

11、refore, patient will wait to ensure that the doctors and dentists are utilized to a very high level.ProblemsProblemType of ProblemDifficultyCheck figure in Appendix DModel 1Model 2Model 3Model 41YesEasyYes2YesDifficultYes3YesEasy4YesModerate5YesModerate6YesModerate7YesModerate8YesModerate9YesModerat

12、e10YesDifficult11YesDifficult152Use model 2.4 = 60/50 per minute R 60/45 per minute60/50a tl - 2(60 / 45)(60 / 45 - 60 / 50)=3.375 minutes.Zs = t +l/ = 3.375 + 1/60/45 = 4.125 minutes2260/502b. Use model 2.4 = 60/50 per minute R 60/45 per minute60/50a tl - 2(60 / 45)(60 / 45 - 60 / 50)=3.375 minutes

13、.Zs = t +l/ = 3.375 + 1/60/45 = 4.125 minutes2260/502b. n. = 4.05 cars2( 2) 2(60 / 45)(60 /45-60/50)Yls = % +，/ = 4.05 + (60/50)/(60/45) = 4.95 carsUse model 3. 2 = 30per hour / =40 per hour Pn11A JSpacesPoUtilizationIncrease in UtilizationIncreasedRevenue/yr23456789100.4324320.3657140.3277850.30412

14、80.2885120.2778130.2702950.2649180.2610240.5675680.6342860.6722150.6958720.7114880.7221870.7297050.7350820.7389760.0667180.0379290.0236570.0156170.0106990.0075180.0053770.003894$7,013.41$3,987.14$2,486.78$1,641.62$1,124.68$790.25$565.18$409.35Use model 2.2= 10 per hour / = 12 per houra.a.a.b.c.22102

15、a.b.c.22102n, = = 2.083 people2(-4) 2(12)(12-10)ns nl = 2.083 + 10/12 = 2.917 people210=.2083 hours 2( 4) 2(12)(12-10)d.ts =t1 +1/Z = .2083 + 1/12 = .2917 hourse.d.e.e.It will cause it to increase, at 2 = 12 per hour, / =e.It will cause it to increase, at 2 = 12 per hour, / =A2122- oo2( 4) 2(12)(12-

16、12)Supplement 114.4.4.Use model 1A = 3 per minute 4 =4 per minutea.= 3 workers /Li X 4 3b.=1 minute / 4 4 34.Use model 1A = 3 per minute 4 =4 per minutea.= 3 workers /Li X 4 3b.=1 minute / 4 4 33-=.75 or 75%4d.Probability of 3 or more is equal to 1 - probatility of 0, 1, 2.25001 3丫3 丫1 = .1875p/ 3丫3

17、 尸2 一匚= .1406Total ofP()+ Pi +P2 = (.2500 + .1875 + .1406) = .5781Therefore, the probability of three or more is 1 - .5781 = .4219If a automatic vendor is installed, use model 2.32n.=1.125 workers2(4)(4-3)ns = /= 1.125 +% =1.875一 43t =.375 minutes2( 2) 2(4)(4-3)t s+1/= .375 + % = .625By converting t

18、o constant service time, the number in line is reduce from 3 to 1.875 people (a reduction of 1.125, and time in system is reduced from 1 minute to .625 minutes (a reduction of .375 minutes or 22.5 seconds).154Use model 1.A = 20 per hour 4=30 per hour20a.=2 people in the system/-A 30-20b.c.= .1() hou

19、rs or 6 minutes4-4 30-20Probability of 3 or more is equal to 1 - probatility of 0, 1, 2p= 1.3333d.4=1 2oYZ230 A 30=.2222竺丫型、30 A 30 2Use model 1.A = 20 per hour 4=30 per hour20a.=2 people in the system/-A 30-20b.c.= .1() hours or 6 minutes4-4 30-20Probability of 3 or more is equal to 1 - probatility

20、 of 0, 1, 2p= 1.3333d.4=1 2oYZ230 A 30=.2222竺丫型、30 A 30 2= .1481Total ofP()+ Pi + P2 = (.3333 + .2222 + .1481) = .7036Therefore, the probability of three or more is 1 - .70362 20p = = .67 or 67%430.2964. Use model 14 = 5 per hour / =6 per houra.=1 hour zl 6 5b.52c.n, =4.17 people on average, therefo

21、re add 5 seats.(-) 6(6-5)2 5It would be busy p = =.833 or 83.3% of the time, 6for a 12 hour day .833(12) = 10 hoursd.i-A40= .167 or 16.7%.75=4 51u4 =F5 = 6.33, therefore, the service rate must be at least 6.33 customers per hour* 7Supplement 11Use Model 2,A = 5 per hour =6.7 per hourt. = .225 hours

22、or 13.5 minutes.2(6.7)(6.7-5)ts = 0 +1/Z = .225 + 1/6.7 = .375 hours or 22.5 minutesUse Model 42= 1/30 per day - 42= 1/30 per day - 42= 1/30 per day - 425 days per case 4 = 1/25 per day2= 1/30 per day - 425 days per case 4 = 1/25 per daya2 =(27 - 25)2 + (26 - 25)2 + (26 - 25/+ (25 - 25)2 + (27 - 25)

23、2 + (24 -25)2+ (27 -25)2+ (23 - 25)2 + (22 - 25)2 + (23- 25)2 )/1()3.2a. 25 days per caseb. b. ntb. ntA?1/30、IM J7n2 i-4I Q1/25J2+ (1/30)2(3.2)2/2 1-b. ntA?1/30、IM J7n2 i-4I Q1/25J2+ (1/30)2(3.2)2/2 1-1/301/25=2.12 cases2. 2(;A2 1-、4/1/30_ (l/25+ (1/30)(32)21/30、2 1I 1/25 J=63.52 daysUse model 12 =

24、70 per hour =(1/40)3600 = 90 per hour70t. = .0389 hours or 2.33 minutes./ ( ) 90(90-70)2 70p = = = . 78 or 78% 90For cars with out stickers:42=.00972 hours or .583 minutes or 35 seconds.Z = .60(70) = 42 per hour 42=.00972 hours or .583 minutes or 35 seconds./(/-2) 90(90 - 42)For cars with stickers:2

25、 = .40(70) = 28 per hour =(1/5)3600 = 720 per hour2828282(720)(720-28)=.0000279 hours or .00167 minutes or .10 seconds.156282(720)(720-28)=.0000279 hours or .00167 minutes or .10 seconds.1562 = 50/8 = 6.25 per hour =(1/3)60= 20 per hour一 46.25cccrt, = .0227 hours or 1.36 minutes./(/-2) 20(20-6.25)2

26、6.25p = =.3125 or 31.25%420Use Model 1, for guest reservations:4 = .75(80/8) = 7.50 per hour /d =(1/2)60= 3() per hour44Mr心4Mr心7.5030(30-7.50)4Mr心7.5030(30-7.50)= .01111hours or .67 minutes.2 7.50p 2 7.50p = . 25 or 25%430Use Model 1, for technical support:2 = .25(80/8) = 2.50 per hour =(1/6)60= 10

27、per hour一 10(10-2.50)2.50 =.0333 hours or 2 minutes.2.50 _10. 25 or 2.50 _10. 25 or 25%Supplement 11Use model 1.4=3 per hourWith 4 people: =60/18 = 3.33 per hour:3n = 9.00 trucks5 4-43.33-3With 5 people: =60/15 = 4.00 per hournv = 3.0U trucks A 4 3With 6 people: / =60/12 = 5.00 per hour23 1s= 1.50 t

28、rucks4 - 4 5 3With 7 people: =60/10 = 6.00 per hour%31nM& = 1.UO trucks - 4 6 3Number of people unloadingService rate per hour3nsCost of waiting per hour1Cost of service per hour2Total cost per hour43.339.00$675$64$73954.003.00$225$8()$30565.001.50$113$96$20976.001.00$75$112$187Note: 1 = cost of waiting is number in system times waiting cost of $75 per hour.2 = cost of service is number of people unloading times $16 per hour.Seven unloading people should be selected as it has the lowest total cost per hour.158Use model 1.4 = 2.5 per hourWith 1 person: =60/15 = 4.00 per hour TOC o 1-5