1 Chapter 17 Standing in line – at the bank, the market, the movies – is the time-waster everyone loves to hate. Stand in just one 15-minute line a day, every day, and kiss goodbye to almost four days of idle time by year’s end.—Kathleen Doheny Waiting Lines (Queues)

2 Queuing Analysis  A queue is a waiting line.  A queuing system involves customers arriv- ing for service who sometimes have to wait.  Queuing analysis provides:  Summary measures for assessing a queuing system in terms of customers and time.  A way to balance the costs of providing service and costs of congestion.  Importance of Queuing Analysis:  Servicing customers can be costly.  Retail environments are plagued with customer congestion. Managing that has benefits.

3 Single-Server Queuing Model (M/M/1)  Assumes exponential distribution for times:  between successive arrivals—mean rate  to complete service—mean rate   Provides important results:  Probability for n customers in system: P 0 = 1   P n =  n P 0  Mean number of customers in system:  Mean customer time spent in system:  L=  W= L = 1 

4 Single-Server Queuing Model (M/M/1)  Important results (continued):  Mean number of customers waiting:  Mean customer waiting time:  Server utilization factor: LqLq =   WqWq = LqLq =   = 

5 Single-Server Queuing Model (M/M/1)  Problem: Customers arrive at a supply room with mean rate = 25/hr. Service take a mean of 2 minutes, so that  = 30/hr. Do a queuing analysis.  Solution:  = 25/30 =.833P 0 = 1 .833 =.167 P 1 = (25/30) 1 (.167) =.139 P 2 = (25/30) 2 (.167) =.116

6 Single-Server Queuing Model (M/M/1)  Problem (continued): Hourly cost of providing service is $10 (clerk’s wages). Lost productivity of employee customers while getting supplies is $7/hr, the congestion penalty. Evaluate total cost.  Solution (continued): The hourly system cost may be used to compare alternatives: TC = hourly queuing cost + hourly service cost = $7×W × + $10 = $7(.20)(25) + 10 = $45  The queuing cost accounts for all customers in the system during the hour, and collective time under a congestion penalty, W ×, applies above.  When the congestion penalty involves just the waiting time, W q × is used for queuing cost. Retail customers resent the wait, not the time receiving service.

7 Multiple-Server Queuing Model (M/M/S)  Multiple servers involve greater complexity.  The model begins with computation of

8 Multiple-Server Queuing Model (M/M/S)  The mean number of customers is then found:  Then the mean customer waiting time is found:  That is followed by:

9 Queuing Spreadsheet Templates and Software  Queuing Templates  Palisade Decision Tools BestFit 4.0

10 Queuing Spreadsheet Templates  M/M/1  M/M/S  Cost analysis for M/M/S  M/M/1 non-exponential service times  Exponential distribution  Poisson distribution  Waiting times for M/M/1  M/M/1 with a finite queue  M/M/1 with a limited population  M/M/1 with a constant service time

11 M/M/1 (Figure 17-4) 2. Queuing summary results are here: L, W, Lq, Wq, and rho. 3. System state probabilities are here, Pn and  Pn. To fit the spreadsheet on one page some of the rows have been hidden. 1. Enter the problem parameters in G6:G7. Remember lambda must be less than mu.

12 M/M/S (Figure 17-6) 3. The number of servers is limited to 100 or less. Caution: For large S and n sometimes the resulting numbers are too large for Excel so error messages occur, such as #NUM! 1. Enter the problem parameters in G6:G8. 2. Remember that lambda must be less than mu times S. Otherwise the results will not be meaningful (negative probabilities).

13 M/M/S Formulas 2. The Sum !A2:C102 in cell D20 refers to the table located on the Sum worksheet, shown next. 3. The formulas in D21:E21 are copied down to the end of the table. 1. All the calculations on this spreadsheet are based on Po found in cell D20.

14 SumWorksheet (Sum Tab) (Figure 17-19) This worksheet is used to compute the sum The sum is used in the P 0 calculation. MMS!$G$6 in cell B2 refers to cell G6 on the MMS worksheet, shown previously. The formulas in B3:C3 are copied down to B102:C102.

15 Cost Analysis for M/M/S (Figure 17-7) Note: In order for the spreadsheet to compute correctly, make sure automatic calculation is selected on both worksheets (Tools, Options, Calculation Tab, and click on the Automatic button under Calculation). 1. Enter the problem parameters in G6:G9. 3. The number of servers is limited to 100 or less. 2. Remember that lambda must be less than mu times S for results to be meaningful. Consequently, rows are hidden (because lambda is not less than mu times s for these rows).

16 Formulas for M/M/S Cost Analysis (Figure 17-20) 1. The Sum !$A$2:$C$102 in cell B22 refers to the table located on the Sum worksheet. It is identical with the one for the M/M/S template shown previously. 2. The formulas in row 22 are copied down to the other rows of the table.

17 M/M/1Non-exponential Service (Figure 17-8) Enter the problem parameters in G6:G8. Remember lambda must be less than mu.

18 Exponential Distribution (Figure 17-9) 1. Enter the problem parameters in D5:D7. 2. The exponential frequency curve and cumulative distribution are here:

19 10 E =IF(E7>0,POISSON(E7-1,E5*56,TRUE),0) 15 E =1-POISSON(E8,E5*E6,TRUE) C =POISSON(E7,E5*E6,TRUE) =POISSON(E8,E5*E6,TRUE)-E10 =POISSON(E8-1,E5*E6,TRUE)-E11 =POISSON(E8,E5*E6,TRUE)-C10 =POISSON(E8-1,E5*E6,TRUE)-C10 =1-POISSON(E8-1,E5*E6,TRUE) B =POISSON(A20,$E$5*$E$6,FALSE) =POISSON(A21,$E$5*$E$6,FALSE) =POISSON(A22,$E$5*$E$6,FALSE) C =POISSON(A20,$E$5*$E$6,TRUE) =POISSON(A21,$E$5*$E$6,TRUE) =POISSON(A22,$E$5*$E$6,TRUE) Poisson Distribution (Figure 17-11) 1. Enter the problem parameters in D5:D8. 2. The Poisson probabilities and cumulative probabilities are here:

20 Waiting Times for M/M/1 (Figure 17-13) 1. Enter the problem parameters in G6:G7. Remember lambda must be less than mu. 2. The cumulative distributions for the waiting times in the line and in the system are here:

21 M/M/1with a Finite Queue (Figure 17-14) Enter the problem parameters in G6:G8. Remember lambda must be less than mu.

22 M/M/1 with a Limited Population (Figure 17-15) Enter the problem parameters in G6:G8. NOTE: The size of the population M is limited to 100.

23 Formulas M/M/1 with a Limited Population 1. All the calculations on this spreadsheet are based on Po found in cell D The Sum !A2:C102 in cell D20 refers to the table located on the Sum worksheet, shown next.

24 SumWorksheet (Sum Tab) (Figure 17-21) 4. Columns B and C will return the error function #NUM! Whenever n > M. This worksheet is used to compute the sum The sum is used in the P 0 calculation. 1. ‘Limited Pop’!$G$8 in cell B2 refers to cell G8 on the Limited Pop worksheet, shown previously. 3. The formula in C3 is copied down to C The formula in B2 is copied down to B12.

25 M/M/1 with Constant Service (Figure 17-16) Enter the problem parameters in G6:G7. Remember lambda must be less than mu.

26 Palisade Decision Tools BestFit The BestFit 4.0 software program on the CD-ROM accompanying this book can be used to find the distribution that best fits a set of data. It compares data with more than 30 different distributions using chi-square, Kolmogorov-Smirnov, and Anderson-Darling tests.A few of the common distributions it checks are beta, binomial, chi-square, exponential, gamma, geometric, hypergeometric, normal, Poisson, triangular, and uniform.

27 BestFit To start BestFit, click on the Windows Start button, select Programs, Palisade Decision Tools, then BestFit 4.0 BestFit will open and the initial screen shown next will appear.

28 Initial Screen 1. Enter the 25 times from Table 17-5 of the text in the Sample column. 2. Click Fitting on the menu bar and select Run Fit. This will give the Fit Results dialog box shown next.

29 BestFit Results Box (Figure 17-17) 1. This figure shows portions of the Fit Results dialog box. 5. Click on the GOF (Goodness of Fit) tab to see the test values of Chi- Sq, A-D, or K-S. 2. The distributions fitted, from the best (Inverse Gaussian) to the worst (Uniform). 3. The distributions are ranked by K-S values. Click on the down arrow to rank by Chi-Square or Anderson- Darling values. 4. Graph of the best fit and the data.

30 Exponential Fit (Figure 17-18) 1. The graph compares the WaySafe data with an exponential distribution. 2. The GOF tab gives the test values. 3. BestFit also gives the critical values for various significance levels. 4. The Chi-Sq and K-S test values are less than all the critical values. The A-D test value is less than the 0.01 critical value but larger than the others.