copia de queuetransient

222126347.xls.ms_office. Instructions, page 1 of 10

A Transient Queuing Model v. 2.3, 25 Oct 2007

John O. McClain

Johnson Graduate School of Management

Sage Hall, Cornell University

Ithaca NY 14853

This spreadsheet is intended for teaching purposes. You are welcome to use it in any

manner and change it as you see fit. This model comes without any guarantee, and is

distributed free of charge.

Contents:

The Model

The Transient Finite Queue Simulation begins with no customers, and simulates a

fixed interval of time (input), using the Gamma probability distribution to generate times

between arrivals and service times.

● If CV (coefficient of variation)= 1.0, Gamma is the Exponential distribution

At the end of a designated time period, the results are recorded. The simulation then beginsagain (with no customers) and repeats. Optionally the start of data collection can be

postponed, so that data pertains to the last part of the run time.Assumptions: Identical Servers, Gamma inter-arrival times, Gamma service times,

All servers arrive at the same time (input), First customer arrival time is known (input).

Arrivals cease when the queue is full. This is "balking". First come, first served queue.

Transient, Defined.

This model gives "Transient" results; it models a fixed time period, not a "long run".

Accurate results are obtained by repeating the simulation many times.

● The outputs are averages over a designated time period.

For example, if the model gives 9% probability that the queue is empty, it means that

9% of the time there will be no one waiting. But that represents an average. For example,

there may be 100% near the beginning and 0% at the end of the time period, after the

queue has had time to build. That fact would not be visible in the output.

● The probability distributions of arrivals and service times do not change with time.

For example, you cannot model variations in the arrivals at different times of day.

● Customers and servers can be made to arrive "late". No business is

transacted until both are present. If servers are late, there may be a queue when they

arrive. If customers are late, there may be idle time for the servers at the outset.

Inputs

The Output Sheet

Example

Solution

Click Here if the worksheets don't seem to work properly.

The Model

Tranisient, Defined

Queue Simulation Worksheet


Inputs

● S : the number of (identical) servers.

● M: the queue cannot hold more than M customers. Arrivals balk when the queue is full.

Therefore the system can hold up to M+S customers (M in queue and S in service).

● l: the average arrival rate of customers

● m: the average service rate capacity for each server. (I.e. total system capacity is Sm per time unit.)

Please be careful with time units. l and m are rates, and they must have the

same time units. For example, suppose the arrival rate is 4 customers per hour, and the

average service time is 10 minutes. To be consistent, the service rate must also be given in customers per hour, which would be 60/10 or 6. The inputs would be l=4 and m=6.

● CV(a) = Coefficient of Variation of Inter-arrival Times (i.e. times between arrivals):

● CV(s) = Coefficient of Variation of Service Times:

Definition: CV = standard deviation divided by the mean.

● Simulated time per repetition, RunLength: Time units per repetition of the simulation.

Time Units are defined by Arrival and Service Rates. For example, if you input the

arrival rate as customers per hour, then you MUST use customers per hour for the

service rate, and the time units for the simulation will be "hours".

● Time when Data Collection begins: DataStartTime = the point in time during the

simulation after which data collection occurs. Before this time, the simulation runs

normally, but averages, frequencies, etc. are not collected. Thus, the output relates

only to the time between DataStartTime and RunLength.

● Repetitions, nReps = the number of times the simulation is repeated.

Data collection occurs after each repetition, and the simulation starts over.

● Time when Servers Arrive: No customers can begin service until this time.

● Time when First Customer Arrives: Subsequent customers arrive randomly after this

time (using the Gamma distribution to generate inter-arrival times).

The Output Sheet, SimResult

After you enter the input parameters, click the "Run Simulation" button.

When the simulation is finished, the results are displayed on a new worksheet.

Most of the output is self-explanatory.

However, there is one "input" on the SimResult worksheet, and it looks like this:

Use it to find how often the queue exceeded a given length, Q. For example, to find how

often 11 or more customers were waiting for service, type 10 in the yellow box.

Q: Frequency of Queue more than 2 customers waiting =

Your inputs always go in the yellow cells, like this: 3


Example:

NearBy Call Center provides over-the-phone help for computer owners. They currently

receive an average of 35 calls per hour during business hours, beginning at 8AM local time.

The CV of inter-arrival times is 1.0. Service times average 6 minutes, with CV of 0.5.

There are currently 2 servers.

The phones are answered by an automatic system, which delivers a message and then

places the customer on hold until a server becomes available. It gives a "busy signal"

if more than 5 customers are already waiting, and those calls are "lost".

Before 8AM it gives a brief message about the opening time, and then hangs up.

The system begins accepting calls at 8AM, but the servers do not begin work until 8:12AM.

This policy is intended to make sure that the servers are "highly utilized" by avoiding the

idle time that would occur if the servers started when no customers were present.

NearBy is considering several changes to the system, and would like you to evaluate

the performance of current and proposed policies during the first 2 hours of each day.

They would like to see the results for 100 of these 2-hour runs.

a. Find the average number waiting and how often more than 2 are waiting.

What is the average waiting time for a customer?

What fraction of customers get a busy signal?

b. How do these answers change if the servers start at 8AM instead of 8:12?

c. How do these answers change if they increase to 3 servers, starting at 8:12AM?

d. How would the current system perform if the CV of arrivals were reduced to 0.3?

e. What should they expect during the third hour, compared to the first 2 hours?

(Run the simulation of part (a) for 3 hours, with data collection starting after 2 hours.)


Solution:

Time units: Hours is a convenient unit for this problem. Arrival rate is already in hours, but

service time averages 6 minutes, so the service rate is 60/6 = 10 per hour.

Servers arrive 12 minutes late, or 12/60 = 0.2 hours after the simulation begins.

a. On the Queue Simulation worksheet, enter S = 2, M = 5, l = 35, m = 10,

CV(a) = 1.0, CV(s) = 0.5, RunLength = 2, DataStartTime = 0, and nReps = 100.

Set "Time when servers arrive" = 0.2 and "Time when first customer arrives" = 0.

Then click the "Simulate" Button.

Your answers will differ because each simulation has different customers.Answers: See SimResult (2)

Average number waiting = 4.0 P(>2) in queue = 96%

Average Waiting Time (Tq) = 0.21 hours. Fraction who balk = 43%

b. Set "Time when servers arrive" = 0. Answers: See SimResult (3)



c. Set S = 3 and "Time when servers arrive" = 0.2. Answers: See SimResult (4)



d. Set CV(a) = 0.3 and S = 2. Answers: See SimResult (5)



e. Since the arrival rate exceeds the service capacity (35 compared to 20 per hour), they

should expect the queue to grow steadily. However, the answers below do not agree

with that analysis. Apparently the 12 minute tardy period causes enough queue

buildup to disguise the effect. And, the queue cannot exceed 5.Set CV(a) = 1, RunLength = 3 and DataStartTime = 2. Answers: See SimResult (6)



SimResult2 SimResult3 SimResult4 SimResult5 SimResult6

19.28 21.29 26.50 20.49 19.26 Average Rate Joining System without Balking

15.57 13.14 7.89 14.35 15.23 Average Rate Balking (Leaving Without Service)

0.38 0.38 0.18 0.54 0.43 Frequency that the System is Full

42.6% 36.4% 21.2% 40.7% 41.5% Fraction who balk

4.04 3.69 2.93 4.46 4.01 Average Number Waiting in Queue (Nq)

1.12 1.45 1.70 0.79 1.15 Standard Dev. of Number Waiting

0.21 0.17 0.11 0.22 0.21 Average Waiting Time (Tq)

96.2% 89.5% 76.2% 98.6% 96.0% More than 2 customers waiting

0.98 0.98 0.95 0.99 1.00 Average Utilization of Servers

1.76 1.97 2.57 1.78 1.99 Average Number Being Served (Ns)

5.81 5.66 5.50 6.25 6.01 Average Number in the System (N)

1.34 1.54 1.96 1.16 1.17 Standard Dev. of Number in System

0.30 0.27 0.21 0.30 0.31 Average Time in System (T)

222126347.xls.ms_office, Queue Simulation, page 5

Transient Finite Capacity Queue Simulation

Number of Servers, S = 2

TRUE Queue Capacity, M = 5

Arrival Rate, l = 35

Coefficient of Variation of Inter-arrival time, CV(a) = 1

Service Rate Capacity of each server, m = 10

Coefficient of Variation of Service time, CV(s) = 0.5

Simulated time per repetition, RunLength = 2

Start collecting data at: DataStartTime = 0

Animation is ON Repetitions, nReps = 100

Time when servers arrive = 0.2

Time when first customer arrives = 0

Service Time and Inter-Arrival Time use the Gamma Distribution.

The Exponential Distribution is Gamma with CV=1

Service Times:CV=0.5

Inter-Arrivals:CV=1

Run Simulation

Run with Animation

222126347.xls.ms_office, Warning, p. 6 of 10

Try these steps. If step 1 does not work, then go to step 2.

STEP 1: Try to enable the macros

For Excel 2007,

A. If you see the Security Warning in your menu bar, proceed with step B.

If the Security Warning is not there, go to STEP 2.

B. In the Security Warning, Click Options.

C. In the Alert Window that appears, click Enable This Content and click OK.

For Excel 2003,

A. Close this file. Then open it again.

B. In the window that appears, click Enable Macros.

If a window like this one

does NOT appear, then go to STEP 2.

STEP 2: If Step 1 does not workIf you DID NOT get a Security Message, then your security setting is too high.

Here is what you should do:

For Excel 2007,

A. Click the Microsoft Office Button at the top-left of the screen:

B. Click Excel Options.

C. Click Trust Center, then Trust Center Settings, and then Macro Settings.

D. Click Disable all macros with notification

WARNING: YOU NEED TO ENABLE THE MACROS IN THIS FILE


E. Exit from Excel. Closing the file is not enough. On the menu bar, select File, and then Exit.

F. Open this file again and follow the instructions in STEP 1 to enable the macros.

For Excel 2003,

A. On the menu bar at the top of this page, select Tools, then Macro, then Security.

B. On the Security Level tab, select Medium and click OK.

C. Then exit from Excel. Closing the file is not enough. On the menu bar, select File, and then Exit.

D. Open this file again and follow the instructions in STEP 1 to enable the macros.

222126347.xls.ms_office, SimResult (2), page 9 of 10

Transient Finite Capacity Queue Simulation

Number of Servers: S = 2

Queue Capacity: M = 5 Simulated n

Arrival Rate: l = 35 34.842 7,179

Coefficient of Variation of Inter-arrival time: CV(a) = 1 1.0027

Service Rate Capacity of each server: m = 10 10.022 3,685

Coefficient of Variation of Service time: CV(s) = 0.5 0.4929 FormRange$C$29:$C$31

Simulated time per repetition, RunLength = 2

Start collecting data at time DataStartTime = 0

Repetitions, nReps = 100

Time when servers arrive = 0.2

Time when first customer arrives = 0

Arrivals: Average Rate Joining System without Balking = 19.277

Average Rate Balking (Leaving Without Service) = 15.565

Frequency that the System is Full = 38.4%

Fraction who balk = 42.6%

Waiting Line (Queue): Average Number Waiting in Queue (Nq) = 4.0445

Standard Dev. of Number Waiting = 1.1193

Average Waiting Time (Tq) = 0.2098

Q: Frequency of Queue more than 2 customers waiting = 96.2%

Servers: Average Utilization of Servers = 97.9%

Average Number of Customers Being Served (Ns) = 1.7618

Total in System: Average Number in the System (N) = 5.8063

Standard Dev. of Number in System = 1.3438

Average Time in System (T) = 0.3012

Simulation: 100 reps, from time 0 to 2 2 Servers, Queue Capacity = 5, Arrival Rate = 35, CV=1, Service Rate = 10, CV=0.5

Total Number in the System: Number Waiting in Queue Inter-Arrival Times Service Times Balksn = 0 1 2 3 4 5 6 7 0 1 2 3 4 5

P(n) = 0.0006 0.0171 0.0187 0.0291 0.0614 0.2032 0.2864 0.3836 0.0087 0.0289 0.0636 0.1539 0.2979 0.4471

±t·SE = 0.0008 0.0041 0.0037 0.0060 0.0098 0.0125 0.0111 0.0190 0.0057 0.0059 0.0101 0.0112 0.0114 0.0199 Mean StDev n Mean StDev n Rate Fraction n

Cumul = 0.0006 0.0177 0.0364 0.0655 0.1269 0.3301 0.6164 1.0000 0.0000 0.0087 0.0376 0.1012 0.2551 0.5529 1.0000 0.0287 0.0288 7179 0.0998 0.0492 3685 15.565 0.4255 3113

1 0.0000 0.0175 0.0109 0.0281 0.0747 0.1610 0.3056 0.4023 5.8766 5.8766 36.246 1.308 0.447 0.0000 0.0332 0.0807 0.1023 0.3104 0.4734 0.0254 0.0248 79 0.0964 0.0452 38 18.5 0.4684 37

2 0.0000 0.0236 0.0032 0.0554 0.0774 0.2640 0.2500 0.3265 5.6108 5.7437 33.369 1.348 0.395 0.0000 0.0341 0.0796 0.2786 0.2509 0.3567 0.0312 0.0273 65 0.0957 0.0584 41 9.5 0.2923 19

3 0.0000 0.0002 0.0059 0.0168 0.0312 0.2478 0.3606 0.3374 5.9517 5.813 36.383 1.2414 0.389 0.0000 0.0002 0.0355 0.1859 0.3623 0.4161 0.0277 0.0236 73 0.0873 0.036 42 13.5 0.3699 27

4 0.0000 0.0088 0.0231 0.0639 0.1262 0.2010 0.2863 0.2908 5.5096 5.7372 32.275 1.2857 0.392 0.0039 0.0525 0.1207 0.1798 0.3110 0.3321 0.0271 0.0286 74 0.0943 0.0551 41 15 0.4054 30

5 0.0000 0.0155 0.0141 0.0504 0.0086 0.0818 0.4100 0.4195 6.0352 5.7968 38.025 1.2872 0.392 0.0000 0.0155 0.0149 0.1014 0.4178 0.4504 0.0319 0.0296 64 0.1037 0.0489 35 12.5 0.3906 25

6 0.0000 0.0244 0.0239 0.0032 0.0114 0.1921 0.2744 0.4706 6.0286 5.8354 38.072 1.2947 0.406 0.0000 0.0244 0.0337 0.1165 0.2760 0.5494 0.0269 0.0215 75 0.1053 0.0465 35 18.5 0.4933 37

7 0.0004 0.0942 0.0555 0.1701 0.2477 0.2833 0.0926 0.0563 4.0719 5.5835 18.982 1.4699 0.371 0.0990 0.1906 0.2675 0.2080 0.0945 0.1404 0.0491 0.0478 41 0.0926 0.0416 36 2.5 0.122 5

8 0.0000 0.0296 0.0244 0.0156 0.0351 0.2288 0.2464 0.4201 5.8287 5.6141 36.004 1.4666 0.374 0.0000 0.0296 0.0550 0.1776 0.2510 0.4869 0.0285 0.0275 71 0.0961 0.0497 38 14 0.3944 28

9 0.0000 0.0037 0.0254 0.0092 0.0033 0.1326 0.3332 0.4926 6.2057 5.6799 39.688 1.4413 0.392 0.0000 0.0037 0.0271 0.0732 0.3348 0.5612 0.0233 0.0221 86 0.1069 0.0416 34 24 0.5581 48

10 0.0000 0.0193 0.0009 0.0422 0.0829 0.1750 0.2696 0.4101 5.8425 5.6961 35.904 1.4314 0.401 0.0000 0.0556 0.0799 0.1085 0.2735 0.4825 0.0265 0.0249 76 0.1083 0.0518 34 18.5 0.4868 37

11 0.0226 0.0313 0.0704 0.0660 0.1018 0.1504 0.2079 0.3496 5.2241 5.6532 30.913 1.4867 0.393 0.0853 0.0271 0.1303 0.1798 0.2119 0.3656 0.0347 0.0369 58 0.0974 0.042 37 9 0.3103 18

12 0.0000 0.0105 0.0178 0.0117 0.0250 0.1343 0.2768 0.5240 6.1812 5.6972 39.584 1.4704 0.397 0.0000 0.0105 0.0266 0.0997 0.2930 0.5702 0.0279 0.0211 72 0.1056 0.0503 35 16 0.4444 32

13 0.0000 0.0126 0.0343 0.0269 0.1271 0.3230 0.2060 0.2701 5.4121 5.6753 31.152 1.4645 0.393 0.0000 0.0391 0.1554 0.2323 0.2120 0.3612 0.0313 0.0361 64 0.0916 0.0412 39 11 0.3438 22

14 0.0000 0.0021 0.0011 0.0396 0.1563 0.1401 0.3290 0.3318 5.7452 5.6803 34.466 1.4478 0.398 0.0000 0.0174 0.1294 0.1199 0.3570 0.3764 0.0285 0.0352 72 0.1046 0.0444 35 17 0.4722 34

15 0.0000 0.0078 0.0093 0.0311 0.0533 0.1968 0.1638 0.5379 6.0648 5.7059 38.349 1.4388 0.401 0.0000 0.0376 0.0338 0.1084 0.1926 0.6275 0.0321 0.0354 64 0.1193 0.0501 31 14 0.4375 28

16 0.0000 0.0054 0.0016 0.0003 0.0142 0.2584 0.3433 0.3768 6.0557 5.7278 37.524 1.4146 0.402 0.0000 0.0054 0.0094 0.1720 0.3497 0.4635 0.028 0.0299 73 0.0966 0.04 38 15.5 0.4247 31

17 0.0000 0.0061 0.0125 0.0253 0.0337 0.1564 0.3004 0.4656 6.0855 5.7488 38.362 1.4031 0.407 0.0000 0.0061 0.0391 0.1321 0.3076 0.5153 0.0274 0.0243 73 0.1129 0.0686 33 18 0.4932 36

18 0.0000 0.0254 0.0034 0.0153 0.0243 0.1693 0.3302 0.4320 6.0274 5.7643 37.855 1.3958 0.414 0.0000 0.0266 0.0256 0.1177 0.3324 0.4978 0.0243 0.0234 84 0.1063 0.0501 35 22.5 0.5357 45

19 0.0000 0.0007 0.0297 0.0202 0.0796 0.1528 0.2574 0.4596 5.9645 5.7748 37.181 1.39 0.418 0.0000 0.0044 0.0837 0.1312 0.2831 0.4976 0.0292 0.0318 69 0.1115 0.0475 33 17 0.4928 34

20 0.0000 0.0408 0.0008 0.0236 0.1579 0.2555 0.2634 0.2580 5.4089 5.7565 31.297 1.3942 0.412 0.0000 0.0547 0.1476 0.2242 0.2746 0.2990 0.0353 0.0331 57 0.097 0.0566 38 8.5 0.2982 17

21 0.0000 0.0190 0.0004 0.0139 0.0802 0.1377 0.3512 0.3976 5.9612 5.7663 36.998 1.3866 0.413 0.0000 0.0294 0.0736 0.0627 0.3582 0.4761 0.0277 0.0268 74 0.0973 0.048 38 16 0.4324 32

22 0.0000 0.0049 0.0103 0.0076 0.0201 0.2432 0.3519 0.3619 5.9896 5.7764 36.918 1.3729 0.414 0.0000 0.0049 0.0217 0.1506 0.3606 0.4621 0.0273 0.027 74 0.097 0.0419 37 16 0.4324 32

23 0.0000 0.0028 0.0237 0.0116 0.0057 0.1557 0.3762 0.4244 6.1141 5.7911 38.524 1.3629 0.417 0.0000 0.0028 0.0237 0.1083 0.3819 0.4833 0.0243 0.0254 83 0.0929 0.0433 40 20 0.4819 40

24 0.0000 0.0051 0.0107 0.0775 0.0367 0.1954 0.2277 0.4469 5.877 5.7947 36.311 1.3617 0.420 0.0051 0.0518 0.0422 0.1650 0.2278 0.5081 0.024 0.0227 85 0.0892 0.0579 41 21.5 0.5059 43

25 0.0000 0.0169 0.0306 0.0437 0.0735 0.2276 0.2444 0.3634 5.6511 5.7889 34.003 1.3651 0.425 0.0290 0.0306 0.0661 0.2112 0.2534 0.4098 0.0287 0.0306 70 0.1293 0.082 28 18.5 0.5286 37

26 0.0000 0.0337 0.0121 0.0151 0.0336 0.1465 0.2306 0.5283 6.052 5.7991 38.607 1.3677 0.431 0.0000 0.0376 0.0434 0.1159 0.2330 0.5701 0.0227 0.0237 89 0.1144 0.0496 32 26 0.5843 52

27 0.0000 0.0142 0.0141 0.0145 0.0371 0.1747 0.2334 0.5120 6.0922 5.8099 38.653 1.3643 0.432 0.0000 0.0142 0.0464 0.0678 0.2381 0.6334 0.0284 0.0286 71 0.101 0.0543 34 17 0.4789 34

28 0.0000 0.0213 0.0014 0.0013 0.0525 0.2785 0.2660 0.3790 5.8793 5.8124 35.986 1.3585 0.433 0.0000 0.0213 0.0376 0.2106 0.2823 0.4481 0.0256 0.0263 80 0.092 0.0369 40 18 0.45 36

29 0.0000 0.0119 0.0234 0.0187 0.0241 0.2145 0.3031 0.4044 5.9327 5.8165 36.747 1.355 0.431 0.0036 0.0299 0.0431 0.1366 0.3038 0.4830 0.031 0.0269 65 0.1033 0.0606 36 12.5 0.3846 25

30 0.0000 0.0030 0.0057 0.0115 0.0203 0.1390 0.3248 0.4956 6.2436 5.8308 39.909 1.3459 0.435 0.0000 0.0077 0.0159 0.0698 0.3349 0.5717 0.0229 0.0267 88 0.1024 0.0572 36 24 0.5455 48

31 0.0000 0.0099 0.0248 0.0017 0.0423 0.1597 0.2952 0.4665 6.0683 5.8384 38.276 1.3423 0.436 0.0000 0.0099 0.0644 0.0769 0.2979 0.5510 0.027 0.0267 75 0.1054 0.0562 34 18 0.48 36

32 0.0000 0.0040 0.0332 0.0437 0.0940 0.2202 0.3309 0.2738 5.5813 5.8304 32.872 1.3421 0.433 0.0000 0.0282 0.1237 0.1902 0.3345 0.3234 0.0305 0.0262 66 0.0917 0.0497 40 11 0.3333 22

33 0.0000 0.0004 0.0038 0.0287 0.0580 0.2299 0.3037 0.3754 5.9258 5.8333 36.278 1.3349 0.435 0.0000 0.0044 0.0549 0.1629 0.3107 0.4671 0.0282 0.0255 71 0.115 0.0672 31 17.5 0.493 35

34 0.0000 0.0035 0.0079 0.0118 0.0120 0.1869 0.2526 0.5253 6.2299 5.845 39.839 1.3283 0.436 0.0000 0.0035 0.0079 0.1160 0.2646 0.6080 0.0275 0.0285 76 0.1083 0.0533 34 18.5 0.4868 37

35 0.0000 0.0017 0.0115 0.0003 0.0392 0.1566 0.3559 0.4349 6.145 5.8535 38.716 1.3205 0.435 0.0000 0.0017 0.0115 0.0899 0.3951 0.5019 0.0308 0.0293 67 0.1002 0.0523 37 12.5 0.3731 25

36 0.0000 0.0103 0.0289 0.0913 0.1152 0.2375 0.2598 0.2569 5.3477 5.8395 30.67 1.3265 0.431 0.0036 0.0800 0.1248 0.1852 0.2755 0.3309 0.0343 0.0331 61 0.093 0.0515 37 9.5 0.3115 19

37 0.0000 0.0005 0.0207 0.0380 0.1106 0.2214 0.2433 0.3654 5.7233 5.8364 34.396 1.3254 0.430 0.0140 0.0346 0.0998 0.1515 0.2609 0.4392 0.0282 0.0288 71 0.0935 0.0613 40 14 0.3944 28

38 0.0000 0.0110 0.0405 0.0488 0.0452 0.2028 0.2929 0.3589 5.7022 5.8328 34.532 1.3282 0.431 0.0418 0.0512 0.0435 0.1118 0.2966 0.4552 0.0247 0.0252 81 0.0919 0.0498 40 19 0.4691 38

39 0.0000 0.0130 0.0003 0.0246 0.0317 0.0585 0.2733 0.5985 6.3367 5.8457 41.375 1.3253 0.433 0.0000 0.0130 0.0003 0.0485 0.3051 0.6332 0.0265 0.0237 76 0.1158 0.0535 32 19.5 0.5132 39

40 0.0000 0.0035 0.0070 0.0254 0.0433 0.1371 0.3246 0.4591 6.114 5.8524 38.564 1.3206 0.435 0.0000 0.0052 0.0425 0.0904 0.3324 0.5295 0.0247 0.0267 82 0.0976 0.0417 38 20 0.4878 40

41 0.0000 0.0069 0.0233 0.0369 0.0275 0.1644 0.3728 0.3682 5.9105 5.8539 36.446 1.3185 0.434 0.0000 0.0069 0.0378 0.1648 0.3857 0.4048 0.0301 0.0259 67 0.0982 0.0369 37 13 0.3881 26

42 0.0000 0.0049 0.0105 0.0005 0.0400 0.1885 0.3586 0.3970 6.0606 5.8588 37.767 1.3125 0.432 0.0000 0.0049 0.0267 0.1249 0.3825 0.4610 0.0267 0.0226 76 0.0825 0.0366 44 13.5 0.3553 27

43 0.0000 0.0052 0.0272 0.0455 0.1946 0.2961 0.2360 0.1954 5.2387 5.8444 29.109 1.3153 0.430 0.0089 0.0431 0.2047 0.1908 0.2442 0.3084 0.0305 0.031 66 0.0855 0.047 41 11 0.3333 22

44 0.0283 0.1228 0.0540 0.0540 0.0279 0.1514 0.2027 0.3589 4.99 5.8249 29.94 1.3496 0.430 0.1610 0.0802 0.0270 0.1150 0.2207 0.3961 0.0253 0.028 79 0.092 0.0494 38 18 0.4557 36

45 0.0000 0.0043 0.0139 0.0076 0.0540 0.2097 0.2733 0.4373 6.02 5.8293 37.5 1.3453 0.432 0.0000 0.0099 0.0678 0.1189 0.2733 0.5301 0.0252 0.0252 80 0.1081 0.04 33 21 0.525 42

46 0.0000 0.0087 0.0029 0.0040 0.0287 0.2354 0.2848 0.4355 6.0758 5.8346 37.995 1.3399 0.434 0.0000 0.0087 0.0214 0.1442 0.2951 0.5307 0.0244 0.0236 83 0.0932 0.0403 39 20.5 0.494 41

47 0.0000 0.0252 0.0298 0.0290 0.0334 0.1306 0.3643 0.3877 5.8578 5.8351 36.315 1.3415 0.434 0.0000 0.0322 0.0375 0.1527 0.3900 0.3877 0.0281 0.0277 72 0.1059 0.0487 34 16.5 0.4583 33

48 0.0000 0.0101 0.0427 0.0065 0.0345 0.3299 0.3118 0.2645 5.625 5.8308 33.225 1.3402 0.433 0.0000 0.0123 0.0711 0.2460 0.3179 0.3527 0.0341 0.0334 61 0.1052 0.0474 33 12 0.3934 24

49 0.0000 0.0064 0.0170 0.0005 0.0286 0.2675 0.1865 0.4935 6.0676 5.8356 38.122 1.3369 0.435 0.0000 0.0064 0.0427 0.1801 0.1894 0.5815 0.0251 0.0268 80 0.1065 0.051 34 21 0.525 42

50 0.0000 0.0433 0.0034 0.0017 0.0867 0.2640 0.2681 0.3327 5.66 5.8321 34.016 1.3385 0.434 0.0000 0.0433 0.0785 0.2130 0.2797 0.3855 0.0334 0.0331 60 0.1111 0.0549 34 11.5 0.3833 23

51 0.0000 0.0089 0.0439 0.0051 0.0697 0.0639 0.3137 0.4948 6.056 5.8365 38.481 1.339 0.434 0.0000 0.0089 0.0451 0.0690 0.3822 0.4948 0.0284 0.0291 72 0.0961 0.0415 38 15.5 0.4306 31

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Number of Customers Waiting for Service

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0.3

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Total Customers in the System (waiting or being served)

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222126347.xls.ms_office, SimResult (2), page 10 of 10

52 0.0000 0.0064 0.0056 0.0108 0.0916 0.2705 0.2657 0.3493 5.8089 5.8359 35.04 1.3354 0.433 0.0000 0.0171 0.0895 0.1708 0.2734 0.4492 0.0278 0.0279 72 0.0958 0.0522 40 14 0.3889 28

53 0.0000 0.0064 0.0269 0.0063 0.0079 0.3267 0.3062 0.3196 5.8185 5.8356 35.148 1.3319 0.431 0.0000 0.0064 0.0333 0.2603 0.3076 0.3923 0.0289 0.026 70 0.0877 0.0436 42 11.5 0.3286 23

54 0.0000 0.0032 0.0109 0.0012 0.0098 0.2102 0.2864 0.4785 6.186 5.8421 39.223 1.3271 0.430 0.0000 0.0032 0.0112 0.1351 0.2958 0.5547 0.0323 0.0293 62 0.1109 0.0604 33 12 0.3871 24

55 0.0000 0.0355 0.0247 0.0446 0.1694 0.1920 0.2508 0.2830 5.3419 5.833 30.941 1.3332 0.428 0.0000 0.0701 0.1855 0.1704 0.2594 0.3146 0.0316 0.0347 64 0.092 0.0428 41 10 0.3125 20

56 0.0000 0.0974 0.0084 0.0042 0.0205 0.0996 0.3077 0.4620 5.7877 5.8322 36.705 1.3427 0.429 0.0000 0.1016 0.0290 0.0996 0.3077 0.4620 0.0252 0.0282 81 0.0947 0.0477 39 19 0.4691 38

57 0.0000 0.0160 0.0073 0.0072 0.0066 0.1253 0.3441 0.4933 6.2236 5.8391 39.911 1.3396 0.431 0.0000 0.0160 0.0084 0.0602 0.3497 0.5657 0.0237 0.0219 85 0.1033 0.049 35 22.5 0.5294 45

58 0.0000 0.0054 0.0070 0.0023 0.0556 0.2309 0.3509 0.3478 5.9437 5.8409 36.394 1.335 0.430 0.0000 0.0054 0.0491 0.1596 0.3645 0.4214 0.0315 0.0296 64 0.1034 0.0595 36 13 0.4063 26

59 0.0000 0.0810 0.0619 0.0681 0.0193 0.2081 0.2385 0.3231 5.2194 5.8303 30.87 1.349 0.431 0.1199 0.0605 0.0328 0.1592 0.2405 0.3871 0.0299 0.0398 67 0.093 0.0542 37 15 0.4478 30

60 0.0000 0.0266 0.0448 0.0005 0.0612 0.0501 0.2414 0.5753 6.089 5.8346 39.325 1.3521 0.431 0.0000 0.0266 0.0448 0.0507 0.3026 0.5753 0.0267 0.029 77 0.1008 0.0609 35 18.5 0.4805 37

61 0.0000 0.0080 0.0035 0.0027 0.0252 0.2144 0.3167 0.4295 6.1026 5.839 38.256 1.3476 0.432 0.0000 0.0080 0.0286 0.1213 0.3168 0.5253 0.0263 0.028 76 0.1015 0.0407 37 17 0.4474 34

62 0.0000 0.0238 0.0007 0.0323 0.0312 0.2525 0.2660 0.3935 5.8598 5.8394 35.986 1.3466 0.432 0.0000 0.0369 0.0294 0.2033 0.2685 0.4619 0.0238 0.0249 86 0.0835 0.0406 43 20 0.4651 40

63 0.0000 0.0405 0.0408 0.0794 0.1170 0.1739 0.2485 0.3000 5.2884 5.8306 30.782 1.3542 0.430 0.0000 0.0956 0.1577 0.1982 0.2485 0.3000 0.0313 0.0304 64 0.0871 0.0503 42 8.5 0.2656 17

64 0.0000 0.0004 0.0144 0.0868 0.0982 0.1854 0.2398 0.3750 5.6731 5.8282 34.052 1.3546 0.429 0.0128 0.0755 0.0939 0.1050 0.2456 0.4671 0.0296 0.0337 72 0.0936 0.043 39 14.5 0.4028 29

65 0.0000 0.0025 0.0013 0.0179 0.0668 0.2399 0.3955 0.2761 5.8312 5.8282 35.002 1.3498 0.429 0.0000 0.0025 0.0449 0.2004 0.4187 0.3334 0.0286 0.0274 72 0.0903 0.0364 41 14 0.3889 28

66 0.0000 0.0005 0.0049 0.0069 0.0192 0.1417 0.2480 0.5788 6.3559 5.8362 41.222 1.3457 0.431 0.0000 0.0005 0.0049 0.0587 0.2672 0.6686 0.0213 0.0237 95 0.1173 0.0577 31 29.5 0.6211 59

67 0.0000 0.0066 0.0043 0.0239 0.1056 0.2795 0.2354 0.3447 5.7319 5.8347 34.279 1.3437 0.432 0.0000 0.0241 0.0933 0.1958 0.2520 0.4348 0.0259 0.0238 79 0.0965 0.0469 38 18.5 0.4684 37

68 0.0000 0.0125 0.0155 0.0268 0.0787 0.1654 0.3537 0.3474 5.8195 5.8344 35.464 1.3425 0.433 0.0000 0.0138 0.0716 0.1605 0.3763 0.3778 0.0271 0.0236 74 0.1111 0.0471 33 18.5 0.5 37

69 0.0000 0.0349 0.0130 0.0180 0.0390 0.2241 0.2562 0.4149 5.8324 5.8344 36.026 1.3436 0.433 0.0000 0.0383 0.0443 0.1997 0.2639 0.4538 0.0273 0.0311 75 0.1008 0.0526 37 16.5 0.44 33

70 0.0000 0.0450 0.0249 0.0323 0.0511 0.2264 0.2984 0.3219 5.5717 5.8306 33.428 1.3471 0.432 0.0122 0.0713 0.0629 0.1734 0.2994 0.3809 0.0299 0.0274 68 0.0935 0.0486 39 13 0.3824 26

71 0.0000 0.0433 0.0715 0.0706 0.1939 0.1872 0.2075 0.2261 4.9369 5.8181 27.294 1.3569 0.431 0.0605 0.0834 0.1861 0.2046 0.2393 0.2261 0.0356 0.0353 58 0.1009 0.0493 36 9 0.3103 18

72 0.0000 0.0031 0.0138 0.0459 0.0471 0.3022 0.2555 0.3324 5.7277 5.8168 34.267 1.355 0.431 0.0000 0.0340 0.0594 0.2099 0.2570 0.4397 0.0321 0.0341 63 0.1016 0.0455 35 13.5 0.4286 27

73 0.0000 0.0145 0.0012 0.0085 0.0370 0.1935 0.3562 0.3890 6.0183 5.8196 37.41 1.352 0.430 0.0000 0.0145 0.0300 0.1274 0.3644 0.4637 0.0319 0.0275 63 0.0952 0.0422 38 11.5 0.3651 23

74 0.0000 0.0054 0.0013 0.0318 0.0156 0.2361 0.2966 0.4132 6.0183 5.8222 37.374 1.3488 0.430 0.0000 0.0054 0.0162 0.1723 0.2973 0.5088 0.029 0.0315 69 0.1066 0.0588 34 15.5 0.4493 31

75 0.0000 0.0133 0.0209 0.0238 0.1111 0.3108 0.2232 0.2968 5.5418 5.8185 32.437 1.3487 0.429 0.0178 0.0324 0.1004 0.2220 0.2371 0.3904 0.0268 0.0272 75 0.0914 0.0457 41 14.5 0.3867 29

76 0.0126 0.0470 0.1150 0.1173 0.0732 0.1678 0.1679 0.2993 4.863 5.8059 27.638 1.3636 0.428 0.1497 0.1273 0.0667 0.0836 0.1856 0.3871 0.0356 0.0344 57 0.1032 0.0375 34 9.5 0.3333 19

77 0.0000 0.0112 0.0104 0.0434 0.0215 0.1314 0.3493 0.4328 6.0308 5.8089 37.856 1.3621 0.428 0.0000 0.0112 0.0104 0.1502 0.3708 0.4574 0.0279 0.0258 72 0.0979 0.0473 37 16 0.4444 32

78 0.0000 0.0095 0.0154 0.0063 0.0790 0.2493 0.2846 0.3558 5.8203 5.809 35.306 1.3601 0.429 0.0000 0.0095 0.0718 0.2014 0.3072 0.4100 0.0264 0.0294 76 0.0959 0.0419 38 17 0.4474 34

79 0.0000 0.0033 0.0035 0.0660 0.0720 0.1561 0.2662 0.4329 5.9042 5.8102 36.46 1.3589 0.429 0.0000 0.0469 0.0680 0.1079 0.2737 0.5036 0.0312 0.0324 68 0.1006 0.0402 37 15 0.4412 30

80 0.0000 0.0099 0.0252 0.0625 0.1081 0.1994 0.3276 0.2673 5.514 5.8065 32.28 1.3595 0.427 0.0000 0.0150 0.1285 0.2460 0.3324 0.2781 0.0313 0.0325 65 0.0887 0.046 41 10 0.3077 20

81 0.0000 0.0019 0.0131 0.0061 0.0465 0.2393 0.3705 0.3226 5.91 5.8078 35.981 1.3559 0.427 0.0000 0.0058 0.0585 0.1563 0.3716 0.4077 0.0288 0.0281 70 0.0976 0.0538 38 15 0.4286 30

82 0.0000 0.0014 0.0045 0.0156 0.0362 0.2952 0.3245 0.3227 5.8836 5.8087 35.612 1.3521 0.428 0.0000 0.0118 0.0323 0.1957 0.3328 0.4274 0.0255 0.0352 80 0.0981 0.0463 37 20 0.5 40

83 0.0000 0.0120 0.0327 0.0100 0.0399 0.1587 0.2944 0.4524 5.9933 5.8109 37.603 1.3516 0.428 0.0000 0.0120 0.0466 0.1446 0.3203 0.4765 0.0298 0.0283 72 0.1004 0.0619 37 15.5 0.4306 31

84 0.0000 0.0202 0.0085 0.0044 0.0352 0.2159 0.2530 0.4629 6.0289 5.8135 37.845 1.3504 0.428 0.0000 0.0202 0.0424 0.1460 0.2542 0.5371 0.0268 0.0254 75 0.0948 0.043 39 15.5 0.4133 31

85 0.0000 0.0112 0.0135 0.0000 0.0385 0.1847 0.2694 0.4826 6.1106 5.817 38.646 1.3485 0.429 0.0000 0.0112 0.0329 0.1154 0.2886 0.5520 0.0272 0.0249 74 0.1063 0.0607 35 18 0.4865 36

86 0.0000 0.0178 0.0136 0.0511 0.0843 0.3119 0.1878 0.3335 5.5565 5.814 32.782 1.3492 0.430 0.0000 0.0628 0.0976 0.2515 0.1881 0.4000 0.0256 0.0309 80 0.1098 0.0557 34 21 0.525 42

87 0.0000 0.0103 0.0056 0.0151 0.0583 0.1281 0.3241 0.4585 6.0947 5.8172 38.439 1.3473 0.431 0.0008 0.0114 0.0372 0.0830 0.3500 0.5176 0.0261 0.0236 77 0.108 0.0552 34 19.5 0.5065 39

88 0.0000 0.0276 0.0015 0.0043 0.0544 0.2876 0.3683 0.2563 5.7033 5.8159 33.953 1.3457 0.430 0.0000 0.0304 0.0500 0.1936 0.3742 0.3518 0.0339 0.0346 62 0.0952 0.0487 38 10 0.3226 20

89 0.0000 0.0225 0.0313 0.0223 0.1691 0.1738 0.2941 0.2868 5.4699 5.812 32.041 1.3475 0.427 0.0000 0.0400 0.1533 0.1787 0.3413 0.2868 0.0351 0.0299 57 0.0926 0.0444 39 6.5 0.2281 13

90 0.0000 0.0160 0.0047 0.0032 0.0532 0.3313 0.3552 0.2364 5.6905 5.8107 33.569 1.3449 0.426 0.0000 0.0160 0.0507 0.2564 0.3624 0.3145 0.0281 0.0278 72 0.0886 0.0416 42 12.5 0.3472 25

91 0.0000 0.0075 0.0413 0.0859 0.1682 0.2728 0.2981 0.1262 5.0567 5.8024 27.374 1.3472 0.424 0.0420 0.0533 0.1652 0.2429 0.3052 0.1913 0.0345 0.0393 58 0.0878 0.0505 43 6 0.2069 12

92 0.0000 0.0415 0.0039 0.0305 0.0010 0.1780 0.2911 0.4541 5.9598 5.8041 37.526 1.3481 0.424 0.0000 0.0415 0.0039 0.1604 0.2921 0.5022 0.0289 0.0264 70 0.1032 0.0388 35 15.5 0.4429 31

93 0.0000 0.0095 0.0145 0.0040 0.0544 0.2914 0.2461 0.3801 5.8624 5.8047 35.743 1.3463 0.424 0.0000 0.0095 0.0610 0.1820 0.2540 0.4935 0.0304 0.0321 71 0.0876 0.0472 40 14 0.3944 28

94 0.0000 0.0081 0.0188 0.0178 0.0099 0.1503 0.3278 0.4673 6.128 5.8082 38.857 1.3447 0.424 0.0000 0.0081 0.0276 0.0591 0.3289 0.5763 0.0331 0.0356 61 0.1147 0.0568 33 13 0.4262 26

95 0.0000 0.0012 0.0495 0.0286 0.1678 0.1351 0.2041 0.4138 5.6536 5.8066 34.141 1.3463 0.424 0.0000 0.0218 0.1911 0.1273 0.2303 0.4296 0.0261 0.0231 77 0.0923 0.0464 40 16 0.4156 32

96 0.0000 0.0101 0.0270 0.0039 0.0042 0.1628 0.2951 0.4970 6.1559 5.8102 39.266 1.345 0.425 0.0000 0.0101 0.0270 0.1016 0.2993 0.5621 0.024 0.0243 86 0.1012 0.0485 36 22.5 0.5233 45

97 0.0000 0.0108 0.0021 0.0382 0.0517 0.3074 0.2952 0.2946 5.7066 5.8091 33.936 1.3434 0.425 0.0000 0.0339 0.0533 0.2132 0.2957 0.4039 0.0287 0.0274 70 0.0982 0.0526 36 15.5 0.4429 31

98 0.0000 0.0244 0.0212 0.0139 0.0068 0.2340 0.2827 0.4170 5.9209 5.8103 36.803 1.3432 0.426 0.0000 0.0244 0.0212 0.1516 0.2896 0.5133 0.0268 0.029 75 0.0992 0.0599 35 18 0.48 36

99 0.0000 0.0182 0.0076 0.1351 0.0992 0.1615 0.3448 0.2337 5.3476 5.8056 30.754 1.3453 0.425 0.0000 0.0215 0.1067 0.2933 0.3448 0.2337 0.0318 0.0435 63 0.1099 0.0536 33 12.5 0.3968 25

100 0.0000 0.0055 0.0075 0.0386 0.0512 0.2202 0.3039 0.3731 5.8772 5.8063 35.929 1.3438 0.426 0.0000 0.0055 0.0236 0.2359 0.3391 0.3959 0.0298 0.0282 69 0.1092 0.0366 34 15.5 0.4493 31

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