operation management and excel

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chapter_7_queue_models_bahanshal.xls
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Model1

Model 1: Single Channel, Poisson Arrival, Exponential Service Time
Arrival rate l = 3
Service rate m = 4
Interarrival Time 1/l = 0.3333
Service time 1/m = 0.2500
System Utilization r = 0.7500
Probability system is empty P0 = 0.2500
Average number in line Lq = 2.2500
Average number in system Ls = 3.0000
Average time in line Wq = 0.7500
Average time in system Ws = 1.0000
n = 2
P(2 units in system) = 0.1406
P(n < 2 units in system) = 0.4375
n P(n) P(n < n)
0 0.2500
1 0.1875 0.2500
2 0.1406 0.4375
3 0.1055 0.5781
4 0.0791 0.6836
5 0.0593 0.7627
6 0.0445 0.8220
7 0.0334 0.8665
8 0.0250 0.8999
9 0.0188 0.9249
10 0.0141 0.9437
11 0.0106 0.9578
12 0.0079 0.9683
13 0.0059 0.9762
14 0.0045 0.9822
15 0.0033 0.9866
16 0.0025 0.9900
17 0.0019 0.9925
18 0.0014 0.9944
19 0.0011 0.9958
20 0.0008 0.9968
21 0.0006 0.9976
22 0.0004 0.9982
23 0.0003 0.9987
24 0.0003 0.9990
25 0.0002 0.9992
26 0.0001 0.9994
27 0.0001 0.9996
28 0.0001 0.9997
29 0.0001 0.9998
30 0.0000 0.9998
31 0.0000 0.9999
32 0.0000 0.9999
33 0.0000 0.9999
34 0.0000 0.9999
35 0.0000 1.0000
36 0.0000 1.0000
37 0.0000 1.0000
38 0.0000 1.0000
39 0.0000 1.0000
40 0.0000 1.0000
41 0.0000 1.0000
42 0.0000 1.0000
43 0.0000 1.0000
44 0.0000 1.0000
45 0.0000 1.0000
46 0.0000 1.0000
47 0.0000 1.0000
48 0.0000 1.0000
49 0.0000 1.0000
50 0.0000 1.0000
51 0.0000 1.0000
52 0.0000 1.0000
53 0.0000 1.0000

Model 2

Model 2: Single Channel, Poisson Arrival, Constant Service Time
Arrival rate l = 10
Service rate m = 12
Interarrival Time 1/l = 0.1000
Service time 1/m = 0.0833
System Utilization r = 0.8333
Average number in line Lq = 2.0833
Average number in system Ls = 2.9167
Average time in line Wq = 0.2083
Average time in system Ws = 0.2917
n P(n) P(n < n)
0 0.1667
1 0.1389 0.1667
2 0.1157 0.3056
3 0.0965 0.4213
4 0.0804 0.5177
5 0.0670 0.5981
6 0.0558 0.6651
7 0.0465 0.7209
8 0.0388 0.7674
9 0.0323 0.8062
10 0.0269 0.8385
11 0.0224 0.8654
12 0.0187 0.8878
13 0.0156 0.9065
14 0.0130 0.9221
15 0.0108 0.9351
16 0.0090 0.9459
17 0.0075 0.9549
18 0.0063 0.9624
19 0.0052 0.9687
20 0.0043 0.9739
21 0.0036 0.9783
22 0.0030 0.9819
23 0.0025 0.9849
24 0.0021 0.9874
25 0.0017 0.9895
26 0.0015 0.9913
27 0.0012 0.9927
28 0.0010 0.9939
29 0.0008 0.9949
30 0.0007 0.9958
31 0.0006 0.9965
32 0.0005 0.9971
33 0.0004 0.9976
34 0.0003 0.9980
35 0.0003 0.9983
36 0.0002 0.9986
37 0.0002 0.9988
38 0.0002 0.9990
39 0.0001 0.9992
40 0.0001 0.9993
41 0.0001 0.9994
42 0.0001 0.9995
43 0.0001 0.9996
44 0.0001 0.9997
45 0.0000 0.9997
46 0.0000 0.9998
47 0.0000 0.9998
48 0.0000 0.9998
49 0.0000 0.9999
50 0.0000 0.9999
51 0.0000 0.9999
52 0.0000 0.9999
53 0.0000 0.9999

Model 3

Model 3: Multichannel, Poisson Arrival, Exponential Service Time
Arrival Rate l = 100.0 100.0 100.0 100.0 100.0 100.0
Calculations:
Service Rate m = 120.0 120.0 120.0 120.0 120.0 120.0
Number of servers S = 2.0 3.0 4.0 5.0 6.0 7.0
M P0 M P0
1.0 0.167 1.0 1.000 5.000 0.167
2.0 0.412 2.0 1.833 0.595 0.412
Average number being served r = 0.833 0.833 0.833 0.833 0.833 0.833 3.0 0.432 3.0 2.181 0.134 0.432
Average number in line Lq = 0.175 0.022 0.003 0.000 0.000 0.000 4.0 0.434 4.0 2.277 0.025 0.434
Average number in system Ls = 1.008 0.856 0.836 0.834 0.833 0.833 5.0 0.435 5.0 2.297 0.004 0.435
Average time in line Wq = 0.002 0.000 0.000 0.000 0.000 0.000 6.0 0.435 6.0 2.300 0.001 0.435
Average time in system Ws = 0.010 0.009 0.008 0.008 0.008 0.008 7.0 0.435 7.0 2.301 0.000 0.435
System Utilization rho = 0.417 0.278 0.208 0.167 0.139 0.119 8.0 0.435 8.0 2.301 0.000 0.435
P(zero units in system) P0 = 0.412 0.432 0.434 0.435 0.435 0.435 9.0 0.435 9.0 2.301 0.000 0.435
Average waiting time Wa = 0.007 0.004 0.003 0.002 0.002 0.001 10.0 0.435 10.0 2.301 0.000 0.435
P(wait) Pw = 0.245 0.058 0.011 0.002 0.000 0.000 11.0 0.435 11.0 2.301 0.000 0.435
12.0 0.435 12.0 2.301 0.000 0.435
Note: The Lq value in Exhibit 7.12 may differ slightly from the Lq
on this template. The value of Lq in this template is
based on a queue formula.
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