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Queuing Model M/M/s (Exponential Service Times)
Input Data Operating Characteristics
Arrival rate (l)2Average server utilization (r)0.6667
Probabilities
Number of Units Probability
Cumulative
Probability
0 0.3333 0.3333 0.6667
7 0.0195 0.9610 0.0390
8 0.0130 0.9740 0.0260
9 0.0087 0.9827
10 0.0058 0.9884
Problem 9-15 (a) to (e)
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20 0.0001 0.9998
Computations
n or s
(lam/mu)^n/n!
Cumsum(n-1)
term2 P0(s) Rho(s) Lq(s) L(s) Wq(s) W(S)
0 1
1 0.666667 1 2 0.3333 0.666667 1.333333 2 0.666667 1
12 1.61E-11 1.947734 1.70369E-11 0.5134 0.055556 5.15E-13 0.666667 2.57E-13 0.333333
13 8.25E-13 1.947734 8.69754E-13 0.5134 0.051282 2.41E-14 0.666667 1.21E-14 0.333333
14 3.93E-14 1.947734 4.12575E-14 0.5134 0.047619 1.06E-15 0.666667 5.3E-16 0.333333
15 1.75E-15 1.947734 1.82758E-15 0.5134 0.044444 4.36E-17 0.666667 2.18E-17 0.333333
16 7.28E-17 1.947734 7.59283E-17 0.5134 0.041667 1.69E-18 0.666667 8.47E-19 0.333333
27 1.62E-33 1.947734 1.65734E-33 0.5134 0.024691 2.15E-35 0.666667 1.08E-35 0.333333
28 3.85E-35 1.947734 3.94248E-35 0.5134 0.02381 4.94E-37 0.666667 2.47E-37 0.333333
29 8.85E-37 1.947734 9.05555E-37 0.5134 0.022989 1.09E-38 0.666667 5.47E-39 0.333333
30 1.97E-38 1.947734 2.01077E-38 0.5134 0.022222 2.35E-40 0.666667 1.17E-40 0.333333
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Problem 9-15 (f)
Queuing Model M/M/s (Exponential Service Times)
Input Data Operating Characteristics
Arrival rate (l)2Average server utilization (r)0.3333
Average waiting time in the queue (Wq)0.0417
Average time in the system (W) 0.3750
2 0.1111 0.9444
3 0.0370 0.9815
4 0.0123 0.9938
5 0.0041 0.9979
6 0.0014 0.9993
17 0.0000 1.0000
18 0.0000 1.0000
19 0.0000 1.0000
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20 0.0000 1.0000
8 9.68E-07 1.947733 1.05569E-06 0.5134 0.083333 4.93E-08 0.666667 2.46E-08 0.333333
9 7.17E-08 1.947734 7.74175E-08 0.5134 0.074074 3.18E-09 0.666667 1.59E-09 0.333333
10 4.78E-09 1.947734 5.12021E-09 0.5134 0.066667 1.88E-10 0.666667 9.39E-11 0.333333
11 2.9E-10 1.947734 3.08314E-10 0.5134 0.060606 1.02E-11 0.666667 5.11E-12 0.333333
12 1.61E-11 1.947734 1.70369E-11 0.5134 0.055556 5.15E-13 0.666667 2.57E-13 0.333333
23 3.45E-27 1.947734 3.54961E-27 0.5134 0.028986 5.44E-29 0.666667 2.72E-29 0.333333
24 9.57E-29 1.947734 9.84779E-29 0.5134 0.027778 1.44E-30 0.666667 7.22E-31 0.333333
25 2.55E-30 1.947734 2.62308E-30 0.5134 0.026667 3.69E-32 0.666667 1.84E-32 0.333333
26 6.55E-32 1.947734 6.71876E-32 0.5134 0.025641 9.08E-34 0.666667 4.54E-34 0.333333
27 1.62E-33 1.947734 1.65734E-33 0.5134 0.024691 2.15E-35 0.666667 1.08E-35 0.333333