Homework Sourse14A 60channel operator calling system to and from a communit
14.A 60-channel operator calling system to and from a community of 1000 independent carrier, each of whom has a conversation of 5% of the time during peak hours. Find the probability that during peak hours more traffic than line can handle will be required.
Solution
mean, m = 1000 * 0.05 = 50
r =60
P(x=r) = e^(-m) * m^r /r!
From that following table is made it,
P(x>60) = 1 - P(x <= 60)
P(x>60) = 1 - [ P(x=0) + P(x=1) + P(x=2) + P(x=3) ...... + P(x=60) ]
P(x>60) = 1 - 0.9278
P(x>60) = 0.0722
| x | P(x) | Cumulative |
| 0 | 1.93E-22 | 1.9287E-22 |
| 1 | 9.64E-21 | 9.8366E-21 |
| 2 | 2.41E-19 | 2.5093E-19 |
| 3 | 4.02E-18 | 4.2692E-18 |
| 4 | 5.02E-17 | 5.4497E-17 |
| 5 | 5.02E-16 | 5.5678E-16 |
| 6 | 4.19E-15 | 4.7424E-15 |
| 7 | 2.99E-14 | 3.464E-14 |
| 8 | 1.87E-13 | 2.215E-13 |
| 9 | 1.04E-12 | 1.2596E-12 |
| 10 | 5.19E-12 | 6.4502E-12 |
| 11 | 2.36E-11 | 3.0044E-11 |
| 12 | 9.83E-11 | 1.2835E-10 |
| 13 | 3.78E-10 | 5.0645E-10 |
| 14 | 1.35E-09 | 1.8568E-09 |
| 15 | 4.5E-09 | 6.358E-09 |
| 16 | 1.41E-08 | 2.0424E-08 |
| 17 | 4.14E-08 | 6.1795E-08 |
| 18 | 1.15E-07 | 1.7672E-07 |
| 19 | 3.02E-07 | 4.7914E-07 |
| 20 | 7.56E-07 | 1.2352E-06 |
| 21 | 1.8E-06 | 3.0353E-06 |
| 22 | 4.09E-06 | 7.1265E-06 |
| 23 | 8.89E-06 | 1.602E-05 |
| 24 | 1.85E-05 | 3.4549E-05 |
| 25 | 3.71E-05 | 7.1607E-05 |
| 26 | 7.13E-05 | 0.00014287 |
| 27 | 0.000132 | 0.00027484 |
| 28 | 0.000236 | 0.00051051 |
| 29 | 0.000406 | 0.00091683 |
| 30 | 0.000677 | 0.00159403 |
| 31 | 0.001092 | 0.00268628 |
| 32 | 0.001707 | 0.00439293 |
| 33 | 0.002586 | 0.00697876 |
| 34 | 0.003803 | 0.01078146 |
| 35 | 0.005432 | 0.01621388 |
| 36 | 0.007545 | 0.02375891 |
| 37 | 0.010196 | 0.03395489 |
| 38 | 0.013416 | 0.04737066 |
| 39 | 0.0172 | 0.06457037 |
| 40 | 0.0215 | 0.08607 |
| 41 | 0.026219 | 0.11228906 |
| 42 | 0.031213 | 0.14350223 |
| 43 | 0.036294 | 0.17979662 |
| 44 | 0.041244 | 0.22104023 |
| 45 | 0.045826 | 0.26686647 |
| 46 | 0.049811 | 0.31667761 |
| 47 | 0.052991 | 0.36966817 |
| 48 | 0.055199 | 0.42486668 |
| 49 | 0.056325 | 0.48119168 |
| 50 | 0.056325 | 0.53751669 |
| 51 | 0.055221 | 0.59273729 |
| 52 | 0.053097 | 0.64583401 |
| 53 | 0.050091 | 0.69592526 |
| 54 | 0.046381 | 0.74230605 |
| 55 | 0.042164 | 0.7844704 |
| 56 | 0.037647 | 0.82211714 |
| 57 | 0.033023 | 0.8551406 |
| 58 | 0.028468 | 0.8836091 |
| 59 | 0.024126 | 0.90773495 |
| 60 | 0.020105 | 0.92783982 |
| 61 | 0.016479 | 0.94431922 |
| 62 | 0.01329 | 0.95760907 |
| 63 | 0.010547 | 0.96815656 |
| 64 | 0.00824 | 0.97639679 |
| 65 | 0.006339 | 0.98273543 |
| 66 | 0.004802 | 0.98753742 |
| 67 | 0.003584 | 0.991121 |
| 68 | 0.002635 | 0.99375599 |
| 69 | 0.001909 | 0.9956654 |
| 70 | 0.001364 | 0.99702926 |
| 71 | 0.00096 | 0.99798973 |
| 72 | 0.000667 | 0.99865672 |
| 73 | 0.000457 | 0.99911357 |
| 74 | 0.000309 | 0.99942225 |
| 75 | 0.000206 | 0.99962803 |
| 76 | 0.000135 | 0.99976342 |
| 77 | 8.79E-05 | 0.99985133 |
| 78 | 5.64E-05 | 0.99990768 |
| 79 | 3.57E-05 | 0.99994335 |
| 80 | 2.23E-05 | 0.99996564 |
| 81 | 1.38E-05 | 0.9999794 |
| 82 | 8.39E-06 | 0.99998779 |
| 83 | 5.05E-06 | 0.99999285 |
| 84 | 3.01E-06 | 0.99999586 |
| 85 | 1.77E-06 | 0.99999763 |
| 86 | 1.03E-06 | 0.99999865 |
| 87 | 5.91E-07 | 0.99999925 |
| 88 | 3.36E-07 | 0.99999958 |
| 89 | 1.89E-07 | 0.99999977 |
| 90 | 1.05E-07 | 0.99999988 |
| 91 | 5.76E-08 | 0.99999993 |
| 92 | 3.13E-08 | 0.99999996 |
| 93 | 1.68E-08 | 0.99999998 |
| 94 | 8.96E-09 | 0.99999999 |
| 95 | 4.71E-09 | 0.99999999 |
| 96 | 2.45E-09 | 1 |
| 97 | 1.27E-09 | 1 |
| 98 | 6.46E-10 | 1 |
| 99 | 3.26E-10 | 1 |
| 100 | 1.63E-10 | 1 |
