Copyright Douglas Rice,2005,doug.h.rice@btinternet.com, last modified 10/06/2012 08:31:48 terms of use The Author gives no warranty of any kind in relation to these Webpages. To the maximum extent permitted by law, the Author Douglas Rice, will not be liable for any loss or damage which you may suffer as a result of or connected to the download or use of these Web pages.
Use this page to calculate Erlang Proability.
This page is influenced by Chapter 4 and equ 4.3, p88, mt hills, Telecommunications Switching Principles. ISBN 0 04 621029 6
The probability that k circuits are in use when there are A Erlangs of traffic, or also the probability that k calls start in a second given an average call rate of A calls per second is:
P(k)=A^k/k!*exp(-A)
Use: A^k/k! = (A/k)*(A/(k-1)*..*(A/(1))..) to prevent working out a^k and k!, which over range floats quickly
This tends towards exp( A ) * exp( -A ) which equals 1 if enough k terms are used.
IAT = InterArrivalTime rand() returns a random number between 0 and 1, so it is a probability. InterArrivalTime,IAT = -ln( 1-rand() )/AverageCallRate exp ( - IAT * CallRate ) = 1 - rand() rand( ) = 1 - exp( -IAT * Call Rate )
There is a question of how many calls do you need to get an accurate average calls per second. This is discussed at the bottom of the page.
| Active Circuits | probability of this number of circuits in a call A^k/k!/exp(A) | accum Prob | GOS | ||
|---|---|---|---|---|---|
| 0 | 0.006737946999085467 | 0.006737946999085467 | 1.0067836549063043 | | |
| 1 | 0.03368973499542734 | 0.040427681994512805 | 1.0421309381647685 | | |
| 2 | 0.08422433748856833 | 0.12465201948308115 | 1.142402818373409 | | |
| 3 | 0.14037389581428059 | 0.26502591529736175 | 1.3605921906819178 | | |
| 4 | 0.17546736976785074 | 0.4404932850652124 | 1.787288647852159 | | |
| 5 | 0.17546736976785074 | 0.6159606548330632 | 2.603899867513088 | | |
| 6 | 0.14622280813987562 | 0.7621834629729388 | 4.204922048319162 | | |
| 7 | 0.10444486295705402 | 0.866628325929993 | 7.497843953545174 | | |
| 8 | 0.06527803934815876 | 0.9319063652781516 | 14.68566047450456 | | |
| 9 | 0.036265577415643756 | 0.9681719426937953 | 31.418819891500387 | | |
| 10 | 0.018132788707821878 | 0.9863047314016171 | 73.01791803616605 | | |
| 11 | 0.008242176685373581 | 0.9945469080869908 | 183.38220150192885 | | |
| 12 | 0.003434240285572326 | 0.9979811483725631 | 495.3311013101041 | | |
| 13 | 0.0013208616482970486 | 0.9993020100208602 | 1432.6853248414054 | | |
| 14 | 0.00047173630296323165 | 0.9997737463238233 | 4419.8176882620355 | | |
| 15 | 0.0001572454343210772 | 0.9999309917581445 | 14491.022711369818 | | |
| 16 | 0.00004913919822533663 | 0.9999801309563697 | 50329.54874971618 | | |
| 17 | 0.000014452705360393125 | 0.99999458366173 | 184626.57798501328 | | |
| 18 | 0.00000401464037788698 | 0.9999985983021079 | 713420.4921356859 | | |
| 19 | 0.0000010564843099702577 | 0.999999654786418 | 2896757.4046510793 | | |
| 20 | 2.6412107749256443e-7 | 0.9999999189074955 | 12331595.950812679 | | |
| 21 | 6.288597083156296e-8 | 0.9999999817934664 | 54925337.31009707 | | |
| 22 | 1.429226609808249e-8 | 0.9999999960857323 | 255475629.10130054 | | |
| 23 | 3.1070143691483673e-9 | 0.9999999991927468 | 1238768775.3474364 | | |
| 24 | 6.472946602392432e-10 | 0.9999999998400415 | 6251621870.087544 | | |
| 25 | 1.2945893204784864e-10 | 0.9999999999695005 | 32787431537.1967 | | |
| 26 | 2.4895948470740124e-11 | 0.9999999999943964 | 178455793290.29367 | | |
| 27 | 4.610360827914837e-12 | 0.9999999999990067 | 1006728429053.4248 | | |
| 28 | 8.232787192705067e-13 | 0.9999999999998299 | 5879372881684.721 | | |
| 29 | 1.4194460677077702e-13 | 0.9999999999999718 | 35461414388744.06 | | |
| 30 | 2.365743446179617e-14 | 0.9999999999999956 | 225179981368524.8 | | |
| 31 | 3.815715235773576e-15 | 0.9999999999999994 | 1801439850948198.5 | | |
| 32 | 5.962055055896213e-16 | 1 | Infinity | | |
| 33 | 9.033416751357898e-17 | 1 | Infinity | | |
| 34 | 1.3284436399055733e-17 | 1 | Infinity | | |
| 35 | 1.897776628436533e-18 | 1 | Infinity | | |
| 36 | 2.635800872828519e-19 | 1 | Infinity | | |
| 37 | 3.5618930713898905e-20 | 1 | Infinity | | |
| 38 | 4.68670140972354e-21 | 1 | Infinity | | |
| 39 | 6.008591550927615e-22 | 1 | Infinity | | |
| 40 | 7.510739438659518e-23 | 1 | Infinity | | |
| 41 | 9.15943833982868e-24 | 1 | Infinity | | |
| 42 | 1.090409326170081e-24 | 1 | Infinity | | |
| 43 | 1.2679178211280012e-25 | 1 | Infinity | | |
| 44 | 1.440815705827274e-26 | 1 | Infinity | | |
| 45 | 1.6009063398080822e-27 | 1 | Infinity | | |
| 46 | 1.7401155867479155e-28 | 1 | Infinity | | |
| 47 | 1.851186794412676e-29 | 1 | Infinity | | |
| 48 | 1.928319577513204e-30 | 1 | Infinity | | |
| 49 | 1.9676730382787798e-31 | 1 | Infinity | | |
| 50 | 1.96767303827878e-32 | 1 | Infinity | | |
| 51 | 1.9290912139988038e-33 | 1 | Infinity | | |
| 52 | 1.8548953980757731e-34 | 1 | Infinity | | |
| 53 | 1.7499013189394087e-35 | 1 | Infinity | | |
| 54 | 1.620278999017971e-36 | 1 | Infinity | | |
| 55 | 1.4729809081981555e-37 | 1 | Infinity | | |
| 56 | 1.3151615251769246e-38 | 1 | Infinity | | |
| 57 | 1.1536504606815128e-39 | 1 | Infinity | | |
| 58 | 9.945262592082006e-41 | 1 | Infinity | | |
| 59 | 8.428188637357632e-42 | 1 | Infinity | | |
| 60 | 7.023490531131359e-43 | 1 | Infinity | | |
| 61 | 5.756959451747015e-44 | 1 | Infinity | | |
| 62 | 4.642709235279851e-45 | 1 | Infinity | | |
| 63 | 3.6846898692697225e-46 | 1 | Infinity | | |
| 64 | 2.878663960366971e-47 | 1 | Infinity | | |
| 65 | 2.2143568925899776e-48 | 1 | Infinity | | |
| 66 | 1.6775431004469528e-49 | 1 | Infinity | | |
| 67 | 1.2518978361544421e-50 | 1 | Infinity | | |
| 68 | 9.205131148194428e-52 | 1 | Infinity | | |
| 69 | 6.670384889995962e-53 | 1 | Infinity | | |
| 70 | 4.7645606357114016e-54 | 1 | Infinity | | |
| 71 | 3.3553243913460576e-55 | 1 | Infinity | | |
| 72 | 2.330086382879207e-56 | 1 | Infinity | | |
| 73 | 1.5959495773145252e-57 | 1 | Infinity | | |
| 74 | 1.078344308996301e-58 | 1 | Infinity | | |
| 75 | 7.188962059975338e-60 | 1 | Infinity | | |
| 76 | 4.729580302615354e-61 | 1 | Infinity | | |
| 77 | 3.0711560406593207e-62 | 1 | Infinity | | |
| 78 | 1.9686897696534103e-63 | 1 | Infinity | | |
| 79 | 1.2460061833249434e-64 | 1 | Infinity | | |
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There is a question of how many calls do you need to get an accurate average calls per second.
You can work this out by working out the interarrival times for a number of calls and working out the average call rate of those calls, and ask by what percentage change to the average call rate will the next call make. I think that for 1 % of need about 700 calls.
We use a stated Average call rate A, work out an InterArrivalTime IAT, the measured Average call rate is 1/IAT.
Work this out for say 100 calls or n calls, work out n IATs, the measured Average call rate is n/sum(IAT).
How many calls are required to get measured AverageCallRate/StatedAverageCallRate so this is 1 +/- 0.01?
Given a hours worth of calls where we see say 3600 calls from a population of say 10,000,000 possible callers, we can say we have an average call rate of 1 call per seconds, possibly with any one caller making one call, totally without knowledge of anybody else making a call.
If you have a sequence of calls from independent calls, you could argue they should have no knowledge of who is going to make the next call.
If you see a sequence of evenly spaced calls, the callers must have a knowledge of the previous calls to know when to make their call so that it is spaced correctly from the last call.
If you see a sequence of evenly spaced calls, the callers must have a knowledge of the previous calls to know when to make their call so that it is spaced correctly from the last call one seconds on. They must also agree with the rest of the population who is going to make the next call.
