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Service of traffic demand

Incoming demands (intensity, holding time). no free resource service principle:. loss. limited delay. delay. no waiting place. overflow. free waiting place. loss. redirection. waiting. Service of traffic demand. Simplified scheme: human factors, queue management, etc.

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Service of traffic demand

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  1. Incoming demands (intensity, holding time) no free resource service principle: loss limited delay delay no waiting place overflow free waiting place loss redirection waiting Service of traffic demand Simplified scheme:human factors, queue management, etc. are missing. nooverflow Infocomm networks' planning - traffic aspects - 2011.09.21

  2. 2.1-1 Erlang’s formula and its’ calculation (The intensity of incoming demands is constant) Infocomm networks' planning - traffic aspects - 2011.09.21

  3. Erlang’smodel –1. • Structure: n identical channels (servers, trunks, slots) – homogeneous group • Strategy: • full accessibility, one demand – one channel • if all channels are busy the demand is lost without any after effect (lost calls cleared) • Erlang’s loss model – Lost Calls Cleared (LCC model) • Traffic: • exp. holding time distribution. μintensity (1/μ mean value, „holding time”) • arrival rate:  intensity (Poisson process) • pure birth and death process Pure Chance Traffic type One  PCT-1 See the Textbook: Chapter 4 Infocomm networks' planning - traffic aspects - 2011.09.21

  4. Erlang’s model –2. • Offered traffic: • offered traffic = carried traffic, if n∞ • Considered cases: • (n = ∞ Poisson distribution) • n < ∞ truncated Poisson distribution • Performance measures • E (time congestion) • B (call congestion) • C (traffic congestion) that is: meanarrivalrate x mean holding time The model is insensitive to the holding time distribution Infocomm networks' planning - traffic aspects - 2011.09.21

  5. Erlang’s model –3. The model is insensitive to the holding time distribution Insensitivity: A system is insensitive to the holding time distribution if the state probabilitiesof the system only depend on the mean value of the holding time. Infocomm networks' planning - traffic aspects - 2011.09.21

  6. Erlang’sdistribution -1. Traffic: PCT-1 Erlang’s distribution(truncated Poisson) [conditional Poisson p(ii n) – see: Textbook] Infocomm networks' planning - traffic aspects - 2011.09.21

  7. Erlang’sdistribution -2. Time congestion All n channels are occupied in a random point of time Erlang B formula Call congestion Rejection of a random demand Infocomm networks' planning - traffic aspects - 2011.09.21

  8. Erlang’s distribution -3. Carried traffic Mean value or expectation Lost traffic Traffic congestion E = B = C since the intensity of demands is state independent PASTA – Poisson arrivals see time averages Infocomm networks' planning - traffic aspects - 2011.09.21

  9. Tabular calculation aid: GG Honlap, Gyakorlatok Erlang B táblázat A(traffic), from any N (number of channels two the Erlang B (congestion) third Erlang’s distribution - 4. Infocomm networks' planning - traffic aspects - 2011.09.21

  10. Erlang B formula is robust Generalisation of Erlang B • It is valid for any holding time distribution (formulas depend only on the average holding time which is included in A, the offered traffic). • The deduction assumed a Poisson arrival process. According to Palm’s theorem this is fulfilled, if the traffic is offered by many indpendent sources. • Mathematical generalization is possible for fractional number of channels. Infocomm networks' planning - traffic aspects - 2011.09.21

  11. Evaluation of Erlang’s B formula - 1. • For large state spaces numerical difficulties may occur in calculating state probabilities. • Easily applicable methods and recursion formulas are available. • [See: Textbook, Chapter 4.5) Infocomm networks' planning - traffic aspects - 2011.09.21

  12. Evaluation of Erlang’s B formula - 2. Not easy to handle since n! and An increase rapidly Useful recursion formula: and where: Infocomm networks' planning - traffic aspects - 2011.09.21

  13. 2.1-2 BPP-traffic models (Generalization of Erlang’s classical loss system) Infocomm networks' planning - traffic aspects - 2011.09.21

  14. BPP-traffic models -1. BPP These models are all insensitive to theservice time distribution. Engset and Pascal models are eveninsensitive to the distribution ofthe idle time of sources. It is important always to use trafficcongestion as the most importantperformance metric. ForthesemodelstherelationshipbetweenE, Band Ccongestionvalues and Zpeakedness is welldefined. See the Textbook: Chapter 5 Infocomm networks' planning - traffic aspects - 2011.09.21

  15. BPP-traffic models -2. Erlang distribution n=number of channels Arrival intensity: λ Seethe Textbook: Chapter 5 Infocomm networks' planning - traffic aspects - 2011.09.21

  16. BPP-traffic models -3. Engset distribution n=number of channels Arrival intensity: (S-i) Seethe Textbook: Chapter 5 Infocomm networks' planning - traffic aspects - 2011.09.21

  17. BPP-traffic models -4. n=number of channels Arrival intensity: (S+i) Seethe Textbook: Chapter 5 Infocomm networks' planning - traffic aspects - 2011.09.21

  18. Z, peakedness Peakedness (Z) The peakedness has dimension: [number of channels] „Number representation” Index of Dispersion for Counts – IDC = peakedness Gives a characterization for the probability distribution of occupiedservers (lines, channels). Poissondistribution: Erlangdistribution: Binomial and Engsetdistribution: In the case of binomial and Engset distribution β(offered traffic of free traffic sources), takes congestion already into account. Infocomm networks' planning - traffic aspects - 2011.09.21

  19. BPP-traffic models -5. Textbook: Fig. 5.7 For applications thetraffic congestion C is the most important, as it is almost a linear function of the peakedness. Infocomm networks' planning - traffic aspects - 2011.09.21

  20. 2.1-3 Engset’s formula and its’ calculation (The intensity of incoming demands depends on the number of occupied traffic sources) Infocomm networks' planning - traffic aspects - 2011.09.21

  21. Engset’smodel -1. • Structure:n identical channels (servers, trunks, slots) – homogeneous group • Strategy: • full accessibility, one demand – one channel • if all channels are busy the demand is lost without any after effect – LCC (lost calls cleared)model • Traffic: • exp. holding time distribution. μintensity (1/μ mean value, „holding time”) • offeredtraffic, A = carriedtraffic, ifthenumber of channels is not limited (independent of thenumberofchannels) • pure birth and death process Pure Chance Traffic type Two PCT-2 • Resultsareindependentfromthe holding timedistributiontheydependonits’ averagevalue. Infocomm networks' planning - traffic aspects - 2011.09.21

  22. Engset’s model -2. S traffic sources offer demands to n fully available channels. The arrival intensity of new demands is: (S-i) Infocomm networks' planning - traffic aspects - 2011.09.21

  23. Engset’smodel -3. The trafficsource Possible states of a traffic source exponentially distributed time intervals (assumption required for the mathematical deduction) Infocomm networks' planning - traffic aspects - 2011.09.21

  24. Engset distribution - 1. S > n Cut equations exist for 0 ≤ i ≤ n . Infocomm networks' planning - traffic aspects - 2011.09.21

  25. Engset distribution - 2. (See: Textbook Chapter 5.3) Normalization: offered trafficof free trafficsource Distribution: (truncated binomial) Engset, 1918 !! Infocomm networks' planning - traffic aspects - 2011.09.21

  26. Engset distribution - 3. Time congestion Call congestion After some transformations: Infocomm networks' planning - traffic aspects - 2011.09.21

  27. Engset distribution - 4. Interpretation: As if the remaining S-1 traffic sources had occupiedall channels. When S increases E is increasing too, therefore: Infocomm networks' planning - traffic aspects - 2011.09.21

  28. Engset distribution - 5. Carried traffic: transformation with cut equations Infocomm networks' planning - traffic aspects - 2011.09.21

  29. Engset distribution - 6. Traffic congestion: designation: relationship was applied Number of calls (traffic demands) per time unit (S – Y) the number of free traffic sources Infocomm networks' planning - traffic aspects - 2011.09.21

  30. Engset distribution - 7. Lost traffic: Duration of state [i] Improvement function Infocomm networks' planning - traffic aspects - 2011.09.21

  31. Engset distribution - 8. Relations between E, B and C Designation: Alreadyderived Infocomm networks' planning - traffic aspects - 2011.09.21

  32. Evaluation of Engset’s formula - 1. There are numerical problems for large values ofS and n. Various numerically stable recursive formulaehave been elaborated. recursion according n: Infocomm networks' planning - traffic aspects - 2011.09.21

  33. Evaluation of Engset’s formula - 2. recursionaccordingS: (See details in Chapter 5.5 of the Textbook) Tabular calculation aid: GG Honlap, Gyakorlatok Engset táblázat S(number of sources), n (number of channels γ (callintesity)μ(releaseintensíty) Engset E, B, C A A-Y Infocomm networks' planning - traffic aspects - 2011.09.21

  34. Evaluation of Engset’s formula - 3. recursion according n and S: Based on the previous calculations Evaluation: If the parameter is increasing recursions by n and by n and S are both good, but not acceptable for decreasing parameters. Recursion by decreasing S is however acceptable. (See details in Chapter 5.5 of the Textbook) Infocomm networks' planning - traffic aspects - 2011.09.21

  35. 2.1-4 Overflow traffic Peakedness Smooth and bursty traffic Infocomm networks' planning - traffic aspects - 2011.09.21

  36. Overflow traffic - model Basic problem: traffic from node A to nodes B or C are directed on different „first choice” routes and if these are fully occupied the overflow traffic might use the „overflow” route Nowadays these type of arrangements are used only in networks. Infocomm networks' planning - traffic aspects - 2011.09.21

  37. Overflow traffic – Example 1a. 16 10 erl PCT-I …… 8 8 1. 10 erl, 16 channels, E16=2,23%, losttraffic0,223 erl. Could this be calculated in two steps ?? PCT-I If yes,how ? 8 8 Infocomm networks' planning - traffic aspects - 2011.09.21

  38. Overflow traffic – Example 1b. PCT-I ?? 8 8 2.10 erl, 8 channels, E8 =33,832%, Alost = 3,3832 erl A’ =3,3832 erl, 8 channels, E8’=0,1457 A’lost= 3,3832 x 0,1457 = 0,0483 erl. 0,223 erl = 0,0483 erl What is the reason ??? Overflow traffic does not have PCT-I/PCT-II character Infocomm networks' planning - traffic aspects - 2011.09.21

  39. Overflow traffic - peakedness –1. • PeakednessZ is a good indicator for the relative loss probabilities of traffics with the same average value (A). • For a given A traffic Z has a maximum as a function of n, the number of channels. • For PCT-I Z = 1. • If Z < 1, the traffic is smooth. • If Z > 1, tha traffic is bursty. • Congestion: smooth < PCT-I < bursty. Infocomm networks' planning - traffic aspects - 2011.09.21

  40. Overflow traffic - peakedness –2. PeakednessZof overflow traffic as a function of the offered PCT-1traffic (A) and the number of channels (n) Textbook: Fig. 6.8 Infocomm networks' planning - traffic aspects - 2011.09.21

  41. IPP principle IPP = Interrupted Poisson Process Principle: the process is in theoff state, if there are free channels in the primary route;if there isn’t any free channel it is in thein state. For practical applications of the method one has to determine the actual values of the parameters involved. Infocomm networks' planning - traffic aspects - 2011.09.21

  42. Dimensioning overflow systems • ERT (Equivalent Random Theory) • an equivalent random traffic is applied which is derived from the average value (expected value) and the variance of the overflow traffic • Modified ERT • calculation is based on a Zpeakedness value which is derived from the average value (expected value) and the variance of the overflow traffic • IPP (Interrupted Poisson Process) • If the primary route is occupied, a random (Poisson) traffic appears temporarily (in an interrupted way) on the secondary route. Textbook: Chapter 6. Infocomm networks' planning - traffic aspects - 2011.09.21

  43. 2.1-5 Multi-dimensional loss systems Example: multi-dimensional Erlang-B loss formula Infocomm networks' planning - traffic aspects - 2011.09.21

  44. Model – 1. • Structure: n uniform channels (trunks, • slots) – homogenous group • Strategy: • full accessibility • LCC - lost calls cleared • Input process: • two independent PCT-I traffic streams with 1and 2intensity • holding times: exp. distribution. μ1and μ2intensity • Offered traffic • A1=1/μ1 and A2 = 2/μ2 Textbook: Chapter 7. Infocomm networks' planning - traffic aspects - 2011.09.21

  45. Model – 2. • In state (i,j) • i channels are occupied by the first, • j channels are occupied by the second • traffic stream. One demand occupies one channel. Restrictions: Statistical equilibrium, (n+1)(n+2)/2 node equations. Infocomm networks' planning - traffic aspects - 2011.09.21

  46. Model – 3. Number of states: (n+1)(n+2)/2 Example of node equation: p(0,1)[1+2+μ2]= p(0,0) 2 + p(1,1) μ1 + p(0,2)2μ2 Infocomm networks' planning - traffic aspects - 2011.09.21

  47. Multi dimensional Erlang distr. – 1. The state space diagram depicts a reversible Markov process with local balance and with product form solution. It can be shown that the solution is: where: p(i) and p(j) are one dimensional, truncated Poisson distributions and Q is the normalisation constant Poisson Arrivals See Time Averages – PASTA !! Időtorlódás Hivástorlódás P(i+j=n) Forgalmi torlódás Infocomm networks' planning - traffic aspects - 2011.09.21

  48. Multi dimensional Erlang distr. – 2. Generalization by Newton’s binomial law Infocomm networks' planning - traffic aspects - 2011.09.21

  49. Multi dimensional Erlang distr. – 3. It can be found, that: This is a truncated Poisson distribution, with offer traffic Infocomm networks' planning - traffic aspects - 2011.09.21

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