Teletraffic and Network Dimensioning. David Falconer Carleton University

Transcription

1 Teletraffic and Network Dimensioning David Falconer Carleton University 1

2 Topics to be Covered Application - why it s needed What is traffic Blocking probability Examples of provisioning 2

3 Teletraffic Theory To telecommunications facility, like a switch or a trunk group, telephone calls appear to start at random times and to have random durations. Teletraffic theory analyzes the aspects of call randomness and congestion of transmission and switching facilities, and calculates the probability that a call is blocked; i.e. that an attempted call cannot be made because all available facilities are already in use. 3

4 Provisioning Fundamental problem is to determine the number of facilities (lines, channels, trunks etc.) that must be provided ( provisioned ) to handle a given average amount of telephone traffic with a specified maximum blocking probability. E.g. maximum acceptable blocking probability (also called the grade of service ) may be about 1% for residential loop access, or about 0.1% or less for switches. 4

5 Example of Random Traffic Fluctuation Traffic density = number of calls in progress or attempted Number of channels provided Blocking occurs Time of day Blocking occurs when a call is attempted, but all available facilities are already in use. 5

6 Traffic as a Sum of Individual Calls Sum of calls in progress τ 2 τ 1 t 1 t 2 t 3.. t 1, t 2, t 3... are call arrival times. τ 1, τ 2... are call durations (holding times) 6

7 Some Definitions Traffic density Number of calls in progress at a given instant. Traffic intensity Traffic density averaged over one hour (usually in the busy hour). Busy hour Hour of the day with highest traffic intensity. Peak busy hour Traffic intensity in busy hour of busiest day (e.g. Mother s Day). Holding time Duration of a call (off-hook to on-hook) 7

8 Some Statistical Assumptions About Telephone Traffic Call arrivals are Poisson-distributed; i.e. number of calls arriving in a given time T has a Poisson probability distribution. p(n calls in T) = (λt)n exp( λt) n! n = 0,1,2, 3,... where λ=average number of arrivals per unit time. p(n) n Related fact: times between arrivals follow an exponential probability density function. 8

9 Some Statistical Assumptions About Telephone Traffic (cont.) Call durations (holding times) are exponentially distributed: Density function of τ is f(τ) = 1 H exp( τ ) (τ 0) H where H = average holding time f(τ) 0 τ 9

10 Measure of Traffic Intensity: The Erlang A=traffic intensity=average number of calls simultaneously in progress (averaged over a period of time - usually the busy hour). Consider a call that starts at t i and has holding time τ i : Unit amplitide call function Traffic density t i t i +τ i During the period T=1 hour, the traffic density curve is the superposition of all the call functions that begin in this time period (neglecting end effects). 10

11 Measure of Traffic Intensity: The Erlang (cont.) So traffic intensity Area under density curve T N T = A τi A = i= 1 where NT = number of calls starting during T ( λt ) H CH So A= = = λh T T where λ = average number of calls per hour H = average holding time (in hours) C = λt = average number of calls in T hours. T. 11

12 Measure of Traffic Intensity: The Erlang (cont.) Traffic intensity A is expressed in Erlangs. Dimension: call hours per hour (or call minutes per minute, or call seconds per second) Really, A is dimensionless. 12

13 Four Equivalent Interpretations of Traffic Intensity in Erlangs *1. A=average number of calls in progress during a specified time (usually one hour). 2. A=average number of calls which originate during a period of time equal to the average holding time of a call. 3. A= total time duration in hours required to carry all calls (one at a time) which originate in the period T-1 hour. *4. For the case of a a single subscriber line, A is the fraction of time the line is busy. In this case, A<1. * - the most important interpretations in practice. 13

14 Another Unit of Traffic Intensity - ccs 1 ccs=cent call seconds=100 call seconds of server occupancy/hour. ( cent means 100). 1 Erlang=36 ccs. 14

15 Erlang B Formula for Blocking Probability Infinite population of potential callers, generating a total traffic intensity= A Erlangs N channels, or trunks to carry the traffic Blocking occurs if a new caller finds all N channels are in use. Blocked calls are cleared (unsuccessful caller disappears) Probability of blocking is called the grade of service (GOS). 15

16 Erlang B Formula for Blocking Probability (cont.) Blocking probability, given traffic intensity =A and number of channels=n: P = A N N! N A k k! k=0 where x! means x.(x -1).(x -2)...1 Assumptions: Poisson arrivals and exponential holding times. Infinite population. Blocked calls cleared. 16

17 Another Blocking Probability Formula: Poisson Formula Blocking probability, given traffic intensity =A and number of channels=n: P = exp( A)[ A k ] k! k=n Assumptions: Poisson arrivals and exponential holding times. Infinite population. Blocked calls held. (Blocked caller immediately re-attempts.) Other formulas exist for finite populations (Engset) and for case of blocked calls delayed (Erlang C). 17

18 Example of Trunk Loading Capacity Based on Erlang B and Poisson Formulas Number of trunks Traffic units in Erlangs. GOS=P=0.001 GOS=P=0.01 Erlang B Poisson Erlang B Poisson

19 Example Example of user traffic demand and channel provisioning: Typically a residential phone generates about 0.1 erlangs. e.g. in an area with 100 phones, about 10 at a time are typically busy (10 erlangs). If an area has 50 erlangs of traffic demand, and you only provide 50 channels, at least 50% of all call attempts will be blocked (no dial tone). Providing 100 channels would probably be overkill. Providing 64 channels would lead to a blocking probability ( grade of service ) of about 1%, which is usually considered acceptable. 19

20 Example (cont.) How was this result arrived at?... Using the Erlang B blocking probability formula or Erlang B tables. pr(blocking)= N A N N! k=0 A k k! With A = 50, N = 64, pr(blocking) (N= 64 channels serve 50 erlang with 1% blocking probability). 20

21 Trunking Efficiency Illustration of Trunking Efficiency ( economies of scale )» Trunking efficiency=(total Erlangs)/(total number of channels)» 60 channels serve 50 erlangs with about 2% blocking probability. (50/60=83% trunking efficiency)» But... need 10 channels to serve 5 erlangs with 2% blocking. (5/10=50% trunking efficiency).» Therefore, trunking efficiency increases with size. Trunking efficiency 100% Traffic intensity (Erlangs) 21

22 Summary What has been learned Basics of teletraffic theory. Definitions of traffic density and traffic intensity - the Erlang Four interpretations of the Erlang. Calculation of blocking probability using Erlang B or Poisson formulas or tables or graphs. Examples of use to dimension facilities to serve a given amount of traffic with specified grade of service (GOS). Illustration of trunking efficiency. 22

23 Where to get more information R. Haughton, The Telecommunications Mosaic, Vol. 1, part II, chapter II.11 E.B. Carne, Telecommunications Primer, Prentice-Hall, 1995, chapter 16 R.L Freeman, Telecommunication System Engineering, second edition, 1989, chapter 1. 23

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