How To Calculate The Power Of A Cluster In Erlang (Orchestra)

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1 Network Traffic Distribution Derek McAvoy Wireless Technology Strategy Architect March 5, 21

2 Data Growth is Exponential 2.5 x 18 98% 2 95% Traffic % 75% 5%.5 Data Traffic Feb 29 25% 1% 5% 2% Page

3 Spectrum Forecasting Inputs Long range forecast attempts to predict future uses of data Short range forecast use Time Series Analysis Subscriber forecast Subscriber usage forecast X Jun-97 Jun-98 Jun-99 Jun- Jun-1 Jun-2 Jun-3 Jun-4 Jun-5 Jun-6 Jun-7 Jun-8 Jun-9 Jun-1 Jun-11 Network Load Jun-12 Jun Time of Day Traffic Pattern Cell Site Traffic Distribution Spectral Efficiency, RF capacity Spectrum Tool Spectrum Requirement Number of Cell Sites: - the greater the number of cell sites, the greater the amount of traffic supported. Page

4 Cell Site Clusters in and around Toronto The average cluster has 4 sectors Urban clusters have ~1 sectors Downtown Toronto Page

5 Example of Traffic Report SiteName No_Cluster Cluster voice Rankin [CDMA 1.9] Bloor St W,154 [CDMA 1.9] 17.4 Blackthorn [CDMA 1.9] Blackthorn [CDMA 1.9] 13.9 Blackthorn [CDMA 1.9] Outlook [CDMA 1.9] Outlook [CDMA 1.9] 9.72 Outlook [CDMA 1.9] 1.97 Lawrence Ave W,144 [CDMA 1.9] Lawrence Ave W, 1577 [CDMA 1.9] Lawrence Ave W, 1577 [CDMA 1.9] 25.7 Boon [CDMA 1.9] 9.5 Boon [CDMA 1.9] 8.96 Boon [CDMA 1.9] 7.26 Wilson Ave, 139 [CDMA 1.9] South Kingsway [CDMA 1.9] 16.1 South Kingsway [CDMA 1.9] South Kingsway [CDMA 1.9] Page

6 Example of Voice Traffic Distribution in a Cluster Keele / Islington Cluster Voice Traffic 18 Number of Sectors Erlangs Average: 22 Erlangs Variance: 8.2 actual Sector traffic were placed into bins to form a histogram for each cluster Page

7 Possible Probability Distribution Functions The random variable (sector traffic) can have only positive values. Hence, the prospective probability distribution function should only be defined for positive values. This rules out Normal distribution function It is preferable that the probability distribution function should be continuous as oppose to discrete Some random variables such as the number of subscribers can be described by positive integer values, but in general most random variables are continuous such as the level of voice or data traffic Candidate probability distribution functions Gamma distribution Log-normal distribution Weibull distribution Truncated Normal distribution Inverted Gamma distribution Log-logistic distribution Gamma and log-normal appears to be the most promising distributions so far Page

8 Gamma Probability Density Function Gamma density functions.5 f(y).4.3 f ( y) = 1 Γ( α) β α y α 1 e y β y>, α, β > otherwise y gamma(1,2.4) gamma(2,1.2) gamma(3,.8) gamma(4,.6) gamma(6,.4) Page

9 Gamma Distribution Keele / Islington Cluster Voice Traffic Number of Sectors Erlangs actual gammadist Gamma pdf parameters determined using Maximum Likelihood estimation. Methods of Moments produce slightly worse fit. Page

10 Log-Normal Probability Density Function Lognormal pdf's 1.2 f(y) f ( y) = πy σ e 2 1 logy µ 2 σ y>, < µ <, σ > otherwise y LN(,1) LN(,2) LN(1,1) LN(1,2) Page

11 Log-Normal Distribution Keele / Islington Cluster Voice Traffic Number of Sectors Erlangs actual lognormal Page

12 Weibull Probability Density Function Weibull pdf's 1.2 f(y) F( y) = 1 exp β y α y y > ( α, β > ) y W(1,1) W(1,2) W(2,1) W(2,2) Page

13 Weibull Distribution Keele / Islington Cluster Voice Traffic Number of Sectors Erlangs actual weibull Page

14 Comparison of Distribution Functions Keele / Islington Cluster Voice Traffic Number of Sectors Erlangs actual gammadist lognormal weibull Page From this chart, it appears that gamma and log-normal are possible distributions to be considered

15 Goodness of Fit Test The chi-square test is a goodness of fit test: it answers the question of how well do experimental data fit expectations. Procedure for performing chi-square goodness of fit test: 1. State a hypotheses based on the fit of the data, e.g. H: the data fits a gamma distribution. H1: the data does not fit a gamma distribution. 2. Make a table of the observed and expected values. You will most likely be given the observed values. 3. Calculate the chi-squared test statistic, this is: 4. Look up the chi-squared critical value from a chi-square table 5. If calculated chi-square value is greater than the critical value from the table, reject the null hypothesis H. If chi-square value is less than the critical value, you fail to reject the null hypothesis H. Page

16 Goodness of Fit Chart (Keele/Islington Cluster) bins observed expected gamma gamma X 2 expected log-normal lognormal X 2 expected weibull weibull X 2 < > sum There are 1 bins, hence there are = 7 degrees of freedom Each probability distribution function has 2 parameters Page

17 Chi-Square Table Page All candidate distributions pass the chi-square test at p=.5

18 Cluster Chi-Square Test Summary Null hypothesis rejected possibly due to large number of sectors (951 sectors) Page

19 Problem of Chi-Square Test With a limited number of data points (<1) the chi-square test accepts the null hypothesis too easily. With a large number of data points (>1) the chi-square test rejects the null hypothesis too easily. Is there another way to confirm that either the gamma distribution or the log-normal distribution is the correct distribution? One can study the spatial distribution of organisms (ecology). Empirical models suggest that the negative binomial distribution works well with aggregated spatial distributions. However, the negative binomial distribution is considered the discrete version of the gamma distribution. One can use the principle of Maximum Entropy. The probability density function that should be chosen is the one that maximizes the entropy subject to given constraints. This avoids accidentally including information that isn t there. Page The probability distribution p(x) maximizing the differential entropy subject to E{x} = m is the exponential distribution (special case of the gamma distribution). The probability distribution p(x) maximizing the differential entropy subject to E{x} = m and E{lnX} = n is the gamma distribution. The probability distribution p(x) maximizing the differential entropy subject to E{lnX} = n and E{(lnX) 2 } = q is the log-normal distribution.

20 Summary Knowing the probability distribution of the network sector traffic helps immensely with traffic capacity planning and spectrum forecast requirements. Given the number of cell sites, the number of subscribers and the average traffic per subscriber, one can calculate the number of congested sectors with a single equation The effects of various network strategies can be easily modeled. What effects will Wi-Fi offload have on network congestion? The effectiveness of network optimization can be analysed. Is it worth while to engineer the network such that each sector carries the same amount of traffic? Sector traffic distribution could play a role in self-organizing network development. The effectiveness of different SON algorithms could be tested against various sector traffic probability distributions. Page