MODELING AND CHARACTERIZATION OF TRAFFIC IN PUBLIC SAFETY WIRELESS NETWORKS

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1 MODELING AND CHARACTERIZATION OF TRAFFIC IN PUBLIC SAFETY WIRELESS NETWORKS Božidar Vujičić, Nikola Cackov, Svetlana Vujičić, and Ljiljana Trajković {bvujicic, ncackov, svujicic, Communication Networks Laboratory Simon Fraser University Vancouver, BC

2 Roadmap Introduction Statistical concepts and analysis tools Analysis of traffic data: call inter-arrival times call holding times Traffic modeling and characterization Conclusions and references SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 2

3 Roadmap Introduction Statistical concepts and analysis tools Analysis of traffic data: call inter-arrival times call holding times Traffic modeling and characterization Conclusions and references SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 3

4 E-Comm network: coverage and user agencies RCMP and Police Fire Agency 1 (Police) Agency 2 (Fire Dept.)... Ambulance Other TG 1 TG 2 TG 3 TG 4 R1 R2 R3 R4 R5 R6 R7 R8 TG: Talk group R: Radio device (user)... TG n... SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 4

5 E-Comm network architecture Users Transmitters/Repeaters PSTN PBX Dispatch console * 8 # Vancouver IBM Network switch Other EDACS systems Burnaby Database server Data gateway Management console SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 5

6 Network characteristics EDACS: Enhanced Digital Access Communications Systems Simulcast: repeaters covering one cell use identical frequencies Trunking: available frequencies in a cell are shared dynamically among mobile users Cell capacity (number of available frequencies in a cell): one radio channel occupies one frequency one call occupies one radio channel SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 6

7 Call establishment Users are organized in talk groups: one-to-many type of conversations Push-to-talk (PTT) mechanism for network access: user presses the PTT button system locates other members of the talk group system checks for availability of channels: channel available: call established all channels busy: call queued/dropped user releases PTT: call terminates SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 7

8 Erlang traffic models Erlang B P B = N x= N A N! x A x! Erlang C N A N P! C = N N A N 1 x N A A N + x! N! N A x= P B : probability of rejecting a call P c : probability of delaying a call N : number of channels/lines A : total traffic volume SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 8

9 Erlang models Erlang B model assumes: call holding time follows exponential distribution blocked call will be rejected immediately Erlang C model assumes: call holding time follows exponential distribution blocked call will be put into a FIFO queue with infinite size SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 9

10 Previous work Simulation: OPNET WarnSim Traffic prediction based on user clusters Seasonal ARIMA model Statistical analysis of traffic three busy hours in 21 [1] N. Cackov, B. Vujičić, S. Vujičić, and Lj. Trajković, Using network activity data to model the utilization of a trunked radio system, in Proc. SPECTS, San Jose, CA, July 24, pp [2] J. Song and Lj. Trajković, Modeling and performance analysis of public safety wireless networks, in Proc. IEEE IPCCC, Phoenix, AZ, Apr. 25, pp [3] H. Chen and Lj. Trajković, Trunked radio systems: traffic prediction based on user clusters, in Proc. ISWCS, Mauritius, Sept. 24, pp [4] D. Sharp, N. Cackov, N. Lasković, Q. Shao, and Lj. Trajković, Analysis of public safety traffic on trunked land mobile radio systems, IEEE J. Select. Areas Commun., vol. 22, no. 7, pp , Sept. 24. SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 1

11 Roadmap Introduction Statistical concepts and analysis tools Analysis of traffic data: call inter-arrival times call holding times Traffic modeling and characterization Conclusions and references SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 11

12 Statistical concepts Probability distribution: probability that outcomes of a process are within a given range of values expressed through probability density (pdf) and cumulative distribution (cdf) functions Autocorrelation: measures the dependence between two outcomes of a process wide-sense stationary processes: autocorrelation depends only on the difference (lag) between the time instances of the outcomes SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 12

13 LRD: definition Slow decay of the autocorrelation function r(k) of a (widesense) stationary process X(n): k = r ( k) = definition r( k) = c (2 2H ) r k, k model f α ( ν ) = ν, ν c f corollary where f(ν) is the power spectral density of X(n), c r and c f are non-zero constants, and <α<1.5 < H < 1 implies LRD LRD: long-range dependence SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 13

14 Wavelet coefficients Discrete wavelet transform of a signal X(t): d( j, k) = X ( t) ψ j, k ( t) dt where j / 2 ψ j, k ( t) = 2 ψ ψ(t): mother wavelet j : octave k : translation ( j 2 t k) Reconstruction formula: X ( t) = d( j, k) ψ ( t j= k j, k ) wavelet coefficients SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 14

15 LRD and wavelets Let X(t) be LRD process (wide-sense stationary) its power spectral density: α f ( ν ) ~ ν, ν c f Mean square value of its wavelet coefficients on octave j satisfies: Ε{ d( j, k) 2 } = 2 jα c f C( α, ψ ) where = α C( α, ψ ) ν Ψ( ν ) dν does not depend on j 2 D. Veitch and P. Abry, A wavelet-based joint estimator of the parameters of long-range dependence, IEEE Trans. on Information Theory, vol. 45, no. 3, pp , April SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 15

16 LRD and wavelets Logarithm: log 2 2 Ε{ d ( j, k) } = α j + c Important property: for given j, d(j,k) does not exhibit long-range dependence (with respect to k) with appropriately chosen mother wavelet Hence: simple estimator for E{d(j,k) 2 } is a sample mean: Ε{ d( j, k) 2 } = 1 n j n j k = 1 d( j, k) n j : number of wavelet coefficients at octave j 2 SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 16

17 Estimation of α and H Logscale diagram: plot of log 2 E{d(j,k) 2 } vs. j (octave) Linear relationship between log 2 E{d(j,k) 2 } and j on the coarsest octaves indicates LRD Estimation of α: linear regression of log 2 E{d(j,k) 2 } on j in the linear region of the logscale diagram H =.5 (α + 1) SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 17

18 Logscale diagram: example 1 wavelet power spectrum regression line 95% confidence intervals log 2 E{d(j,k) 2 } Octave j call inter-arrival times: 22: 23:, α=.576, H=.788 (octaves 4 9) SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 18

19 Test for time constancy of α X(n): wide-sense stationary process α does not depend on n Is α constant throughout the time series X(n)? Approach: divide X(n) into m blocks of equal lengths estimate α for each block compare the estimates If α varies significantly, estimating α for the entire time series is not meaningful In our analysis, m {3, 4, 5, 6, 7, 8, 1} SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 19

20 Kolmogorov-Smirnov test Goodness-of-fit test: quantitative decision whether the empirical cumulative distribution function (ECDF) of a set of observations is consistent with a random sample from an assumed theoretical distribution ECDF is a step function (step size 1/N) of N ordered data points Y, Y, , Y N : n () i E N = n() i N : the number of data samples with values smaller than Y i SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 2

21 Parameters Hypothesis: null: the candidate distribution fits the empirical data alternative: the candidate distribution does not fit the empirical data Input parameters: significance level σ and tail Output parameters: p-value k: test statistic cv: critical (cut-off) value SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 21

22 Input parameters Significance level σ: determines if the null hypothesis is wrongly rejected σ percent of times, if it is in fact true default value σ =.5 σ defines sensitivity of the test: smaller σ implies larger critical value (larger tolerance) tail: specifies whether the K-S performs two sided test (default) or tests from one or other side of the candidate distribution SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 22

23 Output parameters Test statistic k is the maximum difference over all data points: i k = max F( Yi ) 1 i N N where F is the CDF of the assumed distribution The null hypothesis is accepted if the value of the test statistic is smaller than the critical value p-value is probability level when the difference between distributions (test statistics) becomes significant: if p-value σ: test rejects the null hypothesis If test returns critical value = NaN, the decision to accept or reject null hypothesis is based only on p-value SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 23

24 Best-fitting distributions: CDF 1 Cumulative distribution Traffic data Lognormal model Exponential model Gamma model Weibull model Call holding time (s) SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 24

25 Inter-arrival time: complementary CDF Complementary CDF Traffic data Exponential model Lognormal model Weibull model Gamma model Call inter-arrival time (s) SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 25

26 K-S test: call inter-arrival times 21 Significance level σ =.1 Distribution Parameter , 2: 21: , 16: 17: , 15: 16: , 19: 2: , : 1: h exponential p k h 1 1 Weibull p k h gamma p k Significance level σ , 16: 17:: cv , : 1:: cv SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 26

27 Roadmap Introduction Statistical concepts and analysis tools Analysis of traffic data: call inter-arrival times call holding times Traffic modeling and characterization Conclusions and references SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 27

28 Traffic data Records of network events: established, queued, and dropped calls in the Vancouver cell Traffic data span periods during: 21, 22, 23 Trace (dataset) Time span November 1 2, 21 March 1 7, 22 March 24 3, 23 No. of established calls 11,348 37,51 387,34 SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 28

29 Hourly traces Call holding and call inter-arrival times from the five busiest hours in each dataset (21, 22, and 23) Day/hour No. Day/hour No. Day/hour No : 16: 3, : 5: 4, : 23: 4, : 1: 3, : 23: 4, : 24: 4, : 17: 3, : 24: 4, : 24: 4, : 2: 3, : 1: 3, : 3: 4, : 21: 3, : 1: 3, : 2: 4,97 SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 29

30 Example: March 26, call inter-arrival time Call holding times (s) :18: 22:18:2 22:18:4 22:19: Time (hh:mm:ss) SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 3

31 Roadmap Introduction Statistical concepts and analysis tools Analysis of traffic data: call inter-arrival times call holding times Traffic modeling and characterization Conclusions and references SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 31

32 Statistical distributions Fourteen candidate distributions: exponetial, Weibull, gamma, normal, lognormal, logistic, log-logistic, Nakagami, Rayleigh, Rician, t-location scale, Birnbaum-Saunders, extreme value, inverse Gaussian Parameters of the distributions: calculated by performing maximum likelihood estimation Best fitting distributions are determined by: visual inspection of the distribution of the trace and the candidate distributions K-S test on potential candidates SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 32

33 Maximum Likelihood Estimation (MLE) Introduced by R. A. Fisher in 192s The most popular method for parameter estimation Goal: to find the distribution parameters that make the given distribution that follow the most closely underlying data set Conduct an experiment and obtain N independent observations θ 1, θ 2,..., θ k are k unknown constant parameters which need to be estimated SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 33

34 Maximum likelihood estimation SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 34

35 Call inter-arrival times: pdf candidates Probability density Traffic data Exponential model Lognormal model Weibull model Gamma model Rayleigh model Normal model Call inter-arrival time (s) SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 35

36 K-S test results: 23 Distribution Parameter , 22: 23: , 23: 24: , 23: 24: , 2: 3: , 1: 2: h Exponential p k h Weibull p k h Gamma p k h Lognormal p 1.15E E E E E-21 k SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 36

37 Call inter-arrival times, best-fitting distributions: cdf 1 Cumulative distribution Traffic data Exponential model Weibull model Gamma model Call inter-arrival time (s) SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 37

38 Call inter-arrival time: autocorrelation autocorrelation function 99% confidence interval 95% confidence interval Autocorrelation Lag SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 38

39 Logscale diagram, call inter-arrival times: , 22: 23: 1 wavelet power spectrum regression line 95% confidence intervals log 2 E{d(j,k) 2 } LRD: α> (H>.5) Octave j similar logscale diagrams for other traces SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 39

40 Call inter-arrival times: estimates of H Traces pass the test for time constancy of α: estimates of H are reliable Day/hour H Day/hour H Day/hour H : 16: : 5: : 23: : 1: : 23: : 24: : 17: : 24: : 24: : 2: : 1: : 3: : 21: : 1: : 2:.75 SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 4

41 Roadmap Introduction Statistical concepts and analysis tools Analysis of traffic data: call inter-arrival times call holding times Traffic modeling and characterization Conclusions and references SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 41

42 Call holding time: pdf candidates Probability density Traffic data Lognormal model Gamma model Weibull model Exponential model Normal model Rayleigh model Call holding time (s) SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 42

43 Best-fitting distributions: cdf 1 Cumulative distribution Traffic data Lognormal model Exponential model Gamma model Weibull model Call holding time (s) SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 43

44 K-S test results: 23 No distribution passes the test when the entire trace is tested (significance levels =.1 and.1) Lognormal distribution passes test (significance level =.1) for: 5-6 sub-traces from 15 randomly chosen 1,-sample subtraces passes the test for almost all 5-sample sub-traces Test rejects null hypothesis when the sub-traces are compared with candidate distributions: exponential Weibull gamma SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 44

45 Call holding time: autocorrelation.8 Autocorrelation autocorrelation function 99% confidence interval 95% confidence interval Lag SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 45

46 Logscale diagram, call holding times: , 22: 23: wavelet power spectrum regression line 95% confidence intervals log 2 E{d(j,k) 2 } Octave j independence: α (H.5) similar logscale diagrams for other traces SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 46

47 Call holding times: estimates of H All (except one) traces pass the test for constancy of α only one unreliable estimate (*): consistent value Day/hour H Day/hour H Day/hour H : 16: : 5: : 23: : 1: : 23: : 24: : 17: : 24: : 24:.463 * : 2: : 1: : 3: : 21: : 1: : 2:.466 SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 47

48 Roadmap Introduction Statistical concepts and analysis tools Analysis of traffic data: call inter-arrival times call holding times Traffic modeling and characterization Conclusions and references SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 48

49 Call inter-arrival and call holding times Day/hour Avg. (s) Day/hour Avg. (s) Day/hour Avg. (s) inter-arrival holding : 16: : 5: : 23: inter-arrival holding : 1: : 23: : 24: inter-arrival holding : 17: : 24: : 24: inter-arrival holding : 2: : 1: : 3: inter-arrival holding : 21: : 1: : 2: Avg. call inter-arrival times: 1.8 s (21),.86 s (22),.84 s (23) Avg. call holding times: 3.91 s (21), 3.96 s (22), 4.13 s (23) SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 49

50 Distributions Distribution Expression Remark exponential Weibull f f e ( x) = x / µ µ b b 1 ( x / a) ( x) = ba x e I(, ) b ( x) I(, ) ( x) : incomplete beta function gamma f ( x) = x b a 1 a ( x / e Γ( a) b) Γ(a) : gamma function lognormal f ( x) = e (ln x µ ) 2 2σ 2 xσ 2π SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 5

51 Best fitting distributions Busy hour : 16: : 1: : 17: : 5: : 23: : 24: : 23: : 24: : 24: Distribution Call inter-arrival times Weibull Gamma a b a b Call holding times Lognormal µ σ SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 51

52 Estimates of H Hurst parameter estimates call inter-arrival times 22 call inter-arrival times 23 call inter-arrival times 21 call holding times 22 call holding times 23 call holding times Rank call inter-arrival times: H.7.8 call holding times: H.5 SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 52

53 Conclusions We analyzed voice traffic from a public safety wireless network in Vancouver, BC call inter-arrival and call holding times during five busy hours from each year (21, 22, 23) Statistical distribution and the autocorrelation function of the traffic traces: Kolmogorov-Smirnov goodness-of-fit test autocorrelation functions wavelet-based estimation of the Hurst parameter SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 53

54 Conclusions Call inter-arrival times: best fit: Weibull and gamma distributions long-range dependent: H.7.8 Call holding times: best fit: lognormal distribution uncorrelated SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 54

55 References J. Song and Lj. Trajković, Modeling and performance analysis of public safety wireless networks, in Proc. IEEE IPCCC, Phoenix, AZ, Apr. 25, pp D. Sharp, N. Cackov, N. Lasković, Q. Shao, and Lj. Trajković, Analysis of public safety traffic on trunked land mobile radio systems, IEEE J. Select. Areas Commun., vol. 22, no. 7, pp , Sept. 24. N. Cackov, B. Vujičić, S. Vujičić, and Lj. Trajković, Using network activity data to model the utilization of a trunked radio system, in Proc. SPECTS, San Jose, CA, July 24, pp E-Comm, Emergency Communications for SW British Columbia Incorporated. (25, May). [Online]. Available: F. Barcelo and J. Jordan, Channel holding time distribution in public telephony systems (PAMR and PCS), IEEE Trans. Vehicular Technology, vol. 49, no. 5, pp , Sept. 2. W. Leland, M. Taqqu, W. Willinger, and D. Wilson, On the self-similar nature of Ethernet traffic (extended version), IEEE/ACM Trans. Networking, vol. 2, pp. 1 15, Feb P. Abry, P. Flandrin, M. S. Taqqu, and D. Veitch, Wavelets for the analysis, estimation, and synthesis of scaling data, in Self-similar Network Traffic and Performance Evaluation, K. Park and W. Willinger, Eds. New York: Wiley, 2, pp R. B. D'Agostino and M. A. Stephens, Eds., Goodness-of-Fit Techniques. New York: Marcel Dekker, pp , pp , pp SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 55

56 Network architecture Cell Repeater Network management system Dispatch console Channels Central switch Cell controller Cell Cell User radios Cell SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 56

57 Network model central switch 11 cells SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 57

58 Test for constancy: example Scaling exponent α Rejected average of estimates for sub-traces estimate of α for the entire trace 95% confidence intervals Sub-trace index Trace is divided into 12 sub-traces of equal lengths Variation of the scaling exponent indicates that α is not constant Star Wars IV (MPEG-4) SPECTS '5: July 25, 25 Modeling and characterization of traffic in PSWNs 58