Internet Dial-Up Traffic Modelling

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1 NTS 5, Fifteenth Nordic Teletraffic Seminar Lund, Sweden, August 4, Internet Dial-Up Traffic Modelling Villy B. Iversen Arne J. Glenstrup Jens Rasmussen Abstract This paper deals with analysis and modelling of real Internet dial-up traffic based on extensive data, which are described and analysed. When recording the traffic process the time axis is digitalised and analysis and modelling must take this into account. In particular, this is very important for arrival processes. It is also important to distinguish between fast variations within short time intervals and slow variations during the day. The detailed analysis shows that the arrival process is a Poisson process with slow variations. Keywords: scanning principle, time-dependent Poisson process, mean squared difference Introduction Characteristics of dial-up Internet calls are of great interest when dimensioning the local access network [], [7]. Internet calls have characteristics which are different from ordinary voice calls. The field data used for the study are Internet dial-up calls from Tele Danmark. During one week, Monday 8 Sunday 4, January 999, about.8 million calls have been recorded. Every call is recorded with arrival time, holding time, and other useful information such as identity of modem-pool, ISDN, subscription type, etc. Only about.5 % of the call attempts are rejected for various reasons. About 79 % of all calls are PSTN, and the remaining % are ISDN calls. ISDN calls are always counted as one call independent of the number of channels occupied. The accuracy of the time recordings is one second. Several papers deal with the same subject. The daily traffic profile has the maximum in the evening. In Fig. we show a typical profile for the number of calls. The mean value of the holding times is 5 times bigger the the mean value of voice holding times, and a typical pattern is shown in Fig.. Similar patterns are found by other authors. The distribution of the holding time has a coefficient of variation which is bigger than for voice calls, and the modelling has been studied using hyper-exponential, log-normal and Pareto distributions. Dept. of Telecommunication, Technical University of Denmark, DK 8 Denmark. Dept. of Telecommunication, Technical University of Denmark, DK 8 Denmark. Tele Danmark, Sletvej, DK 8 Denmark.

2 4 arrivals Figure : Number of calls per 5 minutes, Tuesday service time (sec) Figure : Mean holding time n seconds for calls arriving during 5 minutes, Tuesday,

3 In this paper we shall focus upon a study of the arrival process. In some papers it is concluded that the arrival process of Internet calls is more bursty than the Poisson process, whereas [7] concludes that within intervals of up to 5 minutes it can be considered as a stationary Poisson process. In larger time intervals we have intensity variations, but still a Poisson process. This corresponds to voice traffic where we have slow variations due to the daily call profile, but where the fast variations are of random (Poissonian) nature. The concept of slow and fast variations was introduced by Palm [8] and studied in detail for voice traffic in [5]. There are two basic ways to study an arrival process. We may keep a time interval fixed and study the number of calls within this interval (number representation), or we may keep a number of calls fixed, and study the time interval to get this number of calls (interval representation). We shall study both approaches in this paper. Interval representation The accuracy of the recordings is one second. That is, the continuous real time is discretized to an integer. This causes additional problems for the interval representation, whereas the number representation fits with this approach. When observing a continuous time process at regular time instants the continuous time intervals are transformed into a discrete time distribution. The important case of the Poisson process is transformed into a discrete time process with correlation between inter-arrival times of duration zero. Thus a model in discrete time based on geometric inter-arrival times (Bernoulli arrival process) is not equivalent to a Poisson process in continuous time. The transformation of traffic processes from continuous time to discrete time has been dealt with in the theory of traffic measurements where we observe the traffic process at regular time intervals. In the Holbæk measurements [5], which were the first computerized measurement, some fundamental aspects were investigated. The discrete time increases the observed burstiness of the traffic process, so it looks heavy-tailed, even if it is Poissonian.. The scanning method We shall only consider regular (constant) scanning intervals. The scanning method is e.g. applied to traffic measurements, call charging, numerical simulations, and processor control. By the scanning method we observe a discrete time distribution for the holding time which in real time usually is continuous. In practice, we usually choose a constant distance h between scanning instants, and we are able to derive a relation between the observed time interval and the real time interval (Fig. ). We notice that there are overlaps between the continuous time intervals, so that the discrete distribution cannot be obtained by a simple integration of the continuous time interval over a fixed interval of length h. If the real holding times have a distribution function F(t), then it can be shown that we will observe the following discrete distribution [5]: p() = h p(k) = h h h F(t)dt () {F(t + kh) F(t +(k )h)}dt, k =,,... ()

4 Observed number of scans Interval for the real time (scan) Figure : By the scanning method a continuous time interval is transformed into a discrete time interval. The transformation is not unique. Interpretation: The arrival time of the call is assumed to be independent of the scanning process. Therefore, the density function of the time interval from the call arrival instant to the first scanning time is uniformly distributed and equal to (/h). The probability of observing zero scanning instants during the call holding time is denoted by p() and is equal to the probability that the call terminates before the next scanning time. For a fixed value of the holding time t this probability is equal to (F(t)/h), and to obtain the total probability we integrate over all possible values t ( t < h) and get (). In a similar way we derive formula () for p(k). For exponential distributed holding time intervals, F(t) = e µt, we will observe a discrete distribution, Westerberg s distribution [5]: p() = ( e µh) () µh p(k) = ( e µh) e (k )µh, k =,,... (4) µh The i th derivative of the probability generating function Z(z) (Z transformed) for the value z = becomes: Z (i) ()= i! ih (e µh ) i (5) from which we find the mean value m and form factor ε (second moment divided by the squared mean value) [6]: m = Z () ()= µh, (6) σ = Z () ()+Z () () Z () (), (7) ( ) σ ε = = µh eµh + e µh. (8) m 4

5 Thus for any distribution function F(t) we will always observe the correct mean value. When using Karlsson charging we will therefore always in the long run charge the correct amount. For a continuous measurement the form factor is. The contribution ε T is thus due to the influence from the measuring principle. In [] the scanning method is generalized to phase-type distributions for both the scanning interval and the inter-arrival time. The form factor is a measure of accuracy of the measurements. Fig. 4 shows how the form factor of the observed holding time for exponentially distributed holding times depends on the length of the scanning interval (8). By continuous measurements we get an ordinary sample. By the scanning method we get a sample of a sample so that there is uncertainty both because of the measuring method and because of the limited sample size. 5 Formfactor "... k = Scan interval [s, ] Figure 4: Form factor for exponentially distributed holding times which are observed by Erlang-k distributed scanning intervals in an unlimited measuring period. The case k = corresponds to regular (constant) scan intervals which transform the exponential distribution into Westerberg s distribution. The case k = corresponds to exponentially distributed scan intervals. The case h = corresponds to a continuous measurement. We notice that by regular scan intervals we loose almost no information if the scan interval is smaller than the mean holding time (chosen as time unit). In Fig. 5 and Fig. 6 we compare for ISDN calls the observed formfactor with the formfactor calculated from Westerberg s distribution using the total number of calls during 5 minutes to calculate the mean arrival rate. We notice the two curves are very similar. The ratio between the two results is shown in Fig. 7. So even though the formfactor is greater than two, we may assume it is a Poisson process within 5 minutes. If we look at high arrival rates and include all calls the observed and calculated formfactors 5

6 .5.5 Palm form factor Figure 5: Observed formfactor of inter-arrival times for ISDN-calls in 5-minutes intervals from observed data, Tuesday Cf. Fig Westerberg form factor Figure 6: Formfactor of inter-arrival times for ISDN-calls calculated from Westerberg s distribution (8) based on the total number of calls observed per 5 minutes, Tuesday Cf. Fig. 5. 6

7 . Palm/Westerberg form factor ratio Figure 7: The ratio between the observed values (Fig. 5) and the theoretical values (Fig. 6). will be identical, because all inter-arrival times will be zero except for 9 inter-arrivals equal to one second due to 9 scans during 5 minutes. An example is shown in Fig. 8. In this case we cannot make any conclusions about the inter-arrival time distribution for the busy periods. Investigation of intervals with fewer calls, e.g. for individual modem pools or during the night, show that Westerberg s distribution is a very good model for inter-arrival times for periods up to one hour. If we include the ISDN-calls as two simultaneous arrivals, the observed and calculated values become different for the cases where we don t have too many calls per scan interval. Number representation In this case we study the number of calls during intervals of fixed duration. The measuring method is exact. If the arrival process is a stationary Poisson process, then the number of calls during a fixed time interval will be Poisson-distributed. We may observe the number of calls per time interval (e.g. one second) during 5 minutes and perform tests for the Poisson distribution. In general, a χ -test will accept a Poisson hypothesis. In the following we shall look at a second order statistics to study intensity variations.. Generalized mean differences Let us consider a discrete distribution function p(i) (i =,,...) with mean value m, second moment m, and variance σ = m m. 7

8 6 4 Palm form factor Figure 8: Observed formfactor of inter-arrival times for all calls in 5-minutes intervals from measurements. The values calculated from Westerberg s distribution will be identical for the busy periods. Tuesday The generalized mean difference of order r is defined by: For r = wehave δ = i = i = i δ r = i i j r p(i) p( j). j (i j) p(i)p( j) j { i p(i) p( j)+ j p( j) p(i) i jp(i) p( j) } j i p(i)+ j p( j) = ( m m ) = σ. j ( ip(i) i )( ) jp( j) j For the discrete distributions applied in the BPP (Binomial Poisson Pascal) traffic model we get the normalized generalized mean difference (NGMD) (In the Binomial and Pascal distributions p denotes the probability of success): 8

9 Binomial: Poisson: Pascal: δ m = ( p). δ m =. δ m = ( + p). In the Binomial and Pascal distributions p denotes the probability of success. A double stochastic Poisson process (a Cox process) is a Poisson process where the intensity λ is a stochastic variable. If for example λ is Γ-distributed, then the interarrival times are Paretodistributed [6]. Let λ have the mean value be m λ and the variance σ λ, then we get for the the number of calls in a fixed time interval: Palm: δ m = + σ λ m λ. The above statistics may be used to distinguish between fast and slow variations in a point process [5]. If we let p(i) and p( j) denote the number of calls in two consecutive intervals of fixed length, then the slow variations have no influence upon the the result, because the statistic is a linear function of the intensity. On the other hand, fast variations within the duration of the two intervals will increase the value of the statistics. In Fig. 9 we show the NGMD for periods of 5 minutes based on consecutive intervals of duration second each. The average is about. indicating that the arrival process is a little more bursty than the Poisson process. In Fig. we show the NGMD for the same period when the consecutive intervals are of length 6 seconds each. The mean value increases to.5 indicating that there are small intensity variations during minutes. In Fig. we show the NGMD for increasing values of the consecutive intervals from to seconds. The curve indicates significant intensity variations during two consecutive periods of each minutes. If we consider individual model pools, the NGMD becomes close to for intervals of length 5 minutes, which indicates a Poisson process. By Palm s theorem [6] the total arrival process should be even more Poissonian. In the above examples the average number of call per second is about 7 for all 4 hours of the day. It seems as if the large number of calls per time interval tends to increase the value of NGMD above, even for a Poisson process. Further detailed investigations have to be carried out. 4 Conclusion Above we have presented results for the analysis of the arrival process of dial-up Internet calls. Most results are based on all recorded calls. In [4] investigations have also been carried out for individual modem-pools. All investigations indicate that the arrival process is a Poisson process with slow variations during the day. Only at particular times, e.g. at 9: in the evening when the tariff is reduced to 5 %, there are rather fast variations (Fig. ). 9

10 .5 generalised mean difference Figure 9: Normalized generalised mean differences using -second intervals for periods of 5 minutes. Average =.79. Tuesday generalised mean difference Figure : Normalized generalised mean differences using 6-seconds intervals for periods of 5 minutes. Average =.58. Tuesday

11 6 5 generalised mean difference interval length (sec) Figure : The mean-squared-deviation for all calls as a function of the period. Tuesday References [] Christensen, T. K. & Nielsen, B. F. & Iversen, V. B. (): Distribution of channel holding times in cellular communication systems. NTS 5, Fifteenth Nordic Teletraffic Seminar, Lund, August. 9 pp. [] Färber, J. & Bodamer, S. & Charzinski, J. (999): Statistical evaluation and modelling of Internet dial-up traffic. IND, Institute of Communication Networks and Computer Engineering, University of Stuttgart 999. Technical Report. pp. [] Fredericks, A. A. (999: Impact of holding time distributions on parcel blocking in multi class networks with application to Internet traffic on PSTN s. ITC 6, Sixteenth International Teletraffic Congress, Edinburgh, June 999. Proceedings pp Elsevier Science B. V [4] Hussain, I. (999): Characterization of dial-up based Internet user traffic. Master s thesis. Department of Telecommunication, Technical University of Denmark, August pp. [5] Iversen, V. B. (97): Analyses of real teletraffic processes based on computerized measurements. Ericsson Technics, vol. 9 (97): 64. [6] Iversen, V.B. (999): Introduction to Traffic Engineering. Textbook. Department of Telecommunication, Technical University of Denmark pp.

12 [7] Naldi, M. (999): Measurement-based modelling of Internet dial-up access connections. Computer networks, Vol. (999) :, 8 9. [8] Palm, Conny (94): Intensitätsschwankungen im Fernsprechverkehr. Ericsson Technics No. 44, pp. English translation by Chr. Jacobæus: Intensity Variations in Telephone Traffic, North-Holland Publ. Co. 987.

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