A source model for ISDN packet data traffic *

Transcription

1 1 A source model for ISDN packet data traffic * Kavitha Chandra and Charles Thompson Center for Advanced Computation University of Massachusetts Lowell, Lowell MA * Proceedings of the 28th Annual Conference on Information Sciences and Systems, p , Princeton University, Abstract The statistical characteristics of packetized data traffic on an Integrated Services Digital Network (ISDN) are examined and modeled. The traffic is shown to be composed of packet arrivals having long and short interarrival times. The combined packet arrivals form structures or events in the traffic stream. The mean and variance in the number of arrivals within an event are estimated using a stationary renewal process model. The occurrence of intermittent clusters of events gives rise to nonstationary traffic features. These clusters are shown to produce an index of dispersion for the number of packet arrivals which increases in time at a power law rate. Simulations based on a stochastic model with added chaotic noise for the ev ent clusters reflect this feature of the traffic. 1.0 Introduction Packetized data traffic on communication networks has been shown to exhibit a high variability in the packet interarrival times. Meier-Hellstern et. al.[1]. in their study of ISDN data traffic have shown that the interarrival times for terminal generated packet traffic can be modeled by superposing a gamma and power-law type probability density functions (pdfs). The gamma and power-law pdfs are used to describe the short and long interarrival times respectively. When taken to the limit, the slow decay rate in the tail of the power-law pdf can give rise to a high variance in the interarrival times. In such a case, estimates of the average number and density of arrivals exhibit a slow convergence with increasing observation time. Additional factors that contribute to the high variability are the intermittency and the temporal correlation between the arriving packets. When these situations occur classical models based on Markov renewal process theory are not adequate[1-3]. In this work, the features of the data traffic responsible for the variability in long-time statistics are identified. These features are incorporated into a source model. The results of analysis and numerical simulations are also presented. The objective in analyzing this type of data traffic in detail, is to determine how and what methods can be used to yield insight into the underlying traffic generation processes. 2.0 ISDN Traffic Characterization 2.1 Measured features in ISDN traffic In this section the data from the IS- DN traffic measurements of Meier- Hellstern et. al.[1] will be analyzed. These measurements involve an office automation application where the traffic

2 2 stream originating from eight user terminals was recorded. The traffic was monitored for a period of one week. The terminal-to-host traffic is composed of single and multiple byte packets. Single byte packets that makeup 74% of the terminalto-host traffic will be the focus of the analysis in this paper. The sequence of packet interarrival times are denoted by the set {x i }, where the member x i is the interarrival time of the i th packet. The analysis in this work is based on measured data having a sample size of 16x10 3 packets. Interarrival times in the range {0. 06: 10} seconds comprise 95% of the total data set. The probability density function f (x) of the resulting set of interarrival times is shown in Fig. (2.1.1) on a logarithmic scale. In the range {x: (0. 35: 3. 0) } seconds, a simple linear regression of the probability density predicts a decay rate of O(x 1.8 ) with a 93% correlation coefficient. The distribution of the interarrival times beyond this range cannot be modeled with the same certainty. Therefore we will limit the maximum interarrival time in our analysis to be 3. 0 seconds. On examination of the data, the omission of arrivals beyond 3. 0 seconds decreases the magnitude in the mean number of arrivals over time. However, the long-time trend of the variance in the arrival statistics is not affected. Therefore the salient features of the traffic can be captured with the aforementioned range of interarrival times. Tw o parameters of interest for data traffic applications are the expected number of arrivals over a given time interval and the variance in this estimate. Denoting the number of packet arrivals in a time interval (0, t] as N t, the expected value and variance in this parameter are denoted by E(N t ) and Var(N t ) respectively. Estimates of E(N t ) and Var(N t ) from the measured traffic data are shown in Fig. (2.1.2) for t ranging from 1 to 30 seconds. These estimates were determined by averaging the number of arrivals over non-overlapping time intervals of duration t. A linear regression analysis of the data yields a power-law exponent of 0. 9 and 1. 5 for E(N t ) and Var(N t ) respectively. Therefore the index of dispersion I D (N t ), which is ratio of the variance and the expectation of N t, increases in time at a power-law rate. In the case of independent and identically distributed interarrival times the index of dispersion would converge to a constant value with time. The power law increase in the index of dispersion is characteristic of distribution functions that behave as O(x α ) in the limit as x tends to infinity. For 1<α < 0, the interarrival times are characterized by infinite expectation and variance. In such a case, it has been shown by Feller[4] that the expected number of arrivals increases as O(t α ) and the variance as O(t 2α ) in the limit as the interarrival time x tends to infinity. If the range of interarrival times is finite, the power law increase can be observed over a short time interval. The long time behavior in this case is characterized by an asymptotic convergence of the index of dispersion to a constant value. This value is given by the variance to mean ratio of {x i }. We observe that long time variations in the statistics persists when a finite range of interarrival times are retained. Therefore we conclude that the presence of power law decay in the interarrival time distribution in itself does not offer an explanation for such variations. For the analysis of the short time statistics, an empirical probability density function is constructed for the set of interarrival times {0. 06: 3. 0}. This function denoted as f (x), is a mixture of two density functions, f (x) = c 1 f 1 (x) + c 2 f 2 (x) 2.1.1

3 3 The function f 1 is a gamma distribution f 1 (x) = c n1 b a Γ(a) xa 1 e bx (1 + c +α)d 1+c x c f 2 (x) = c n2 Γ(β)Γ(1 β) 1 + (Dx) 1+c+α and f 2 is the power-law distribution where β=(1 + c)/(1 + c +α). The parameters are (a, b, c, d): (8. 0, 45. 0, , 2. 85). The unit normalizing constants are c n1 and c n2. The weights of the pdfs are (c 1, c 2 ): (0. 7, 0. 3) and α=0. 8. The parameters for the empirical model were determined by matching the mean value and variance of the interarrival times and by goodness of fit tests. The hypothesis tests[5] were found to pass at the 20% significance level. The mean value and the variance of the interarrival times assume values of 0. 4 and respectively. The index of dispersion variations in the packet arrivals using the empirical distribution are in agreement for time intervals less then 2. 0 seconds. Outside this range the model index of dispersion converges to the expected asymptotic value of No such asymptotic value for the index of dispersion is evident for the ISDN data. Hence the iid assumption must be modified if the analysis and observations are to be brought into agreement. The lack of independence beyond a characteristic time scale motivates one to consider the packet generation process and user characteristics more carefully. A typical pattern for a user generating packets would be to generate a burst of activity for a finite time duration followed by a period of inactivity. This pattern would then be repeated at random time intervals. We define the burst of activity to be a successive arrival of packets characterized by short interarrival times. The period of inactivity can constitute one or more packets characterized by long interarrival times. Such periods have been referred to as "think time" in the literature. The hypothesis is that the repetitive arrival of these patterns over time can generate the long term dependence that is observed in the packet traffic measurements. Predicting the time scale of dependence between successive arrivals is crucial for performance modeling and resource allocation if quality of service is to be maintained. To this end, the remaining part of this paper is focussed on developing a model that simulates the aforementioned statistics in the number of packet arrivals N t. 2.2 Event characterization Upon examination, the ISDN data was found to be comprised of patterned sequences of packet arrivals. These sequences will be called events. Each event features a number of packet arrivals having short interarrival times followed by packets arriving at longer time intervals. A schematic of the packet arrival pattern and event structure is given in Figure Short interarrival times fall into the range (0: 0. 35] seconds. Each packet arrival in this range is referred to as a burst arrival. Packet arrivals having interarrival times greater than seconds are referred to as jump arrivals. The threshold time value of seconds corresponds to the intersection point of the pdfs f 1 and f 2. Interarrival times drawn from f 1 yield burst arrivals whereas those drawn from f 2 yield jump arrivals. With the pdfs given in Eqns (2.1.2) and (2.1.3), the probability that one misclassifies an arrival is less than The arrival process is comprised of a sequence of events. Each ev ent onset is initiated by the terminating jump arrival from the previous event. The first generated arrival is a burst. After this arrival a number of burst arrivals ensue. The burst arrivals are then followed by a sequence

4 4 of jump arrivals. This process terminates with the onset of a new event. The burst arrivals occupy 70% of the total distribution f (x). The mean and variance of these interarrival times are and 3. 8x10 3 respectively. The mean and variance of the jump interarrival state distribution which occupy the range ( ] seconds are and respectively. These values are in agreement with the corresponding mean and variance values in the empirical pdf. Interarrival time partitioning of the ISDN data yielded a total of 2517 events. The run-length or number of arrivals in an event will be denoted as n r. The number of burst arrivals in an event is denoted as n b. The variables n b and n r are found to be correlated. Typically long run-lengths incorporate a large number of burst arrivals relative to the number of jump arrivals. Based on analyzing successive events in the ISDN data, the joint probability density function f n (n r, n b ) can be modeled by a hyperexponential function. The range of n r is {2: N r } where N r is the maximum runlength observed. Since each event has a minimum of one jump arrival, the range 12 f n (n r, n b ) = c 4 Σ a1 i b1e i bi 1 n 12 r i=1 Σ a j 2 b j 2 e b j 2 (n r n b 1) j=1 of the burst run-length n b is {1: n r 1}. In the model, (a 1, b 1, a 2, b 2 ): (0. 85, , 0. 85, ). The coefficient c 4 is a normalizing constant of the pdf. The parameters of the model are determined by matching the first two moments with respect to n r and n b. The agreement between the model and the ISDN data are shown in Figs. (2.2.2 a,b) in terms of the marginal statistics f (n r ) and f (n b ) respectively. As seen in this figure, the tail probabilities in the ISDN data for run lengths ranging from { 21: 74 } are not adequately modeled by the empirical pdf despite close agreement of the first two moments. Run lengths in this region account for 2% of the total distribution. The mean and variance of the burst arrivals {µ b, v b } are {4. 43, 24. 4}. The corresponding values for the total run lengths {µ r, v r } are {6. 36, 26. 8}. The marginal statistics for the empirical model exhibit a close agreement with moment values of { µ b, v b, µ r, ṽ r }:{ 4. 52, 21. 2, 6. 44, 24. 5}. The identification of the aforementioned ev ent structure as the basic unit in the traffic stream allows us to characterize the traffic on a local scale. The local time scale corresponds to the mean duration of an event which was found to be 2. 5 seconds. The long time arrival statistics determined from a sequence of events is governed by the features of the event onset times or correspondingly the sequence of event run lengths. Analysis of the ev ent run lengths n r as a function of the ev ent index e i, for i = 1, demonstrates nonstationary features in the mean and variance estimates of n r (e i ). The typical fluctuation in the mean value is computed from the relation, µ nr (e i, M) = 1/M M 1 Σ n r (e i + j) j=0 where M = 100 is the local averaging interval. The mean value was found to exhibit significant local deviations from the first moment of the marginal pdf f (n r ). We find that this nonstationarity is the result of intermittent clustering of events from the tail of the burst run length distribution f (n b ). Typically, burst run lengths greater than eight exhibit this feature. These burst arrivals exhibit an almost linear correlation with the corresponding ev ent run length n r. In addition, this set of events are uncorrelated in the traffic stream. By extracting this set of events, which correspond to 15% of the total sample size, we find that the index of dispersion for the residual set of events exhibit stationarity.

5 5 The arrival of uncorrelated intermittent clusters of events in an otherwise stationary traffic stream, can be modeled by a chaotic noise signal ˆk m. The additive noise variable is the solution of the difference equation [6] where L = (1 ε D)/D 2 and ˆk m = Ak m. The parameters ε=10 6, A = 15 and D = 0. 9 are found to yield the correct first and second moments of the run lengths that comprise the intermittent clusters. Therefore, to determine the run length at a given trial m = e i, one draws the burst length from the pdf f (n b ) and adds the contribution ˆk m. The resulting burst run length is denoted as ˆn b. The corresponding run length ˆn r is then sampled from the marginal f (n r n b ) for ˆn b < 8. For ˆn b 8, the following linear regression relation is found valid for the ev ent run length. ˆn r = c ˆn b + d where c = and d = Source model 3.1 Source model for event arrivals In this section, we will present a stationary renewal model for the generation of burst and jump interarrival times within an event. The influence of additive noise in the run length is not considered. The mean and variance in the number of arrivals versus time are determined by averaging over the event statistics. Each event is taken to be initiated by a triggering arrival which yields a burst and run length. These variables are drawn from the joint run length distribution given in Section 2.2. The arrivals within each event will be modeled using the pdfs f 1 and f 2 which generate the burst and jump arrivals respectively. Initially n b burst arrivals occur. These arrivals are followed by jump arrivals for the remainder of the run. At the end of each run, the process is reinitialized with a burst arrival. Using the run-length pdf given in k m+1 = Section 2.2 the expected number of arrivals can be determined. The probability ε+k m + Lk 2 m k m < D (k m D)/(1 D) 1 > k m D that greater than m arrivals occur at a time t is g m (t) = m 1 Σ f j* (m j)* 1 * f 2 p(m, j) + f1 m* Σ p(m, j) j=1 j=m The function p(m, j) represents the cumulative probability that the run-length n r is greater than m for a given burst runlength j. This can be expressed in terms of the run-length pdf as p(m, j) = Σ f n (i, j) i=m The summation on the index j in Eqn is the result of taking the expected value over the burst run-lengths. The operation f j* denotes the j 1 fold convolution of the function f. Whereas a * b denotes the convolution of a and b. The density of the expected number of arrivals for a single event of duration t is denoted as h(t) and is given by the first moment of the probability that m arrivals have occurred in a time interval t. Therefore h(t) = Σ m(g m (t) g m+1 (t)) m=0 The expected number of arrivals E(N t )is obtained by integrating h(t) with respect to time. The variance density in the number of arrivals is given by the second moment of the probability that m-events have occurred in a time t. v(t) = Σ m 2 ( g m (t) g m+1 (t) ) 2h(t) h(t)dt m=0 t o

6 6 The variance in the number of arrivals is obtained by integrating v(t) with respect to time. The expected number and variance in the number of event arrivals in the IS- DN data were determined by averaging the counting process N t over the ensemble of the data events. A comparison of the corresponding results for the ISDN data ensemble and the analytical results obtained by integrating Eqns. (3.1.3) and (3.1.4) is shown in Fig for t ranging from 1 to 10 seconds. The results from the renewal model and the measurements show reasonable agreement. The initial part of the curve is governed by the burst arrivals. Whereas the slow growth is evidence of jump arrivals. 3.2 Simulation results In this section the simulation results for the long time behavior in the arrival statistics are presented. A set of burst state run lengths {n b (e i )} are first drawn as independent variables from their marginal density function f (n b ) determined from Eqn. (2.2.1). The intermittency in the traffic stream is generated by an additive chaotic noise ˆk m given by Eqn. (2.2.3). The resulting sequence of burst arrivals is therefore ˆn b (e i ) = n b (e i ) + ˆk ei The corresponding run length variables ˆn r (e i ) are then determined by the conditional distribution f (n r n b ) and the regression relation discussed in Eqn. (2.2.4). The arrival process is constructed by the sequence of burst and run length pairs ( ˆn b,ˆn r ) by drawing ˆn b and ˆn r random variables from the interarrival time distributions given by Eqns. (2.1.2) and (2.1.3) respectively. In Fig. (3.2.1) the arrival statistics obtained from the simulation are compared with the ISDN data results for time intervals ranging from 1 to 30 seconds. The results show good agreement. The power law increase in the index of dispersion is the cumulative result of a finite set of jumps that occur at each instant of arrival of a cluster from the superposed noise. The influence of the simulated data traffic on queue statistics will be examined next. A packet arrival rate of λ=2. 5 is taken based on the mean value of the sample interarrival times. The nonstationarity in the traffic arrivals preclude a meaningful discussion of long time average estimates of queue parameters. We therefore observe the queue behavior for fixed time intervals. The average buffer occupancy in a G/D/1 system is denoted as B i, i = 1,... N. Here the index i refers to the ith segment of the arrival process of duration 10 seconds. Results of Section 3.0 have demonstrated the stationary features in the arrival statistics for this duration. The mean value of B i computed over the N ensembles of packet segments is plotted as a function of the load factor ρ in Fig. (3.2.2). The simulation results are found to be in good agreement with the ISDN data results. As the observation time is increased, the buffer occupancy exhibits a high variability. This can be attributed to the intermittent arrival of event clusters that overload the system for a finite duration. 4.0 Conclusions A source model that captures the long time dependence features in measured ISDN data traffic has been presented. A basic structural unit in the traffic has been identified to exhibit stationary features. The arrival statistics within this unit are described by a classical renewal model. The long range dependence is shown to be the result of a deterministic component that results in an intermittent clustering of a specific class of the component units. Modeling this class of ev ents as additive chaotic noise, one can

7 7 demonstrate the long time dependence in the traffic. Acknowledgement The authors wish to thank P.E. Wirth and K. Meier-Hellstern of AT&T Bell Laboratories for providing the traffic data and for useful discussions. References [1] K. S. Meier-Hellstern, P. E. Wirth, Y. Yan, and D.A. Hoeflin, "Traffic models for ISDN data users: Office automation application," in Teletraffic and Data traffic, A period of change, Eds. A. Jensen and V.B. Iversen, Elsevier Science Publishers, p , [2] A. Erramilli, W. Willinger, "Fractal Properties in Packet Traffic Measurements," Proc. of ITC Regional Seminar, St. Petersburg, [3] W. E. Leland, M. S. Taqqu, W. Willinger and D.V. Wilson, "On the Self- Similar Nature of Ethernet Traffic," preprint, [4] W. Feller, "Fluctuation Theory of Recurrent Events," Trans. Am. Math. Soc., 67, , [5] B. Ostle, "Statistics in Research," Chap. 15, The Iowa State University Press, Ames, Iowa, [6] H.G. Schuster, "Deterministic Chaos: An Introduction",second revised edition, VCH, 1988.

8 Fig Interarrival time distribution function for ISDN data traffic Fig Mean and variance in number of arrivals vs time. Fig Schematic of the event structure O denotes an event onset B denotes a burst arrival J denotes a jump arrival Fig (a)Fitted event run length distribution and distribution for ISDN data run lengths. Fig (b) Fitted burst run length distribution and distribution for ISDN burst run lengths. 8