Statistical Multiplexing Strategies for Self-Similar Traffic

Statistical Multiplexing Strategies for Self-Similar Traffic Linawati and Nyoman Putra Sastra Electrical Engineering Department Udayana University Pho...
Author: Clarence Dorsey
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Statistical Multiplexing Strategies for Self-Similar Traffic Linawati and Nyoman Putra Sastra Electrical Engineering Department Udayana University Phone/Fax: +62 361 703315 Email: [email protected] and [email protected] Abstract-Traffic-shifting and shuffle methods are proposed as statistical multiplexing strategies for self-similar traffic sources. The purpose of these strategies is to decrease the burstiness of the traffic. This paper explores the Hurst parameter, and autocorrelation of the homogenous and heterogenous traffic. The simulation initially investigates the traffic with and without strategies implementation. Finally the comparisons between both methods are explored. The simulation results show that the strategies can decline the burstiness of the traffic. When contrasting the traffic performances achieved by both strategies, no large differences of H-parameters and autocorrelation coefficients of the traffic are observed.

I. INTRODUCTION Relatively few studies have analysed statistical multiplexing implementation on self-similar traffic. Theoretical results (by assessing all parameters in Norros formula [10 - 11]) and simulation results using a cell-level ATM network simulator regarding the statistical multiplexing of self-similar traffic sources were investigated in [2]. These theoretical results argued that less effective bandwidth was needed for the aggregated sources than the sum of the effective bandwidths of individual stream. The simulation results demonstrated a qualitative and quantitative agreement with the theoretical results. A priority strategy which was strictly First In First Out (FIFO) was implemented for self-similar traffic before it went to the multiplexer [1]. The priority level of each cell was based on the linear prediction of the number of cells per time unit offered to the multiplexer by each input. This priority information managed the discarding of incoming cells. Therefore bandwidth allocation with reasonable values of cell loss and delay were obtained. Random swapping packets orders could not simply remove Long Range Dependent (LRD) characteristics from selfsimilar traffic with high-degree Hurst parameter [14]. The authors also argued that multiplexing self-similar traffic with non self-similar traffic will not eliminate the long-range dependent characteristics. Therefore they proposed shuffle procedure for self-similar traffic. Shuffling can be used to decrease the burstiness of the traffic. This procedure has been implemented on self-similar traffic and its effect has been analysed on cell loss and buffer size [7].

The effective bandwidth formulation by Norros has also been reviewed by [3] for statistical multiplexing of self-similar video streams in a video-on-demand scenario. The statistical multiplexing gain was achieved but it was not as large as for Markovian traffic flows. In Ref. [4], a study of self-similar traffic impact on statistical multiplexing using the pseudo selfsimilar model to approximate self-similar traffic has been carried out. The Poisson, ON-OFF and pseudo self-similar models with respect to the cell loss probability at an ATM statistical multiplexer has been compared. The merging and splitting of self-similar traffic streams, on the other hand, have been examined mathematically by [5]. The authors concluded that merging or splitting self-similar streams also produced a self-similar stream. In addition the Hurst parameter of merged traffic has been determined by the largest Hurst parameter of the input stream. Ref. [6, 8] proposed an envelope process for statistical multiplexing of independent self-similar sources. The authors showed that the multiplexing gain increased with the number of sources, but decreased for the higher H-parameter. The effect of statistical multiplexing on the long-range dependence of the Internet packet traffic was reviewed by [9]. The authors showed that the number of active connections could change components of long-range dependence at all levels. II. STATISTICAL MULTIPLEXING SCENARIO The multiplexer is modelled as a finite capacity queuing system with buffer size B and one server with fixed output rate C. Service discipline is assumed to be FIFO. The input of the multiplexer consists of N independent self-similar traffic sources, which is generated based on the Random Midpoint Displacement (RMD) algorithm [12] and each source generates a variable number of cells per second. The multiplexer switch samples each input source in an interval of T seconds in a multiplexer cycle. After collecting information from the input, the multiplexer passes traffic onto the common buffer. The multiplexer then samples the next source and passes it onto the common buffer and likewise for all N sources, until all the cells in a particular multiplexer cycle are sampled. The common buffer is used to store incoming cells, which cannot be instantly transported at the time of arrival. The cells remain in the buffer until they reach the head of the buffer and the output link becomes available.

The asynchronized multiplexing model is applied in this multiplexer. The asynchronized multiplexing model is shown in Fig. 1. III. STATISTICAL MULTIPLEXING STRATEGIES Traffic-shifting method is proposed and shuffle method [13, 14] is adopted as statistical multiplexing strategies for selfsimilar traffic sources A. Traffic-shifting Method A simple method to prevent the network from the worst case of the multiplexed traffic is to shift the subsequent sources in the multiplexer input. The worst case occurs if all the peak rates of the traffic sources come at the same time. This will lead to the burstier aggregated traffic. With this rearrangement of sources, the peak rates will not all arrive at once. Since the service strategy is FIFO, the first traffic that arrives has a higher priority than the traffic arriving later. Therefore the main idea of the proposed traffic-shifting method is to shift the lower priority traffic. The proposed strategy is described briefly below.

With this composition, the peak rates of traffic sources will be evenly spread over the time axis among different sources. B. Shuffle Method The shuffle strategy for long-range dependent traffic was initially introduced by [13]. Although randomly shuffling packets will not succeed in removing the long-range dependence completely, it still can reduce the burstiness of traffic. Fig. 2 shows how this strategy can reduce the burstiness of a traffic source. The traffic source is shuffled for a certain time interval to decrease the burstiness. This strategy is adopted to implement on multiplexing self-similar traffic sources. In multiplexing traffic sources, two ways of implementations of the shuffle strategy are compared. In the first implementation, each traffic source is shuffled and then all shuffled traffic is multiplexed. In the second implementation, all traffic sources are multiplexed and then shuffle the multiplexed traffic.

Step 1 Sum some incoming traffic sources. For example, for 4 traffic sources, we sum three arriving traffic sources, Asum=3(t). There are four possibilities of Asum=3(t) must be calculated, i.e.,

Time

Source 1

Source 2

. . .

Buffer B Output C Time

Source N

Fig. 2 Original traffic (top) and shuffled traffic (bottom) Fig. 1 Asynchronized multiplexing traffic sources

Asum=3(t)=A1(t)+A2(t)+A3(t), Asum=3(t)=A1(t)+A2(t)+A4(t), Asum=3(t)=A1(t)+A3(t)+A4(t), Asum=3(t)=A2(t)+A3(t)+A4(t). Step 2

Set the fixed rate control R, where R > [(N-1) x λthe lowest H-value traffic input]. λ is the mean rate of traffic.

Step 3

At time t, if any of Asum=3(t)>R then A4(t) is shifted to be A4(t+1) or delayed its transmission. Otherwise the n=4

aggregate of all traffic sources, A(t ) = ∑ An (t ) can n =1

be transmitted at that time simultaneously.

The general idea of the shuffle method is explained below: Step 1 A time series of traffic is divided into blocks or range of time, RT. Step 2 Then the blocks are shuffled. However, the structure of the time series inside a block remains unchanged. Thus, this shuffling removes correlation from the series beyond a lag equal to the length of a block. IV. SIMULATION RESULTS Simulations are carried out to evaluate the effect of the proposed strategy. The simulation process is modelled as close to the real process as possible. The starting time of the traffic sources are random with a normal random distribution. Therefore all traffic sources are treated fairly. They are then

H-parameter

0.8 0.7 0.6 0.5 0.0

0.5

0.7

0.9

1.0

1.3

1.6

1.8

1.9

lo g 1 0 (m) mux H=0.9 traffic

mu x H=0.8 traffic

mux H=0.7 traffic

mu x H=0.6 traffic

Fig. 4 Estimated H-values of multiplexed homogeneous self-similar traffic when the traffic- shifting method is applied 0.7 0.6 autocorrelation

A. Traffic-Shifting Method Fig. 3 compares the aggregated traffic from multiplexing H=0.9 homogeneous self-similar traffic sources without strategy implementation to the multiplexing traffic sources with the traffic-shifting strategy implementation. The traffic-shifting implementation produces smooth aggregated traffic than the multiplexing the traffic sources without strategy. In general, Fig. 4 shows that the Hurst parameters of multiplexed traffic declines by one magnitude of order from the traffic source Hurst parameter, excluding for H=0.6. The traffic source with the highest Hurst parameter experiences the most decline and stable of Hurst parameter when the strategy was implemented. In addition, Fig. 5 demonstrates that the autocorrelation coefficients of multiplexed traffic of the traffic sources with H=0.9 are much less than the traffic source coefficients. The same results also occur for multiplexing heterogeneous self-similar traffic sources, as it is shown in Fig. 6.

0.9

0.5 0.4 0.3 0.2 0.1 0 1

7

13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 lag k traffic source (H=0.9) multiplexed traffic

Fig. 5 Autocorrelation coefficients of multiplexed homogeneous self-similar traffic when the traffic-shifting method is applied 0.9 0.8 H-parameter

multiplexed in the multiplexer and the Hurst parameter and autocorrelation coefficients of the multiplexer output are assessed. Self-similar synthetic traffic has mean rate of 5000 cells/sec, peakedness factor of 800 cells.sec and the Hurst parameters of 0.6, 0.7, 0.8 and 0.9. In this simulation, multiplexing is performed for homogeneous self-similar traffic sources and heterogenous self-similar traffic sources. Multiplexing homogeneous self-similar traffic sources means multiplexing the traffic sources with the same Hurst parameters. Multiplexing heterogenous self-similar traffic sources is limited for traffic sources with different Hurst parameters. Only four traffic streams are applied as traffic sources because the simulation is designed only for four Hurst parameters, namely, 0.6, 0.7, 0.8 and 0.9. The fixed rate control R is set to 18000cells/sec. The time length of the simulation is 8100 second. The Hurst parameters of multiplexed traffic are estimated using the variance-time plot. As well the autocorrelation coefficients of multiplexed traffic are compared to the traffic source autocorrelation coefficients, which have the largest Hurst parameter.

0.7 0.6 0.5 0.0

0.5

0.7

0.9

mux three H=0.6 with one H=0.9 traffic mux two H=0.6 with two H=0.8 traffic mux H=0.6,0.7,0.8,0.9 traffic

1.0 log 10 (m)

1.3

1.6

1.8

mux two H=0.6 with two H=0.9 traffic mux two H=0.6 with two H=0.7 traffic

Fig. 6 Estimated H-value of multiplexed heterogeneous self-similar traffic when the traffic-shifting method is applied

Fig. 3 Multiplexing homogeneous self-similar traffic sources (H=0.9) Without strategy (left), with the implementation of traffic-shifting strategy (right)

1.9

B. Shuffle Method As shown in Fig. 7, all the estimated H-parameters of multiplexed traffic decrease steadily and their values approximate one order of magnitude less than the Hparameters of the traffic sources. The largest decline and difference between the estimated H-parameter of the multiplexed traffic and the H-parameter of the traffic source is happened to multiplexed H=0.9 homogeneous self-similar traffic. However multiplexing H=0.6 traffic sources produces

the smallest decrease of H-parameter and its estimated H-value is just below 0.6. Fig. 8 also demonstrates that when the shuffle strategy is implemented, the autocorrelation coefficients of multiplexed homogeneous traffic are, generally smaller than the FGN autocorrelation coefficients of the same H-parameter as traffic source. 1

H-parameter

0.9 0.8 0.7 0.6

similar traffic sources show that the shuffle strategy can decrease the burstiness of self-similar traffic. It is shown by the estimated H-parameters of aggregated traffic using the variance-time plot and by comparing the autocorrelation coefficients of aggregated traffic to the FGN autocorrelation coefficients. The H-value of multiplexed homogeneous traffic is approximately one order of magnitude below the H-value of its input. Moreover the H-parameter and the autocorrelation coefficients of multiplexed heterogeneous traffic are far less than the largest H-parameter and the autocorrelation coefficients of the traffic source. Again, as in the trafficshifting strategy implementation, in this shuffle strategy implementation, a traffic source with a high H-parameter should be multiplexed with many traffic sources with a low Hparameter to further decreasing the traffic burstiness, as shown in Fig. 10.

0.5 0.60

0.85

1.00 log 10 (m)

1.60

mux four H=0.9 traffic

mux four H=0.8 traffic

mux four H=0.7 traffic

mux four H=0.6 traffic

1

1.85

Fig. 7 Estimated H-values of multiplexed homogeneous self-similar traffic when the shuffle method is applied

0.9 H-parameter

0.00

0.8 0.7

0.5 0.00

0.60

0.85 log 10 (m) 1.00

1.60

1.85

mux two H=0.6 with two H=0.7 traffic

mux two H=0.6 with H=0.8 traffic

mux two H=0.6 with H=0.9 traffic mux H=0.6,0.7,0.8,0.9 traffic

mux three H=0.6 with one H=0.9 traffic

Fig. 9 Estimated H-value of multiplexed heterogeneous self-similar traffic when the shuffle method is applied 1

7

13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 lag k mux H=0.9 traffic

FGN(H=0.9)

Multiplexing H=0.9 self-similar traffic sources Fig. 8 Autocorrelation coefficients of multiplexed homogeneous self-similar traffic when the shuffle method is applied

Commonly, all estimated H-parameters of multiplexed heterogeneous traffic fall slightly and are less than the largest H-values of self-similar traffic source after the implementation of the shuffle strategy, as shown in Fig. 9. Unexpectedly, the H-parameter of multiplexing three H=0.6 with one H=0.9 traffic sources is smaller than the H-parameter of multiplexing two H=0.6 with two H=0.9 traffic sources. This signifies that the shuffle strategy implementation on multiplexing less traffic sources of a high H-value with more traffic sources of a low H-value produces in a smaller H-value of the aggregated traffic. Generally, all the autocorrelation coefficients of aggregated traffic are smaller than the FGN autocorrelation coefficients, which the FGN’s H-parameter is the same as the largest Hparameter of traffic source. Similar to the traffic-shifting strategy, the simulation results of implementing the shuffle strategy on multiplexing self-

mux two H=0.6 with two H=0.9 traffic mux three H=0.6 with one H=0.9 traffic FGN(H=0.9)

0.8 Autocorrelation

Autocorrelation

0.6

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1

0.65 0.5 0.35 0.2 0.05 -0.1 1

7

13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 lag k

Multiplexing two H=0.6 with two H=0.9 self-similar traffic sources, and multiplexing three H=0.6 with one H=0.9 self-similar traffic sources Fig. 10 Autocorrelation coefficients of multiplexed heterogeneous self-similar traffic when the shuffle method is applied

C. Comparisons between Traffic-Shifting and Shuffle Methods Generally, both strategies are able to reduce the burstiness of multiplexed self-similar traffic that is caused by the largest Hparameter of traffic input. They can decrease the H-value of

multiplexed homogeneous traffic approximately one order of magnitude from the largest H-value of traffic input. As shown in Table 1, the traffic-shifting and shuffle methods produce exactly the same average H-parameters of aggregated traffic for all multiplexing homogeneous selfsimilar traffic sources, except for multiplexing H=0.6 traffic sources. In this case, the shuffle method produces slightly smaller average H-parameter than the traffic-shifting method.

When contrasting the traffic performances achieved by both methods, no large differences of H-parameters and autocorrelation coefficients of multiplexed traffic were observed. The average differences were up to 0.07 of the Hparameter, and up to 0.08 of the autocorrelation coefficients. REFERENCES [1]

TABLE I THE AVERAGE H-PARAMETERS OF MULTIPLEXED HOMOGENEOUS SELF-SIMILAR TRAFFIC OF THE SHUFFLE AND TRAFFIC-SHIFTING METHODS IMPLEMENTATIONS H- sources 0.6 0.7 Method TS S TS S H-average 0.62 0.57 0.63 0.63 TS stand for Traffic-Shifting Method S stand for Shuffle Method

0.8 TS 0.7

0.9 S 0.7

TS 0.78

S 0.78

TABLE II THE AVERAGE H-PARAMETERS OF MULTIPLEXED HETEROGENEOUS SELF-SIMILAR TRAFFIC OF THE SHUFFLE AND TRAFFIC-SHIFTING METHODS IMPLEMENTATIONS 0.6,0.6, 0.6,0.9 TS S 0.72 0.69

[3]

[4]

Table 2 compares the average H-parameters of multiplexed heterogeneous self-similar traffic of the traffic-shifting method and the shuffle method. In general, the average H-parameter of the shuffle strategy is slightly less than the average Hparameter of the traffic-shifting strategy.

H-sources Method H-average

[2]

0.6,0.7, 0.8,0.9 TS S 0.75 0.71

0.6,0.6, 0.9,0.9 TS S 0.75 0.73

[5] [6] [7]

[8] [9] [10]

H-sources Method H-average

0.6,0.6, 0.8,0.8 TS S 0.7 0.66

0.6,0.6, 0.7,0.7 TS S 0.66 0.59

[11] [12]

V. CONCLUSION The results of multiplexing self-similar traffic sources with no strategy implementation support many previous studies. Multiplexing self-similar traffic sources without any strategy cannot decrease the burstiness of traffic. The H-parameter of multiplexed traffic approached or was more than the largest Hparameter of traffic source. However implementing the traffic-shifting method and the shuffle method on both multiplexing self-similar traffic sources produced significant improvements. These methods declined the burstiness of aggregated self-similar traffic. They were indicated by the estimated H-parameters and the autocorrelation coefficients of multiplexed self-similar traffic. The H-parameter of multiplexed traffic was smaller than the largest H-parameter of the traffic source. The decrease was approximately one order of magnitude. In addition the autocorrelation coefficients of multiplexed traffic were less than the coefficients of the largest H-parameter of traffic input.

[13] [14]

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