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Modelling and Simulation in Engineering
Volume 2012 (2012), Article ID 408395, 6 pages
http://dx.doi.org/10.1155/2012/408395
Research Article

A Study of the Source Traffic Generator Using Poisson Distribution for ABR Service

Department of Computer Engineering, Wadi International University, The Valley, Homs, Syria

Received 8 April 2012; Accepted 18 June 2012

Academic Editor: Agostino Bruzzone

Copyright © 2012 Mohsen Hosamo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper describes modeling of the available bit rate (ABR) source traffic in asynchronous transfer mode (ATM) network using BLPos/GTEXP traffic generator, which employs Poisson distribution for modeling the burst length (BLPos) and exponential distribution for modeling the gap time (GTEXP). This traffic generator inherits the advantages of both Poisson and exponential distribution functions to achieve enhanced link performance. Analytical and simulation results for BLPos/GTEXP traffic generator have been presented and compared.

1. Introduction

The Poisson process is an extremely useful process for modeling purposes in many practical applications, such as, for example, to model arrival processes for queueing models or demand processes for inventory systems. It is empirically found that in many circumstances the arising stochastic processes can be well approximated by a Poisson process. From the standpoint of the network users, the messages within a session are typically triggered by particular events. But from the standpoint of the network, these message initiations are somewhat arbitrary and unpredictable. Therefore, the sequence of times at which messages arrive for a given session is a random process. Further, the arrival process of one session has to be independent of others. The poisson process has been proved to be the most appropriate for such purpose and therefore has been considered in the present work.

In this paper source traffic modeling and simulation have been carried out for ABR service in ATM networks using Poisson distribution for modeling the burst length (BLPos) and exponential distribution for modeling the gap time (GTExp). The corresponding traffic generator is designated as BLPos/GTExp.

2. BLPos/GTEXP Traffic Generator

BLPos/GTEXP traffic generator generates cells sent at a fixed rate (ACR) during BL and no cells are sent during GT. BL is assumed to be Poisson distributed (BLPos) whereas GT is exponentially distributed (GTExp). BLPos/GTEXP traffic generator was created and parameterized as explained below.

When events occur according to an exponential distribution, they are said to occur completely at random. Thus the arrival time of the next event is not affected by the time elapsed since the previous event [15]. An exponential random variable with a mean value of 𝜆Exp (where  𝜆Exp>0) is given by the following probability density function (PDF): 𝑓(𝑥)Exp=𝜆Expe𝜆Exp𝑥𝑥00𝑥<0,(1) and the cumulative density function (probability that time between events is x) can be obtained by integrating (1): 𝐹(𝑥)Exp=1e𝜆Exp𝑥𝑥00𝑥<0.(2) The mean and variance of the exponential distribution are 𝐸(𝑥)Exp=0𝑥𝜆Exp𝑒𝜆Exp𝑥1𝑑𝑥=𝜆Exp,Var(𝑥)Exp=0𝑥2𝜆Exp𝑒𝜆Exp𝑥1𝑑𝑥𝜆Exp2=1𝜆2Exp.(3) A Poisson random variable 𝑥 with parameter 𝜇Pos  (where  𝜇Pos>0) has PDF given by [1, 2] 𝑓(𝑥)Pos=𝜇𝑥Pos𝑒𝑥!𝜇Pos,𝑥=0,1,2,.(4) The mean and variance are identical for the Poisson distribution [1]: 𝐸(𝑋)Pos=Var(𝑋)Pos=𝜇Pos,(5) where a Poisson distribution can be developed by involving the definition of a Poisson Process for a fixed time (𝑡), so that 𝜇Pos=𝜆Exp𝑡. For Poisson distributed events, the time between successive events is exponentially distributed with mean 1/𝜇Pos. For an event that occurs at a time interval with an exponential distribution, the rate of occurrence of the event is Poisson distributed with mean 𝜆Exp. Required distribution modeling involves a transformation function for converting a random variable of uniform distribution into the required distribution. Considering the fundamental transformation law of probabilities for two probability density functions 𝑓(𝑥) and 𝑝(𝑢)||||=|||||||𝑓(𝑥)𝑑𝑥𝑝(𝑢)𝑑𝑢or𝑓(𝑥)=𝑝(𝑢)𝑑𝑢|||,𝑑𝑥(6) where 𝑝(𝑢) is the PDF of random variable 𝑢 and 𝑓(𝑥) is another PDF of random variable 𝑥. Since 𝑢 is a random variable of a uniform distribution in the range 0 to 1, 𝑝(𝑢) is a constant (=1) and hence 𝑓(𝑥)=𝑑𝑢𝑑𝑥andtherefore𝑢=𝐹(𝑥)=𝑥0𝑓(𝑧)𝑑𝑧.(7) Equation (7) can be used to find source random variable 𝑥=𝐺(𝑢) through inverse transformation of 𝑢=𝐹(𝑥). For the required distribution, the inverse can easily be found from (7) with 𝑓(𝑧) corresponding to the required distribution. 𝑢 is uniformly distributed in the range 0𝑢1. It can be generated by using the function 𝑟𝑎𝑛𝑑() provided by the standard Linux library or using Mersenne Twister (MT) [6].

For modeling the GTExp, (7) is used with 𝑢=𝐹(𝑥)=1𝑒𝜆Exp𝑥or1𝑢=𝑒𝜆Exp𝑥.(8) Therefore, the required transformation is 1𝑥=𝜆Expln(1𝑢).(9) By changing the variables 𝑥 and (1𝑢) of (9) with 𝑋Exp and 𝑈 respectively, we get 𝑋Exp1=𝜆Expln(𝑈).(10)

2.1. Estimation of the Load (𝐿𝑖) for the Traffic Generator

The load variation of the traffic can be realized by synthesizing predefined load such that the resulting load 𝐿=𝑁𝑖=1𝐿𝑖, where 𝐿𝑖 is the traffic load due to 𝑖th source. Therefore, the aggregate traffic from 𝑁 sources will generate the load 𝐿 on a link with rate 𝑅 Mbps giving average throughput of 𝑅𝐿 Mbps. The load 𝐿𝑖 generated by an individual source can be expressed as 𝐿𝑖=BLPos𝐾BLPos𝐾+𝑃𝑟+GLExp,(11) where BLPos, GLExp, K, and 𝑃𝑟 are the mean BLPos, mean gap length, cell size, and minimum intercell gap length (Preamble), respectively in bytes, and then the load 𝐿𝑖 can be found from (11).

2.2. Estimation of the Minimum Gap Time (𝑀GTExp)

The minimum GTExp(𝑀GTExp) is a secondary parameter dependent on load. Given a desired load, 𝑀GTExp is calculated by the source automatically using exponential distribution. Using (11) the GTExp can be expressed as GLExp=BLPos𝐾1𝐿𝑖𝐿𝑖𝑃𝑟.(12) The BLPos can be written as BLPos=𝐸(𝑥)BLPos=𝑀BLPos,(13) where 𝑀BLPos is the minimum BLPos.

GLExp can be written as GLExp=𝐸(𝑥)GLExp=𝑀GLExp.(14) Substituting the values of BLPos and GLExp from (13) and (14), respectively, in (12) 𝑀GLExp can be written as 𝑀GLExp=𝑀BLPos𝐾𝐿𝑖𝐾+𝑃𝑟.(15) Considering the link rate and using the following relation: ByteTime=ByteSize(bits)𝑏LinkRate(bits/sec)orByteTime=𝑅,(16) the 𝑀GTExp now can be computed as 𝑀GTExp=𝑏𝑅𝑀BLPos𝐾𝐿𝑖𝐾+𝑃𝑟.(17) Now the value of 𝑃𝑟=1/ACR is readily available, depending upon the selected value(s) of ACR that can be separately taken as variable, and thus (17) can be rewritten as 𝑀GTExp=𝐾𝑏𝑅𝑀BLPos1𝐿𝑖1.(18) Therefore, (18) can be used for computing the value of 𝑀GTExp that would result in link load closer to 𝐿𝑖 using the selected values of 𝐿𝑖 of the 𝑖th source, 𝐾, and Poisson distribution parameters. Considering the same parameter values for burst and gap lengths and 𝑀BLPos=1, (18) can be simplified as 𝑀GTExp=𝐾𝑏𝑅𝑀BLPos1𝐿𝑖1.(19)

2.3. Generating the GTEXP

Bearing in mind that 𝜆Exp=ACR, since in ABR service a source sends its data with a rate equal to ACR, defining 𝜆GTExp as the exponential mean arrival rate for GTExp, and considering (14) in terms of time, we get (20) 𝐸(𝑥)GTExp=1𝜆GTExp=𝑀GTExp.(20) Now the BLPos/GTEXP traffic generator can compute the GTExp using the relation GTExp=𝑀GTExpln(𝑈).(21)

2.4. BLPos Generating

BLPos can be modeled as follows: for ABR service in ATM networks the value of ACR should be within the range MCRACRPCR. The source first starts transmitting a random-sized burst of cell at ICR. It then waits for a random amount of time, which follows exponential distribution. The source will go on repeating this cycle BLPos/GTExp until completing transmission. In every cycle the number of cells is determined by ACR, which is changed due to the feedback used in the switch. The source repeats the calculation of Poisson distribution, (4) with 𝜇Pos=ACR, 𝑘 times, where 𝑘 is changing from 0 to peak cell rate (PCR). The results will be saved in an array with size 𝑘. The number of cells inside the burst can be found by generating a random number u [6], applying the cumulative distribution function (cdf) for all array values, so that the cumulative values are smaller than or equal to a random number u. The following relation gives the BLPos that is used by the BLPos/GTExp traffic generator: BLPos=𝑥=𝑢𝑥=0𝑒𝜇Pos𝜇xPos𝑥!,𝜇Pos=ACR,(22) where the number of cells inside the burst should be at least one (𝑀BLPos=1).

3. Analytical Results

Consider the values of 𝐿𝑖, 𝑅, 𝐾, 𝑀BLPos, Poisson mean arrival rate (𝜇BLPos), and 𝜆GTExp for BLPos/GTExp traffic generator as given in Table 1. The value of 𝑀GTExpwas determined using (19) (Table 1).

tab1
Table 1: The evaluated parameters for the BLPos/GTExp traffic generator.

The analytical result of BLPos/GTExp traffic generator for 1000 count values of 𝑈, generated by the uniform distribution, are shown in Figure 1, and the corresponding computed values of mean, variance, maximum, and minimum values of BL and GT, and are given in Table 2.

tab2
Table 2: BLPos (cells) and GTExp (𝜇sec) for BLPos/GTExp traffic generators.
408395.fig.001
Figure 1: BLPos/GTExp traffic generator for 1000 count values of U.

The variations in BLPos, GTExp as functions of 𝜇BLPos and 𝜆GTExp for BLPos/GTExp traffic generator for 100 count values of U are shown in Figures 2 and 3 respectively. The increment steps for 𝜆BLPos(1–110) cells/sec and 𝜇GTExp(1–110) cells/sec are 10 for each. Referring to Table 2 it is seen that the minimum values of BLPos/GTExp are greater than their corresponding values of 𝑀BLPos/𝑀BLExp. Referring to Figure 2 it can be concluded that the Poisson mean arrival parameter 𝜇BLEXP should not be a very large value, because BLPos will, consequently, be very large as well, and the source will spend most of its time sending only the burst cells with a smaller number of gap intervals for BLPos/GTExp traffic generator resulting in less-bursty traffic. Referring to Figure 3 it can be concluded that the exponential mean arrival parameter 𝜆GTExo should be selected between 2 and 30 cells/sec for simulation of real bursty traffic because it offers higher peak values of GTExp. This is further supported by the observation that for 𝜆GTExo in the range 30 to 100 cells/sec, the peak values of GTExp have the least variation indicating smoothest traffic.

408395.fig.002
Figure 2: BLPos versus 𝜇BLPos and U.
408395.fig.003
Figure 3: GTExp versus 𝜆GTExo and U.

4. Simulation Results

The ATM network simulation was carried out under Linux network programming. The parameters specified in Table 1 were used for this simulation. Six sources 𝑆𝑖(𝑖=1,2,6) sending their data at the rate ACR𝑖(𝑖=1,2,6) between minimum cell rate (MCR) and peak cell rate (PCR) were considered. The performance of the relative rate marking (RRM) switch was evaluated for traffic generator with respect to the allowed cell rate (ACR), switch input rate (SWIR)/switch output rate (SWOR), memory access time (MAT), queue length (𝑄), and cell transfer delay (CTD). The initial value of ACR for sources 𝑆𝑖 was taken as PCR/2 whereas the final ACR value was kept between 200 to 700 cells/sec in incremental steps of 100 for 𝑖=1,2,6 and taking buffer size=1000 cells, higher queue threshold (𝑄𝐻)=200 cells, lower queue threshold (𝑄𝐿) = 100 cells, and assuming that each source has to send a total of 1000 cells. The variations in ACR, SWIR/SWOR, MAT, 𝑄, and CTD as a function of time are given, respectively, in Figures 48.

408395.fig.004
Figure 4: The variation of ACR.
408395.fig.005
Figure 5: The variation of SWIR/SWOR.
408395.fig.006
Figure 6: The variation of MAT.
408395.fig.007
Figure 7: The variation of 𝑄.
408395.fig.008
Figure 8: CTD using BLPos/GTExp traffic generator.

The simulation results for mean, variance, maximum and minimum values of ACR, SWIR, SWOR, MAT, 𝑄 and CTD are given in Tables 3, 4, and 5 respectively.

tab3
Table 3: The values of the ACR (cells/sec).
tab4
Table 4: The values of the SWIR, SWOR, MAT, and 𝑄.
tab5
Table 5: The values of the CTD (sec).

It can be seen from Figure 4 that ACR changes due to the feed back received from the switch. When the switch is heavily loaded (congestion) there is a decrease in the ACR and when it is lightly loaded (no impeding congestion) the ACR is either kept constant or it is increased. The relative changes between the SWIR and SWOR are shown in Figure 5 are responsible for the variations in the MAT (Figure 6) and 𝑄 (Figure 7). When the SWIR becomes greater than SWOR the buffer starts filling and reaches a specified threshold level. The switch then signals the source to start reducing its data rate. Consequently, source ACR reduces, and its effect appears at the queue, which causes a reduction in the rate of increase in the queue size. For a SWIR smaller than a SWOR, the buffer starts becoming empty and when 𝑄 reaches its minimum value, the source is signaled to start increasing its data rate. There is a time lag between the switch experiencing a traffic load variation, effect of switch feedback control, and the occurrence of the new load due to the feedback.

Referring to Figure 8, and Table 5 it is noticed that the CTD between any source and its corresponding destination changes from minimum value of 0.2 sec to a maximum value of 1.04 sec.

Examining the simulation results for ACR, SWIR, SWOR, MAT, 𝑄 and CTD, which indicate the performance of RRM switch under BLPos/GTExp traffic generator, it is seen that the switch offers the best performance.

5. Conclusion

In this paper a mathematical modeling of the BLPos/GTExp source traffic generator has been carried out. The analytical results showed that the minimum values of BLPos/GTExp are greater than their corresponding values of MBLPos/MBLExp and the Poisson mean arrival parameter 𝜇BLExp should not be a very large value, because BLPos will, consequently, be very large as well, and the source will spend most of its time sending only the burst cells with a smaller number of gap intervals for BLPos/GTExp traffic generator resulting in less-bursty traffic. It can be concluded also that the exponential mean arrival parameter 𝜆GTExo should be selected between 2 and 30 cells/sec for simulation of real bursty traffic because it offers higher peak values of GTExp. This is further supported by the observation that for 𝜆GTExo in the range 30 to 100 cells/sec, the peak values of GTExp have the least variation indicating smoothest traffic. The analytical results have been verified through simulation for six sources.

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