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

Abstract

This paper describes modeling of the available bit rate (ABR) source traffic in asynchronous transfer mode (ATM) network using BL_Pos/GT_EXP traffic generator, which employs Poisson distribution for modeling the burst length (BL_Pos) and exponential distribution for modeling the gap time (GT_EXP). This traffic generator inherits the advantages of both Poisson and exponential distribution functions to achieve enhanced link performance. Analytical and simulation results for BL_Pos/GT_EXP 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 (BL_Pos) and exponential distribution for modeling the gap time (GT_Exp). The corresponding traffic generator is designated as BL_Pos/GT_Exp.

2. BL_Pos/GT_EXP Traffic Generator

BL_Pos/GT_EXP 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 (BL_Pos) whereas GT is exponentially distributed (GT_Exp). BL_Pos/GT_EXP 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 [1–5]. An exponential random variable with a mean value of 𝜆_Exp (where  𝜆_Exp>0) is given by the following probability density function (PDF): 𝑓(𝑥)_Exp=⎧⎪⎨⎪⎩𝜆^Exp⋅0e^{−𝜆Exp⋅𝑥}𝑥𝑥≥<00⎫⎪⎬⎪⎭,(1) and the cumulative density function (probability that time between events is x) can be obtained by integrating (1): 𝐹(𝑥)_Exp=⎧⎪⎨⎪⎩1−e^{−𝜆Exp⋅𝑥}0𝑥𝑥≥<00⎫⎪⎬⎪⎭.(2) The mean and variance of the exponential distribution are 𝐸(𝑥)_Exp=^∞0𝑥⋅𝜆_Exp⋅𝑒^{−𝜆Exp⋅𝑥}𝑑𝑥=1𝜆_ExpVar(𝑥)_Exp,=^∞0𝑥²⋅𝜆_Exp⋅𝑒^{−𝜆Exp⋅𝑥}𝑑𝑥−1𝜆_Exp²=1𝜆²_Exp.(3) A Poisson random variable 𝑥 with parameter 𝜇_Pos  (where  𝜇_Pos>0) has PDF given by [1, 2] 𝑓(𝑥)_Pos=𝜇^𝑥Pos𝑥!𝑒^−𝜇Pos,𝑥=0,1,2,….(4) The mean and variance are identical for the Poisson distribution [1]: 𝐸(𝑋)_Pos=Var(𝑋)_Pos=𝜇_Pos,(5) where a Poisson distribution can be developed by involving the definition of a Poisson Process for a fixed time (𝑡), so that 𝜇_Pos=𝜆_Exp⋅𝑡. For Poisson distributed events, the time between successive events is exponentially distributed with mean 1/𝜇_Pos. 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 𝜆_Exp. 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)𝑑𝑥||=||𝑝(𝑢)𝑑𝑢||or𝑓(𝑥)=𝑝(𝑢)|||𝑑𝑢𝑑𝑥|||,(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 0≤𝑢≤1. It can be generated by using the function 𝑟𝑎𝑛𝑑() provided by the standard Linux library or using Mersenne Twister (MT) [6].

For modeling the GT_Exp, (7) is used with 𝑢=𝐹(𝑥)=1−𝑒^{−𝜆Exp⋅𝑥}or1−𝑢=𝑒^{−𝜆Exp⋅𝑥}.(8) Therefore, the required transformation is 𝑥=−1𝜆_Exp⋅ln(1−𝑢).(9) By changing the variables 𝑥 and (1−𝑢) of (9) with 𝑋_Exp and 𝑈 respectively, we get 𝑋_Exp=−1𝜆_Exp⋅ln(𝑈).(10)

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

The load variation of the traffic can be realized by synthesizing predefined load such that the resulting load 𝐿=∑^𝑁𝑖=1𝐿_𝑖, where 𝐿_𝑖 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 𝑅⋅𝐿 Mbps. The load 𝐿_𝑖 generated by an individual source can be expressed as 𝐿_𝑖=BL_Pos⋅𝐾BL_Pos⋅𝐾+𝑃_𝑟+GL_Exp,(11) where BL_Pos, GL_Exp, K, and 𝑃_𝑟 are the mean BL_Pos, mean gap length, cell size, and minimum intercell gap length (Preamble), respectively in bytes, and then the load 𝐿_𝑖 can be found from (11).

2.2. Estimation of the Minimum Gap Time (𝑀_GTExp)

The minimum GT_Exp(𝑀_GTExp) is a secondary parameter dependent on load. Given a desired load, 𝑀_GTExp is calculated by the source automatically using exponential distribution. Using (11) the GT_Exp can be expressed as GL_Exp=BL_Pos⋅𝐾⋅1−𝐿^𝑖𝐿_𝑖−𝑃_𝑟.(12) The BL_Pos can be written as BL_Pos=𝐸(𝑥)_BLPos=𝑀_BLPos,(13) where 𝑀_BLPos is the minimum BL_Pos.

GL_Exp can be written as GL_Exp=𝐸(𝑥)_GLExp=𝑀_GLExp.(14) Substituting the values of BL_Pos and GL_Exp from (13) and (14), respectively, in (12) 𝑀_GLExp can be written as 𝑀_GLExp=𝑀_BLPos⋅𝐾𝐿_𝑖−𝐾+𝑃_𝑟.(15) Considering the link rate and using the following relation: ByteTime=ByteSize(bits)LinkRate(bits/sec)orByteTime=𝑅𝑏,(16) the 𝑀_GTExp now can be computed as 𝑀_GTExp=𝑏𝑅⋅𝑀^BLPos⋅𝐾𝐿_𝑖−𝐾+𝑃_𝑟.(17) Now the value of 𝑃_𝑟=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 𝑀_GTExp=𝐾⋅𝑏𝑅𝑀^BLPos𝐿1_𝑖−1.(18) Therefore, (18) can be used for computing the value of 𝑀_GTExp that would result in link load closer to 𝐿_𝑖 using the selected values of 𝐿_𝑖 of the 𝑖th source, 𝐾, and Poisson distribution parameters. Considering the same parameter values for burst and gap lengths and 𝑀_BLPos=1, (18) can be simplified as 𝑀_GTExp=𝐾⋅𝑏𝑅𝑀^BLPos𝐿1_𝑖−1.(19)

2.3. Generating the GT_EXP

Bearing in mind that 𝜆_Exp=ACR, since in ABR service a source sends its data with a rate equal to ACR, defining 𝜆_GTExp as the exponential mean arrival rate for GT_Exp, and considering (14) in terms of time, we get (20) 𝐸(𝑥)_GTExp=1𝜆_GTExp=𝑀_GTExp.(20) Now the BL_Pos/GT_EXP traffic generator can compute the GT_Exp using the relation GT_Exp=−𝑀_GTExp⋅ln(𝑈).(21)

2.4. BL_Pos Generating

BL_Pos can be modeled as follows: for ABR service in ATM networks the value of ACR should be within the range MCR≤ACR≤PCR. 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 BL_Pos/GT_Exp 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 𝜇_Pos=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 BL_Pos that is used by the BL_Pos/GT_Exp traffic generator: BL_Pos=^𝑥=𝑢_𝑥=0𝑒^−𝜇Pos⋅𝜇^−xPos𝑥!,𝜇^Pos=ACR,(22) where the number of cells inside the burst should be at least one (𝑀_BLPos=1).

3. Analytical Results

Consider the values of 𝐿_𝑖, 𝑅, 𝐾, 𝑀_BLPos, Poisson mean arrival rate (𝜇_BLPos), and 𝜆_GTExp for BL_Pos/GT_Exp traffic generator as given in Table 1. The value of 𝑀_GTExpwas determined using (19) (Table 1).

Table 1 

The evaluated parameters for the BL_Pos/GT_Exp traffic generator.

The analytical result of BL_Pos/GT_Exp 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.

Table 2 

BL_Pos (cells) and GT_Exp (𝜇sec) for BL_Pos/GT_Exp traffic generators.

Figure 1 

BLPos/GTExp traffic generator for 1000 count values of U.

The variations in BL_Pos, GT_Exp as functions of 𝜇_BLPos and 𝜆_GTExp for BL_Pos/GT_Exp traffic generator for 100 count values of U are shown in Figures 2 and 3 respectively. The increment steps for 𝜆_BLPos(1–110) cells/sec and 𝜇_GTExp(1–110) cells/sec are 10 for each. Referring to Table 2 it is seen that the minimum values of BL_Pos/GT_Exp are greater than their corresponding values of 𝑀_BLPos/𝑀_BLExp. Referring to Figure 2 it can be concluded that the Poisson mean arrival parameter 𝜇_BLEXP should not be a very large value, because BL_Pos 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 BL_Pos/GT_Exp traffic generator resulting in less-bursty traffic. Referring to Figure 3 it can be concluded that the exponential mean arrival parameter 𝜆_GTExo should be selected between 2 and 30 cells/sec for simulation of real bursty traffic because it offers higher peak values of GT_Exp. This is further supported by the observation that for 𝜆_GTExo in the range 30 to 100 cells/sec, the peak values of GT_Exp have the least variation indicating smoothest traffic.

Figure 2 

BLPos versus 𝜇BLPos and U.

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 𝑆_𝑖(𝑖=1,2,…6) sending their data at the rate ACR_𝑖(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 𝑆_𝑖 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 (𝑄_𝐻)=200 cells, lower queue threshold (𝑄_𝐿) = 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 4–8.

Figure 4 

The variation of ACR.

Figure 5 

The variation of SWIR/SWOR.

Figure 6 

The variation of MAT.

Figure 7 

The variation of 𝑄.

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.

Table 3 

The values of the ACR (cells/sec).

Table 4 

The values of the SWIR, SWOR, MAT, and 𝑄.

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 BL_Pos/GT_Exp traffic generator, it is seen that the switch offers the best performance.

5. Conclusion

In this paper a mathematical modeling of the BL_Pos/GT_Exp source traffic generator has been carried out. The analytical results showed that the minimum values of BL_Pos/GT_Exp are greater than their corresponding values of M_BLPos/M_BLExp and the Poisson mean arrival parameter 𝜇_BLExp should not be a very large value, because BL_Pos 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 BL_Pos/GT_Exp traffic generator resulting in less-bursty traffic. It can be concluded also that the exponential mean arrival parameter 𝜆_GTExo should be selected between 2 and 30 cells/sec for simulation of real bursty traffic because it offers higher peak values of GT_Exp. This is further supported by the observation that for 𝜆_GTExo in the range 30 to 100 cells/sec, the peak values of GT_Exp have the least variation indicating smoothest traffic. The analytical results have been verified through simulation for six sources.