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

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.


INTRODUCTION
The Poisson process is an extremely useful process for modelling purposes in many practical applications, such as, e.g. 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.Poison 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 study have been carried out for ABR service in ATM networks using Poisson distribution for modeling the Burst Length ( Pos BL ) and Exponential distribution for modeling the Gap Time ( Exp GT ).The corresponding traffic generator is designated as / Pos Exp BL GT .

BL GT
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][2][3][4][5].An exponential random variable with a mean value of (where 0) is given by the following Probability Density Function (PDF).
-x e x 0 () and the cumulative density function (probability that time between events is x) can be obtained by integrating equation (1): The mean and variance of the Exponential distribution are: The mean and variance are identical for the Poisson distribution [1]: Equation ( 8) can be used to find source random variable . For the required distribution, the inverse can easily be found from equation (8) with ).It can be generated by using the function ( ) rand provided by the standard Linux library or using Mersenne Twister (MT) [6].

For modeling the
Exp GT , equation ( 8) is used with Therefore the required transformation is

Estimation of the Load ( i L ) for the Traffic Generator
The load variation of the traffic can be realized by synthesizing predefined load such that the resulting load Now the value of r P =1/ACR is readily available, depending upon the selected value(s) of ACR that can be separately taken as variable, and thus equation ( 18) can be re-written as where the number of cells inside the burst should be at least one ( 1   Referring to Fig. 3 it can be concluded that the Exponential mean arrival parameter

SIMULATION RESULTS
The ATM network simulation was carried out under Linux network programming.The Parameters specified in Table 1   It can be seen from Figs. 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 Fig. 5 are responsible for the variations in the MAT (Fig. 6) and Q (Fig. 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 the Fig. 8, and Fig. 1: / Pos Exp BL GT Traffic Generator for 1000 Count Values of U.
between 2 and 30 cells/sec for simulation of real bursty traffic because it offers higher peak values of Exp GT .This is further supported by the observation that for Exo GT  in the range 30 to 100 cells/sec, the peak values of Exp GT has the least variation indicating smoothest traffic.

EXP BL GT Traffic Generator
/ Pos EXP BL GT 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 ( Pos BL ) whereas GT exponential distributed ( Exp GT )./ Pos EXP

Estimation of the Minimum Gap Time (
where i L is the traffic load due to i th source.Therefore, the aggregate traffic from N sources will generate the load L on a link with rate R Mbps giving average throughput of L R  Mbps.The load Exp GT M is calculated by the source automatically using Exponential distribution.Using equation (12) the Pos BL .

Table 1 :
The Evaluated Parameters for the / Poisson mean arrival rate ( R 149.76 Mbps

Table 2 :
PosBL 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 / Pos BL (cells) andExpGT( sec  ) for /

Table 5
Examining the simulation results for ACR, SWIR, SWOR, MAT, Q and CTD, which indicate the performance of RRM switch under / is seen that the switch offers the best performance.
it is noticed that the CTD between any source and its corresponding destination changes from minimum value of 0.2 sec to a 14 maximum value of 1.04 sec.

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).