1. Performance of Network of Queues with Traffic Modeled by Heavy-tailed Distributions
Weldisson Ferreira Ruas
José Marcos C. Brito
Inatel – National Institute of Telecommunications
Good afternoon, my name is Weldisson, I will present the work with the title: Performance of Network of Queues with Traffic Modeled by Heavy-tailed Distributions. oriented by Jose Marcos C. Brito.

2. Introduction
Traffic in telecommunications networks has evolved from voice traffic to multimedia traffic, including voice, data and video.
In this new scenario, the traditional Markov models are not suitable to characterize the traffic in the network.

3. Introduction
Several works demonstrates that traffic in some telecommunications networks is statistically self-similar
New models to characterize traffic can be classified in three categories:
a) Based on measurements.
b) Based on fractal models.
c) Based on generic models.

4. Generic Models Less complex than fractal models
Arrival processes is modeled by a heavy-tailed distribution
Pareto, Lognornal or Weibull distributions
Service time can be modeled by:
Exponential distribution (G/M/1 queue)
Heavy-tailed distribution (G/G/1 queue)
Considered constant (G/D/1 queue).

5. Introduction
Several works have analyzed the performance of isolated single server queues with the traffic modeled by a heavy-tailed distribution, but there is a lack of analysis for networks of queues in this scenario.

6. Goal of this paper
Evaluate, based on simulations, the performance of networks of queues with the traffic modeled by Pareto, Lognormal and Weibull distributions.
Software ARENA for simulations
Two scenarios have been considered:
a) Scenario I: an open network of queues without add/drop traffic.
b) Scenario II: an open network of queues with add/drop traffic after each queue.

7. Performance Parameters
Mean waiting time of each queue, as a function of the position of the queue
Total network delay
For both parameters, we present the results as a function of the utilization factor in each queue
To vary the utilization factor we vary the service time of the server

8. Scenario I
The shape parameters of the heavy-tailed distributions used in packet generators are: Pareto,  = 1.3; Lognormal,  = and  = 2; Weibull,  = and  = 1.

9. Scenario I – Pareto/M/1
We can see that as we walk away from the traffic generator, the queue tends do behave like an M/M/1 queue.

10. Scenario I – Pareto/Pareto/1
Comparing with previous figure, we can see that, in this case, the performance tends to M/M/1 system in a very slow way.

11. Scenario I – Lognormal/M/1
Similar conclusions have been obtained for Lognormal and Weibull distributions. Here are the results for Lognormal distribution.

12. Scenario I – Lognormal/Lognormal/1
As said, the results are similar Lognormal/Lognormal/1

13. Scenario I – Total Network Delay
We can see that, for G/G/1 model, the Lognormal distribution results is closer to the M/M/1 model than the other distributions. Considering G/M/1 model, the performances for all distributions are similar.

14. Scenario II
An open network of queues with add/drop traffic after each queue

15. Pareto/M/1 - 50% add-drop after each queue
Figure 7

16. Pareto/M/1 - 5% add-drop after each queue
Figure 8
We can see that the model Pareto/M/1 has performance closer to the M/M/1 model when the percentage of add/drop is smaller.

17. Pareto/Pareto/1 – 50% add-drop after each queue
Figure 9

18. Pareto/Pareto/1 – 5% add-drop after each queue
Figure 10
Again, we can see that the performance is closer to the M/M/1 model when the percentage of add/drop is smaller. Similar conclusions are obtained for Lognormal and Weibull.

19. Conclusions
In this paper we analyzed the performance of networks of queues under traffic modeled by heavy-tailed distributions.
We consider open networks with and without add/drop traffic after each queue.
The mean waiting time in each queue tends to the performance of a M/M/1 system as we move away from the first traffic source, with the velocity of the trend depending of the type of the queue (G/M/1 or G/G/1), of the type of the distribution and of the percentage of the add/drop traffic.

20. Thank You