Aggregate TCP traffic presents strong correlations as seen in previous section. However, it can not be modelled using usual LRD traffic models (such as M/G/∞), because of the closed-loop behaviour of the underlying transport protocol TCP that influences packet inter-arrival correlations.
Two approaches are proposed to model aggregate Web traffic. First, we consider another simulation technique that allows simulating TCP connections quickly using dif- ferential analytical modelling (see Chapter 6 for more details). Second, we keep the key idea of superposing Web sources (ON-OFF processes), while trying to reduce the num- ber of superposed ON-OFF processes using equivalent ON-OFF process. The second approach will be developed hereafter.
Equivalent ON-OFF process Consider a simple ON-OFF process with TOF F ≫ TON
(a typical Web application). We need to generate N simple ON-OFF processes simulta- neously (in other words, we superpose N simple Web applications). The goal is to replace
Chapter 5. Traffic Models for Multimedia Applications 97
the N simple ON-OFF processes by M ≪ N equivalent ON-OFF processes to reduce the computational complexity. However, the traffic generated by the M equivalent ON-OFF processes should preserve the same statistical behaviour and the same performances as the traffic generated by the N simple ON-OFF processes.
The idea is to aggregate the long OFF periods in the simple ON-OFF process into shorter OFF periods in the equivalent ON-OFF process. This should be done under the condition: the average resulting throughput is the same in both cases. Indeed, the aggregation factor of OFF periods and the aggregation factor of superposed processes are calculated under this condition. Consider:
• TOF F is the duration of the OFF period in the simple process.
• TOF FEqu is the duration of the OFF period in the equivalent process.
• AOF F is the aggregation factor of OFF periods.
• AAgg is the aggregation factor of ON-OFF processes.
Then we have:
TOF FEqu = TOF F AOF F
(5.29) Notice that the equivalent ON-OFF process have the same ON period characteristcis as the simple ON-OFF process (no aggregation for ON periods). AAgg can be written as:
AAgg =
N
M (5.30)
The average throuput of superposed simple ON-OFF processes and superposed equivalent ON-OFF processes must be equal:
λEqu= AAgg∗ λ (5.31)
We replace λ = λON ∗ PON and λEqu= λEquON ∗ PONEqu. With:
• λON is the throughput during the ON period of the simple ON-OFF process.
• λEquON is the throughput during the ON period of the equivalent ON-OFF process.
• PON = TONT+TONOF F is the occurrence probability of ON period in the simple ON-OFF
process.
• PONEqu = TONT+TONEqu OF F
is the occurrence probability of ON period in the equivalent ON-OFF process.
However, λON = λEquON as we conserve the same ON period while we perform aggregation
on OFF periods. So, we get:
AAgg = λEqu λ = PONEqu PON (5.32)
Chapter 5. Traffic Models for Multimedia Applications 98
AAgg = MN represents the gain we obtain by applying the equivalent ON-OFF process.
We can rewrite the last equation as:
N ∗ PON = M ∗ PONEqu (5.33)
This equation can be interpreted like this: The superposition of simple ON-OFF processes generate the same average number of active periods (ON) as the superposition of equivalent ON-OFF processes. Recall N ∗PON is the average of a binomial distribution
(which is the distribution of ON periods in ON-OFF processes).
The algorithm Practically, we need to determine the following parameters: AAgg,
PON, PONEquand M in function of N and AOF F in order to substitute N simple ON-OFF
processes by M equivalent ON-OFF processes. The algorithm is presented hereafter:
1. Choose the value of AOF F so that TOF FEqu > TON
2. Calculate PON = TONT+TONOF F 3. Calculate PONEqu= TON TON+TOF FEqu 4. Calculate AAgg = P Equ ON PON 5. Calculate M = AN Agg
6. Generate M ON-OFF processes with the same ON period and equivalent OFF period with TOF FEqu duration
Numerical validation We use the previous algorithm to substitute N superposed ON-OFF processes (using the W1 web model defined in Table 5.10) by M ≪ N equiv- alent ON-OFF processes. The numerical values are listed in Table 5.12.
Table 5.12: Aggregated Web Sessions Example N AOF F TOF FEqu AAgg M
1000 100 0.3 sec 17.5 57
We generate the N superposed simple processes as well as the corresponding M equivalent processes. We compare the traffic generated in both cases statistically using the IDI index. Then we compare their performance in a queuing system of deterministic service.
The IDI evolution for both simple and equivalent processes traffic is depicted on Figure 5.22. The statistical evaluation shows a similar behaviour of simple and equiva- lent processes. The IDI of superposed equivalent processes follows very well the IDI of superposed simple processes.
Chapter 5. Traffic Models for Multimedia Applications 99
Figure 5.22: Evolution of IDI: Simple vs Equivalent Processes
The performance of the generated traffic is evaluated in a queuing system of deter- ministic service. Both simple and equivalent traffic performances are compared under high queue load. Results are good and validate the use of the equivalent ON-OFF process (Table 5.13).
Table 5.13: Performance of the Equivalent ON-OFF Process Traffic Bit Rate (Kbps) Average Load (Packets) ρ
Simple 2425 2.4 0.9
Equivalent 2490 1.9 0.9
A good approximation is realized using the equivalent ON-OFF process. The aggre- gation of OFF periods performs well and can be used to reduce computational complexity when large number of identical Web sessions needs to be simulated (in the previous ex- ample we reduced the number of sessions to be simulated from 1000 to 57). However, its performance decreases when very large number of sessions is considered as the number of equivalent processes may still be large. A better solution to the aggregated TCP mod- elling issue will be detailed in Chapter 6 when a new differential simulation technique is presented.
Chapter 5. Traffic Models for Multimedia Applications 100
5.5
Conclusion
In this Chapter, we presented application models for Audio, Video and Data applica- tions. Application models take into consideration the user behaviour and provide reliable packet generation process according to application type. The detailed application models are used to characterize multimedia applications traffic. Hence, the packet inter-arrival process is studied on single and superposed applications. We found that the exponential approximation of packet inter-arrival process is valid for the superposition of audio appli- cations and MPEG videos under light to medium loads of traffic. This approximation is very convenient as it allows analyzing analytically the performance of superposed traffic in queuing networks. However, we find that this approximation is not valid under heavy loads of traffic and we propose other approximations instead. On the other hand, the superposition of data applications present dynamic correlation structure in function of packet losses. This is due to packet retransmission mechanisms implemented by TCP (the underlying transport protocol). We model the superposition of web applications by an equivalent ON-OFF process. The equivalent ON-OFF process reduces the simulation complexity by aggregating OFF periods. Although the proposed model performs well, its performance is limited when very large number of sources is considered as the number of equivalent processes to superpose may still be very large. In Chapter 6 we present a more efficient solution via differential simulation technique of TCP/IP. The proposed technique allows simulating TCP/IP sources by fluid rate propagation, while preserving the transient behaviour of TCP sources.
Chapter 6
TCP/IP Differential Analytical
Modelling
6.1
Introduction
Transmission Control Protocol on IP (TCP/IP) plays an important role in the Internet. Most end-to-end reliable connections on the Internet are established by TCP/IP. In- order delivery of packets, lost packets retransmission and the efficient use of bandwidth are functionalities implemented in TCP/IP, and they are behind its success. However, the numerous functionalities of TCP/IP resulted in a sophisticated algorithm. Hence, from a traffic modelling point of view, the reliability of TCP/IP generates “elastic” traffic because of packet retransmission mechanisms. No simple traffic models could be used to generate TCP/IP traffic unless the TCP/IP loss process is reproduced.
Event-driven technique is widely used to simulate TCP/IP. Unfortunately, the in- crease in the number of generated events makes it unsuitable for large scale network simulations. Many other techniques to simulate the behaviour of TCP/IP analytically are proposed in the literature. Most of them are based on analytical stationary approx- imations of rate and loss process. However, such approaches do not reproduce the tran- sient behaviour of TCP/IP. Our objective is to model TCP/IP analytically to overcome scaling problems while preserving the TCP/IP transient behaviour for more precision. We achieve this using the Differential Traffic Theory [GGB+01].
In this Chapter, we present a differential model for TCP/IP. The model describes precisely the behaviour of TCP/IP by fluid differential equations mixed with control events. Control events pilot the simulation to pass from one differential equation to an- other. Network nodes are represented by D(t)/D/1/N queues, while D(t) means tran- sient deterministic arrival. Losses and delays are evaluated analytically. The Chapter is organized as follows: in section 6.2, we give an overview of TCP/IP with its different operation modes. In section 6.3, we present the differential analytical modelling tech- nique as well as its application to TCP/IP. Finally, in section 6.4, we validate the model by comparing differential simulation results with event-driven simulations in different topologies.
Chapter 6. TCP/IP Differential Analytical Modelling 103