# Large Deviation Techniques for Traffic Engineering

This page contains some references, both theoretical and experimental using real traffic, on the application and evaluation of large deviation techniques, and in particular the *many sources asymptotic* and its related *effective bandwidth* definition, for traffic engineering in broadband networks.

It also describes and makes available the programs (executables) used for obtaining the results published in [CSS99, CSS98]. A description of the methods and usage of the tools is available in [CS00b,CS99].

Also available is an interface providing Web-based access to the tools.[CS00]

E-mail suggestions to vsiris "at" ics "dot" forth "dot" gr

## Contents:

- Motivation
- The theory
- and its application
- Some FAQs for the s, t parameters
- Software (executables)
- References

## Motivation

Analysis of traffic measurements has shown that a large variety of traffic types exhibit a self-similar or fractal behavior. Such behavior can have implications on the performance of a link that mulitplexes such traffic. Hence, there is a need to understand the impact of the various time scales of traffic burstiness on the performance of a link and on its resource sharing capabilities. Moreover, problems in admission control, network resource dimensioning, and pricing can be simplified with the use of an accurate and unifying definition of resource usage.

Some specific questions in traffic engineering that we seek to answer are the following:

- How is the loss probability (or probability of traffic exceeding some maximum delay) affected by the traffic mix (percentage and characteristics of the multiplexed traffic types) and the link resources (capacity and buffer)?
- For a link with some amount of resources (capacity and buffer), what combinations of traffic sources can be accepted, while still ensuring the target quality of service?
- What is the effect of traffic control mechanisms, such as traffic shaping and priority scheduling, on a link's multiplexing capability and the amount of resources used by a bursty source?
- Measurement platforms enable the collection of detailed packet level traces on high speed links. Such traces can lead to a huge amount of data. Can one collect a smaller amount of traffic statistics, such as the amount of traffic in time periods of equal length, that nevertheless contains all the necessary information for performance analysis? What is the sufficient time granularity (length of the measurement period) of such measurements?

Traditional queueing theory based on elaborate traffic models cannot answer such questions in the context of large multi-service networks carrying traffic of different type. Our objective is to address the above questions and other problems in traffic engineering by applying, to real broadband traffic and a real networking environment, some powerful results from large deviations theory.

## The theory

This work is based on the following definition of effective bandwidths of a source of type ` j` given by Kelly [Kel96]:

where ` X_{j}`[

`0,`] is the amount of workload produced by a source of type

*t*`in a time interval of length`

*j*`. The above definition summarizes the statistical characteristics of a source over different time and space scales.`

*t*When used for the analysis of a link (multiplexer) that guarantees some level of performance or *quality of service* (QoS), the *space* and *time* parameters ` s`,

`characterize the`

*t**context*of the source, which includes the link resources (capacity and buffer), scheduling discipline, QoS, and traffic mix (percentage and characteristics of the different source types).

In this case, parameters ` s`,

`can be computed using the`

*t**many sources asymptotic*[CW96,BD95]:

where ` Q(Nc,Nb,Nn)` is probability of overflow in a link with capacity

`and buffer`

*Nc*`which multiplexes`

*Nb*`sources, with`

*Nn=N*(*n*)_{1}, ... n_{J}`the percentage of sources of type`

*n*_{j}`. We will refer to the last equation as the`

*j*`formula.`

**supinf**At this point you're probably asking why there is a need for an effective bandwidth definition which takes into account the context of the source. Here's an answer.

## and its application

Particular values of `( s,t)` can be taken to characterize the

*operating point*of a link. Here's why.

We have implemented the numerical solution of the ` supinf` when the source trace is in the form of aggregate statistics (number of bytes/cells) in fixed time intervals (epochs). A sample trace file looks like this

`# epoch_in_msecs = 40
65
4
5
8
...
`

In the above trace an epoch has duration `40 msec`. In the first epoch ( `40 msec `) the source produces 65 cells, in the second ( `40-80 msec `) 4 cells, etc.

To numerical solve the ` supinf` formula we replace the expectation in the effective bandwidth with the empirical estimate. Hence the effective bandwidth is estimated from traffic traces using the following equation:

where

`is the total duration of the trace (for simplicity we have assumed that it is an integer number of the time parameter`

*T*`).`

*t*`is the amount of work (e.g., number of cells) produced in the interval`

*X*[(_{j}*i-1*)*t,it*]`((`.

*i-1*)*t,it*]
The ` supinf` formula involves two optimization procedures: the first consists of, for fixed

`, a minimization with respect to`

*t*`, and the second a maximization with respect to`

*s*`.`

*t*The minimization with respect to

`(`

*s***inf**) can be solved numerically in an efficient manner by taking into account the convexity of the product

_{s}`times the effective bandwidth, in`

*s*`. In particular, for a specific value of`

*s*`, the`

*t*`is found using a`

**inf**_{s}*golden section search*starting from an initial uncertainty interval which contains the minimum. The search procedure decreases the uncertainty interval in successive steps, and stops when the width of the uncertainty interval is less than some small number.

The maximization with respect to

`(`

*t*`) is found by linear searching a range of values for`

**sup**_{t}`, in steps equal to the epoch of the trace file (there is no general property that we can take advantage of to use a faster approach, as we did for the`

*t*`). The range of`

**inf**_{s}`that is searched is determined heuristically and depends, among others, on the buffer size: the extremizing value of`

*t*`is larger for larger buffer sizes.`

*t*I have made available some typical values of ` s,t` for MPEG-1 video, Internet WAN traffic, and a traffic mix of MPEG-1 video and voice traffic.
There is also a FAQs for the

`parameters`

*s,t*## Software (executables)

The executables for the `msa` tool, along with other traffic engineering related executables, are available here.

There exists a man page, and a collection of frequently encountered problems.

We have started developing a WEB-based interface for our traffic analysis tools. A prelimenary version of the interface is available here.

A sample trace file (MPEG-1 compressed video of Star Wars) can be found here. The trace contains 40000 epochs, with each epoch having duration `40 msec`. Note that the trace file MUST have the extension .tr

If you choose to download and use the software, I'ld appreciate if you send me an email at vsiris@ics.forth.gr, possibly with some of your results !

## References

**Theoretical backround**

**[Kel96]**F.P. Kelly. "Notes on effective bandwidths". In Stochastic Networks: Theory and Applications (Editors F.P. Kelly, S. Zachary and I.B. Ziedins) Oxford University Press, 1996. 141-168.**[CW96]**C. Courcoubetis, R. Weber. "Buffer Overflow Asymptotics for a Buffer Handling many Traffic Sources".*Journal of Applied Probability*, vol. 33, 1996. [.ps.gz]**[CKW97]**C. Courcoubetis, F.P. Kelly, R. Weber. "Measurement-based usage charges in communication networks". Statistical Laboratory Research Report 1997-19, University of Cambridge. To appear in Operations Research.**[BD95]**D.D. Botvich and N.G. Duffield. "Large deviations, the shape of the loss curve, and economies of scale in large multiplexers".*Queueing Systems*, 20:293-320, 1995.**[LM98]**L. Likhanov and R.R. Mazumdar. "Cell loss asymptotics for buffers fed with a large number of independant stationary processes". In*Proc. of IEEE INFOCOM'98*.**[Wis99]**D. Wischik. "The output of a switch, or, effective bandwidths for networks". To appear in Queueing Systems.

## Application and experimental results

**[CS00b]**C. Courcoubetis and V.A. Siris. "Procedures and tools for analysis of network traffic measurements". Submitted, September 2000.**[CS00]**C. Courcoubetis and V.A. Siris. "A web-based tool for advanced statistical analysis of network traffic measurements". July 2000. [pdf, .ps.gz]**[CS99]**C. Courcoubetis and V.A. Siris. "Measurement and analysis of real network traffic". In Proc. of 7th Hellenic Conference on Informatics (HCI'99), Ioannina, Greece, August 1999. Available as ICS-FORTH TR-252 [pdf, .ps.gz]**[CSS99]**C. Courcoubetis, V.A. Siris, and G.D. Stamoulis. "Application of the Many Sources Asymptotic and Effective Bandwidths to Traffic Engineering".*Telecommunication Systems*, 12(2-3):167--191, 1999. [Abstract] Preprint: [pdf, .ps.gz, html]**[CDS99]**C. Courcoubetis, A. Dimakis, and G.D. Stamoulis. "Traffic Equivalence and Substitution in a Multiplexer". In*Proc. of Infocom'99*, New York, USA, March 1999. [.ps.gz]**[CSS98]**C. Courcoubetis, V.A. Siris, and G.D. Stamoulis. "Application and Evaluation of Large Deviation Techniques for Traffic Engineering in Broadband Networks". In*Proc. of ACM SIGMETRICS '98/ PERFORMANCE '98*, Madison, Wisconsin, June 1998. Superseded by [CSS99] above.**[Gib96]**R. Gibbens. "Traffic characterisation and effective bandwidths for broadband network traces". Statistical Laboratory Research Report 1996-9, University of Cambridge.

## Related

**[CLW94]**G.L. Choudhury, D.M. Lucantoni, and W. Whitt. "On the effectiveness of effective bandwidths for admission control in ATM networks". In*Proc. of ITC-14*, 1994.**[Kni97]**E. Knightly. "Second moment resource allocation in multi-service networks". In*Proc. of ACM Sigmetrics'97*, 1997.

For more info: **Vasilios Siris** Tel: +30 2810 391726 Email: vsiris "at" ics "dot" forth "dot" gr