Network Traffic Characterisation

Transcription

1 Modeling Modeling Theory

2 Outline 1 2 The Problem Assumptions 3 Standard Car Model The Packet Train Model The Self - Similar Model 4 Random Variables and Stochastic Processes The Poisson and Exponential Distributions Fractals References

3 Traffic Flow - Importance Modelling traffic flow can be seen to be important for a variety of reasons Understanding as an aid in understanding fundamental characteristics of networks Simulation as input to simulations employed to assist in: evaluating protocol behaviour predicting how a network will behave under general conditions predicting how a network will behave under extreme conditions...

4 Problem and Features The Problem Assumptions Problem: How to model traffic flow for a given network at a router. Features: packet/cell arrivals packet/cell buffering packet/cell departures quality of service - priorities multiplexing behaviour determinate, indeterminate, bursty time...

5 Assumptions The Problem Assumptions We will consider the case of packet/cell arrivals Assumptions: packets/cells will be of same length packet arrival times will be independent of each other ignore packet/cell buffering ignore packet/cell departures ignore quality of service - priorities ignore multiplexing issues assume behaviour indeterminate and so probabalistic in nature, so arrival times will be random non bursty

6 Question The Problem Assumptions What if we assume that arrival times are deterministic?

7 Characteristics The Problem Assumptions Traffic flow is a non-deterministic concept. By this we mean that we cannot guarantee that we can observe traffic at a particular point in space and time. However we can talk about the likelihood, the probability, of traffic being observed. For events occuring randomly in space and time we (traditionally) first explore the suitability of the Poisson distribution.

8 Arrivals The Problem Assumptions Network traffic can be modelled as a sequence of arrivals of packets or cells leading to counting processes as a sequence of arrivals of packets or cells leading to interarrival time processes.

9 Car Model Standard Car Model The Packet Train Model The Self - Similar Model The standard Car Model employs the Poisson distribution. This is motivated by it s use in the telephony model (A. K. Erlang). The Poisson distribution is often used for describing events: that can occur in randomly in both time and space as single independent objects Here the event being considered is the arrival of single unpredictable, independent packets at a router Poisson models are examples of counting processes. In our model the count involved will relate to the number of packets that arrive in a given time interval

10 Standard Car Model The Packet Train Model The Self - Similar Model Testing Results against the Car (Poisson) Distribution The χ 2 Test Comparing Histograms for the interarrival times against a decreasing exponential distribution

11 Findings Standard Car Model The Packet Train Model The Self - Similar Model The car model fails to predict natwork traffic behaviour under bursty conditions can be improved using compound Poisson distributions for the batch size and batch arrival times the packet train model the self-similar model We consider the packet train model and the self similar model.

12 Packet Train Model Standard Car Model The Packet Train Model The Self - Similar Model Assumptions Groups of packets travel together packet sizes are limited to available buffer size and backward compatibility file sizes are getting larger, resulting in greater numbers of smaller packets, hence trains of packets between a sender and receiver arrive at routers routing decisions can be made at the head of the train with no routing overhead for the rest of the train

13 Packet Train Model Standard Car Model The Packet Train Model The Self - Similar Model Model trains are grouped together by comparing interarrival times for cars. These are less than a selected maximum value M adjacent cars with interarrival times greater than the maximum allowed time belong to different trains interarrival times between cars must be smaller than interarrival times between trains the direction of communication sender A to receiver B and sender B to receiver A are immaterial

14 Findings Standard Car Model The Packet Train Model The Self - Similar Model The Train model addresses bursty shortcomings of Poisson model (see last comment) found to be particularly applicable in some (for example) star topology networks not widely employed for LAN or WAN networks does not address multiple time scale burstiness of internet traffic data We consider the self similar model.

15 Self - Similar Model Standard Car Model The Packet Train Model The Self - Similar Model The self - similar model Poisson model applicable for data with limited variability in time and space and no or limited negligible burstiness data networks have been found to exhibit high/extreme variability large variability in space and time tends to cause associated traffic to exhibit fractal behaviour, by which we mean that some statistical proerties (relating to burstiness) repeat themselves at various time scales

16 Random Variable Random Variables and Stochastic Processes The Poisson and Exponential Distributions Fractals References Definition Random variables are functions X t :Ω R with an associated probability distribution defined on each outcome or alternatively element in Ω. Id the number of possible values that X can take is finite or countably infinite, then the variable is said to be a discrete random variable. Otherwise we describe the random variable as a continuous random varaible. Definition A counting Process is a discrete random variable that takes values in N

17 Stochastic Processes Random Variables and Stochastic Processes The Poisson and Exponential Distributions Fractals References Definition A Stochastic Process is a family {X t } t R + of Random Variables X t that vary with respect to time. So we have a collection of rendom variable that for our purposes describe traffic flow for a given region of a network.

18 Poisson Distribution Random Variables and Stochastic Processes The Poisson and Exponential Distributions Fractals References Definition A random variable X is said to have the Poisson distribution with mean λ 0if r N P(X = r) = λr r! e r

19 The Exponential Distribution Random Variables and Stochastic Processes The Poisson and Exponential Distributions Fractals References Definition A random variable X is said to be exponentially distributed with parameter λ>0 if its pdf (probability density function) is given by x R + {0}, f (x) =λe λx and zero otherwise

20 Fractal Random Variables and Stochastic Processes The Poisson and Exponential Distributions Fractals References Definition Fractals are sets, which, when magnified over and over again,always resmble the original image. For pictures of Julia sets nd Mandelbrots, the dark regions represent non chaotic points, the coloured regions divergence to infinity and the interface between - chaos.

21 References Random Variables and Stochastic Processes The Poisson and Exponential Distributions Fractals References [1] J. Banks et al, Discrete-Event System Simulation, Prentice Hall, [2] Michela Bechi, From Poisson Processes to Self-Similarity: A Survey of Network Traffic Models, http : // jain/cse567 06/ftp/traffic m odels1/index.html [3] Robert L. Devaney, A First Course in Chaotic Dynamical Systems, Perseus Books, 1992.