Some new concepts I have learned through this investigation are noted in this post.
Basic concepts
There are two common representations for modeling network traffic [1].
- Counting process: how many packets arrive for a specific period of time
- Inter-arrival time process: how long did time pass between two successive arrivals (named as An)?
In case of compound traffic of which the arrivals may happen in batches (several arrivals happen at the same moment), an additional sequence (which element is the number of arrivals in the batch) is introduced. However, I don't mention it at this instant.
Poisson Model
The traffic arrivals are assumed to have the following properties:
- Independent
- Exponential distribution (Note that exponential distribution is the probability distribution describing the time between events - the moment packets arrive - in a Poisson process)
The Poisson process exhibits an important property:
- "Stateless" or memoryless, in other words, means that the number of arrivals in disjoint intervals is statistically independent.
Traffic Burstiness: the limitations of the Poisson model
The authors referred to the concept of the auto-correlation function of the arrivals {An}. As far as I understand, according to the Poisson process, the arrivals in any renewal process have no relationship with the former ones due to stateless (or memoryless) property, whereas the burstiness possibly gives rise to connections among arrivals in processes => not fit to Poisson model.
Let's keep continuing with a simpler traffic model that overcomes other limitations of the Poisson model before considering the self-similarity one.
Packet train model
The authors in [2] proposed an arrival model introducing dependence between packets traveling between the same end-nodes.
In the study, the author used mathematically statistical tools, namely mean, standard deviation, coefficient of variation on time series to evaluate. They also involved an auto-correlation function (ACF) to show the relationship of an element of a time series tn with a previous one tn-1. Another tool is kth percentile which is a value in a data set that splits the data into two pieces: the lower piece contains k elements and the upper contains the rest of the data (which amounts to 100 -k elements. "90th percentile = a" means that there are 90% elements of the data set which have a value greater than "a".
The model introduces new parameters:
In my opinion, with the introduction of inter-car (and inter-train) time, this model does not directly address the issue of multiple time scale burstiness of network traffics. That inter- time makes the proposed model not randomly enough to fit the random occurrence of burstiness.
Self-similar model
Now, let's go to another model. I mostly read a number of references to have a little understanding of this. Its name exactly associates with an intuitive property of Fractal geometry which is a great tool to imitate nature or "chaos" like network traffic.
The model introduces new parameters:
- inter-train time: a user parameter, depends on the frequency with which applications use the network
- inter-car time: a system parameter, depends on the network hardware and software
In the Poisson model, these are merged into a single parameter: mean inter-arrival time.
In my opinion, with the introduction of inter-car (and inter-train) time, this model does not directly address the issue of multiple time scale burstiness of network traffics. That inter- time makes the proposed model not randomly enough to fit the random occurrence of burstiness.
Self-similar model
Now, let's go to another model. I mostly read a number of references to have a little understanding of this. Its name exactly associates with an intuitive property of Fractal geometry which is a great tool to imitate nature or "chaos" like network traffic.
Many studies proved that the data network has very high variability in terms of time and "space" (actually, I don't get what space means). [3] is such an example.
Because network data usually have a long-range dependence temporal structure, the conventional time series model is not appropriate (this statement is what I know, I have no idea in detail). The self-similar model is a candidate for modeling network data dynamics with such a long correlation.
The authors in [4] also insisted on three implications of the self-similar nature of network traffic for traffic engineering purposes: modeling individual Ethernet source, the inadequacy of conventional notions of burstiness, and effects on congestion management for packet networks.
References
[1] Traffic Modeling for Telecommunications Networks Link
[2] From Poisson Model to Self-Similarity: a Survey of Network Traffic Models Link
[3] High time-resolution measurement and analysis of LAN traffic: Implications for LAN interconnection Link
[3] High time-resolution measurement and analysis of LAN traffic: Implications for LAN interconnection Link
[4] On the Self-Similar Nature of Ethernet Traffic Link
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