Section 18.6. Self-Similarity and Non-Markovian Streaming Analysis

18.6. Self-Similarity and Non-Markovian Streaming Analysis

Multimedia applications are delay sensitive and loss tolerant, unlike static-content communications, which are delay and loss intolerant. Distributed multimedia networks must be able to support the exchange of multiple types of information, such as voice, video, and data among users while also satisfying the performance requirements of each individual application. Consequently, the expanding diversity of high-bandwidth communication applications calls for a unified, flexible, and efficient network to prevent any congestion.

A network handling heavy video streaming must reserve some resources based on the source QoS. With reservation techniques, lossless transmission is guaranteed for the entire duration of the block; otherwise , the block is lost. However, sudden changes in the total volume of traffic at a node can impact the performance of streaming transmission. Multimedia networks are expected to support a large number of bursty sources with different characteristics. This fact enforces the use of processes other than Poisson for describing network traffic. The aggregated arrivals of packets are assumed to form stream batches of packets.

18.6.1. Self-Similarity with Batch Arrival Models

Section 11.5.3 explained that some traffic patterns indicate significant "burstiness," or variations on a wide range of timescales. Bursty traffic, such as a video stream, can be viewedasa batch of traffic units and described statistically using self-similarity patterns . In a self-similar process, that packet loss and delay behavior are different from those in traditional network models using Poisson models. Self-similar traffic can be constructed by multiplexing a large number of ON/OFF sources that have ON and OFF intervals. This mechanism corresponds to a network of streaming servers, each of which is either silent or transferring video stream at a constant rate. Using this traffic, the distributions of transmission times and quiet times for any particular session are heavy tailed , which is an essential characteristic of traffic self-similarity.

The discrete-time representation of a communication system is the natural way to capture its behavior. In most communication systems, the input process to a queue is not renewal but correlated. A renewal process is a process in which the interval between consecutive occurrences of a certain event are independently and identically distributed. For example, the Poisson process is a renewal case with an exponential distribution. In a practical environment, the input process of packetized voice, data, or video traffic to a multiplexer does not form a renewal process but is bursty and correlated.

In streaming-traffic analysis, a batch of packets may arrive simultaneously . With the relatively more accurate traffic model being presented here, maximum and average traffic rates over a number of given intervals are determined. Using maximum rate and average rate parameters in our analysis can be efficient in capturing the burstiness characteristics of streaming sources. This method is especially effective for realistic sources, such as a compressed streaming video. In such situations, a source can even transmit at its peak rates when sending its large- size frames immediately followed by smaller frames . In the performance analysis, the packet-loss probability and the impact of increase in switching speed to link-speed ratio on the throughput are of particular interest.

In this analysis, consider a small buffered multiplexer or router, as large queuing delays are not expected in a multimedia network with a real-time transmission. A bursty arrival is modeled as a batch , or packets with identical interarrivals. This model captures the multirate burstiness characteristic of realistic sources. One property of self-similarity s that an object as an image is preserved with respect to scaling in space or time. In this environment, the traffic-relational structure remains unchanged at varying timescales. For any time t> 0 and a real number a> 0, a self-similar process, X( t ), is a continuous-time stochastic (random) process with parameter 0.5 <H< 1 if it satisfies

Equation 18.3

where parameter H is known as the Hurst parameter , or self-similarity parameter . The Hurst parameter is an important factor in bursty traffic, representing a measure of the dependence length in a burst. The closer H is to its maximum, 1, the greater the persistence of long-range dependence.

The expected values of both sides in Equation (18.3) must then be related as

Equation 18.4

This result indicates that a self-similar process when H = 0.5 can also be obtained from the Brownian random process discussed in Section C.4.2. Based on Equation (C.34), and for any time increment ƒ , the increment of process, X(t + ƒ) - X(t), has the following distribution:

Equation 18.5

To better understand the behavior of aggregated batch sequences of traffic, we can also present the self-similar process, X( t ), in terms of a discrete-time version, X _n,m, defined at discrete points in time. In this process, n ˆˆ{1, 2, ...} is discrete time and m is the batch size. Thus:

Equation 18.6

Note that for X _n,m , the corresponding aggregated sequence with a level of aggregation m , we divide the original series X( t ) into nonoverlapping blocks of size m and average them over each block where index n labels blocks.

Example.

Consider traffic with batch size m = 4. Self-similar process averaging is expressed by

Equation 18.7

Heavy-Tailed Distributions

Self-similarity implies that traffic has similar statistical properties in a timescale range, such as milliseconds , seconds, minutes, hours, or even days. In a practical situation, in which bursts of streams are multiplexed, the resulting traffic tend to produce a bursty aggregate stream. In other words, the traffic has a long-range dependence characterized by a heavy-tailed distribution . A random variable has heavy-tailed distribution if for 0 <±< 2, its cumulative distribution function (CDF)

Equation 18.8

as x

Heavy-tailed distributions are typically used to describe the distributions of burst lengths. A simple example of heavy-tailed distributions i the Pareto distribution , which is characterized by the following CDF and probability density function (PDF):

Equation 18.9

where k is the smallest possible value of the random variable. For this distribution, if a 1, the distribution has infinite mean and variance; if a 2, the distribution has infinite variance. To define heavy tailed , we can now use a comparison over PDFs of a Pareto distribution and exponential distribution. Making this comparison shows how the tail of the curve in a Pareto distribution takes much longer to decay. A random variable that follows a heavy-tailed distribution may be very large with a probability that cannot be negligible.