Abstract
This article addresses the use of Markovian models, based on discrete time MMPPs (dMMPPs), for modeling IP traffic. In order to describe the packet arrival process, we will present three traffic models that were designed to capture self-similar behavior over multiple time scales. The first model is based on a parameter fitting procedure that matches both the autocovariance and marginal distribution of the counting process (Salvador 2003). The dMMPP is constructed as a superposition of two-state dMMPPs (2-dMMPPs), designed to match the autocovariance function, and one designed to match the marginal distribution. The second model is a superposition of MMPPs, each one describing a different time scale (Nogueira 2003a). The third model is obtained as the equivalent to a hierarchical construction process that, starting at the coarsest time scale, successively decomposes MMPP states into new MMPPs to incorporate the characteristics offered by finer time scales (Nogueira 2003b). These two models are constructed by fitting the distribution of packet counts in a given number of time scales.
Key Terms in this Chapter
Batch Markovian Arrival Process: The BMAP further extends the MAP by additionally associating rewards (i.e., batch sizes of arrivals) to the corresponding arrival times.
Poisson Process: A poisson process is an integervalued continuous-time stochastic process whose increments are independent random variables following the poisson distribution.
Markov Modulated Poisson Process (MMPP): A process, belonging to the class of markov renewal processes, where arrivals occur according to a statedependent poisson process with different rates governed by a continuous-time markov chain.
Markovian Arrival Process (MAP): The MAP extends the MMPP model by allowing arrivals not only during holding times in states, but also at state transitions.
Long-Range Dependence (LRD): A process is said to be long-range dependent if its autocovariance function decays hyperbolically (slower than exponentially) and the area under it is infinite.
Self-Similar: When small parts of an object are qualitatively the same or similar to the whole object. In the context of tele-traffic theory, a self-similar traffic process that presents the same statistical characteristics independently of the considered time scale.
Hurst Parameter: It is an index that quantifies the self-similarity degree. In fact, similarity characteristics of self-similar processes with stationary increments depend only on this parameter.