Limits of on/off hierarchical product models for data transmission
A hierarchial product model seeks to model network traffic as a product of independent on/off processes. Previous studies have assumed a Markovian structure for component processes amounting to assuming that exponential distributions govern on and off periods. However, if the number of factor processes grows and input rates are stabilized by allowing the period distribution to change suitably, a limiting on/off process can be obtained which has exponentially distributed on periods and whose off periods are equal in distribution to the busy period of an M/G/\(\infty\) queue. The authors give a complete study of the possible limits of the product process as the number of the factors grows. A process \(\{ Z^{(n)}(t) = \prod_{j=1}^n I_j^{(n)} (t),\, t \geq 0 \}\) where \(I_j^{(n)} (\cdot)\), \(j=1, \ldots, n\) are i.i.d. on/off processes is considered. The authors propose the general theory of processes \(Z^{(n)} (t).\) Under some assertions they give the nessesary and sufficient conditions for \(Z^{(n)} (\cdot)\) to converge to a limiting approximation \(Z^{(\infty)} (\cdot)\) in the sense of convergence of finite dimensional distribution and they provide various interpretations for the limiting process \(Z^{(\infty)} (\cdot).\) Information about dependence structure of the limits process \(Z^{(\infty)} (\cdot)\) is presented as well. In particular various facts about the decay of the correlation functions are given.
scientific article; zbMATH DE number 2063742
scientific article