Title: Colloquium: On fractional counting processes
Speaker: Kuldeep Kumar Kataria (IIT Bhilai)
Date: 12 January 2022
Time: 4 pm
Venue: Microsoft Teams (online)
Advances in various fields of modern studies have shown the limitations
of traditional probabilistic models. The one such example is that of the
Poisson process which fails to model the data traffic of bursty nature,
especially on multiple time scales. The empirical studies have shown that
the power law decay of inter-arrival times in the network connection
session offers a better model than exponential decay. The quest to improve
Poisson model led to the formulations of new processes such as non-homogeneous
Poisson process, Cox point process, higher dimensional Poisson process, etc.
The fractional generalizations of the Poisson process has drawn the attention
of many researchers since the last decade. Recent works on fractional
extensions of the Poisson process, commonly known as the fractional
Poisson processes, lead to some interesting connections between the areas
of fractional calculus, stochastic subordination and renewal theory. The
state probabilities of such processes are governed by the systems of
fractional differential equations which display a slowly decreasing memory.
It seems a characteristic feature of all real systems. Here, we discuss
some recently introduced generalized counting processes and their fractional
variants. The system of differential equations that governs their state
probabilities are discussed.