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Continuous-state branching processes with migration

Vidmar Matija

Probability and Mathematical Statistics

2022

Vol. 42, Fasc. 2

227--249

Continuous-state branching processes (CSBPs) with immigration (CBIs), stopped on hitting zero, are generalized by allowing the process governing immigration to be any Lévy process without negative jumps. Unlike CBIs, these newly introduced processes do not appear to satisfy any natural affine property on the level of the Laplace transforms of the semigroups. Basic properties of these processes are described. Explicit formulae (on neighborhoods of infinity) for the Laplace transforms of the first passage times downwards and of the explosion time are derived.

Extremes of chi-square processes with trend

Liu P., Ji L.

Probability and Mathematical Statistics

2016

Vol. 36, Fasc. 1

1--20

This paper studies the supremum of chi-square processes with trend over a threshold-dependent-time horizon. Under the assumptions that the chi-square process is generated from a centered self-similar Gaussian process and the trend function is modeled by a polynomial function, we obtain the exact tail asymptotics of the supremum of the chi-square proces with trend. These results are of interest in applications in engineering, insurance, queueing and statistics, etc. Some possible extensions of our results are also discussed.

Transient and stationary characteristics of a packet buffer modelled as an MAP/SM/1/b system

Rusek K., Janowski L., Papir Z.

International Journal of Applied Mathematics and Computer Science

2014

Vol. 24, no. 2

429--442

A packet buffer limited to a fixed number of packets (regardless of their lengths) is considered. The buffer is described as a finite FIFO queuing system fed by a Markovian Arrival Process (MAP) with service times forming a Semi-Markov (SM) process (MAP/SM/1/b in Kendall’s notation). Such assumptions allow us to obtain new analytical results for the queuing characteristics of the buffer. In the paper, the following are considered: the time to fill the buffer, the local loss intensity, the loss ratio, and the total number of losses in a given time interval. Predictions of the proposed model are much closer to the trace-driven simulation results compared with the prediction of the MAP/G/1/b model.