TCP and iso-stationary transformations

Abstract

We consider piecewise-deterministic Markov processes that occur as scaling limits of discrete-time Markov chains that describe the Transmission Control Protocol (TCP). The class of processes allows for general increase and decrease profiles. Our key observation is that stationary results for the general class follow directly from the stationary results for the idealized TCP process. The latter is a Markov process that increases linearly and experiences downward jumps at times governed by a Poisson process. To establish this connection, we apply space-time transformations that preserve the properties of the class of Markov processes.

On time reversal of piecewise deterministic Markov processes

“…This is in accordance with countless observations of absolutely continuous stationary distributions for PDMP type stochastic models in the literature (e.g. [3,12,13,27,6,5,40]). However, in general, i.e.…”

“…Hence, for the trivial case of the renewal process our formulas yield the correct results. In [40] a generalized model for the TCP window size has been studied (see also [2,24,16,39,4,42]). We assume that X t is a process with λ(x) = λx β , r(x) = rx α , β > α − 1 and µ x ((0, y]) = (y/x) γ , which means that Q k = U 1/γ K · W k , where the U k are independent random variables with uniform distribution on (0, 1).…”

“…Piecewise deterministic paths can be observed for Markov processes in a variety of applications. We mention risk process ( [1,48,19,20,25]), growth collapse and stress release models ( [9,53,58,12,41]), queueing models ( [14,11]), earthquake models ( [46]), repairable systems ( [38]), storage models ( [15,27,28]) and TCP data transmission ( [39,40,24]). The mathematical framework that is used in this paper was introduced by Davis [22,21].…”

Some simple but challenging Markov processes

“…See also [12] for a simplified TCP windows size model. See [40,43,52,53,54,51,33] for other works dedicated to this process. Generalization to interacting multi-class transmissions are considered in [29,30].…”

Shot-noise queueing models

“…(77)Since (•) is actually a stochastic process, we conclude that the number of customers has a mixed Poisson distribution, i.e., Poisson with a random parameter, viz. the expression in Eqn (77)…”