Brownian meander factor for the spatial axis must be square root of the scaling factor for the time axis. The process resulting from this snip-and-scale procedure is a Brownian meander. As the name suggests, it is a piece of Brownian motion that spends all its time away from its starting point formula_5. The transition density formula_12 of Brownian meander is described as follows: For formula_13 and formula_14, and writing we have and In particular, i.e. formula_19 has the Rayleigh distribution with parameter 1, the same distribution as formula_20, where formula_21 is an exponential random variable with parameter 1. Brownian meander In the
Reflected Brownian motion steady state RBMs. Feller described possible boundary condition for the process Reflected Brownian motion In probability theory, reflected Brownian motion (or regulated Brownian motion, both with the acronym RBM) is a Wiener process in a space with reflecting boundaries. RBMs have been shown to describe queueing models experiencing heavy traffic as first proposed by Kingman and proven by Iglehart and Whitt. A "d"–dimensional reflected Brownian motion "Z" is a stochastic process on formula_1 uniquely defined by where "X"("t") is an unconstrained Brownian motion and with "Y"("t") a "d"–dimensional vector where The reflection matrix describes boundary behaviour. In the interior of