  We consider a version of random motion of hard core particles on the
semi-lattice $ 1, 2, 3,...$, where in each time instant one of three possible
events occurs, viz., (a) a randomly chosen particle hops to a free neighboring
site, (b) a particle is created at the origin (namely, at site 1) provided that
site 1 is free and (c) a particle is eliminated at the origin (provided that
the site 1 is occupied). Relations to the BKP equation are explained. Namely,
the tau functions of two different BKP hierarchies provide generating functions
respectively (I) for transition weights between different particle
configurations and (II) for an important object: a normalization function which
plays the role of the statistical sum for our non-equilibrium system. As an
example we study a model where the hopping rate depends on two parameters ($r$
and $\beta$). For time $\time\to\infty$ we obtain the asymptotic configuration
of particles obtained from the initial empty state (the state without
particles) and find an analog of the first order transition at $\beta=1$.
