We describe a process where two types of particles, marked red and blue, arrive in a domain at a constant rate. When a new particle arrives into the domain, if there are particles of the opposite color present within a distance of 1 from the new particle, then, among these particles, it matches to the one with the earliest arrival time, and both particles are removed. Otherwise, the particle is simply added to the system. Additionally, particles may lose patience and depart at a constant rate. We study the existence of a stationary regime for this process, when the domain is either a compact space or a Euclidean space. In the compact setting, we give a product-form characterization of the stationary distribution, and then prove an FKG-type inequality that establishes certain clustering properties of the particles in the steady state.
