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We study the finite zero-range process with occupancy-dependent rate function g • . Under the invariant measure, which can be written explicitly in terms of g, particles are distributed over sites and we regard all particles at a fixed site as a cluster. In the density one case, that is, equal numbers of particles and sites, we determine asymptotically the size of the largest cluster, as the number of particles tends to infinity, and determine its dependence on the rate function.

Binary exponential backoff is a randomized protocol for regulating transmissions on a multipleaccess broadcast channel. Ethernet, a local-area network, is built upon this protocol. The fundamental theoretical issue is stability: Does the backlog of packets awaiting transmission remain bounded in time, provided the rates of new packet arrivals are small enough? It is assumed n t 2 stations share the channel, each having an infinite buffer where packets accumulate while the station attempts to transmit the first from the buffer. Here, it is established that binary exponential backoff is stable if the sum of the arrival rates is sufficiently small. Detailed results are obtained on which rates lead to stability when n = 2 stations share the channel. In passing, several other results are derived bearing on the efficiency of the conflict resolution process. Simulation results are reported that, in particular, indicate alternative retransmission protocols can significantly improve performance.

We analyze and simulate a two dimensional Brownian multi-type particle system with death and branching (birth) depending on the position of particles of different types. The system is confined in the two dimensional box, whose boundaries act as the sink of Brownian particles. The branching rate matches the death rate so that the total number of particles is kept constant. In the case of m types of particles in the rectangular box of size a, b and elongated shape a ≫ b we observe that the stationary distribution of particles corresponds to the m-th Laplacian eigenfunction. For smaller elongations a > b we find a configurational transition to a new limiting distribution. The ratio a/b for which the transition occurs is related to the value of the m-th eigenvalue of the Laplacian with rectangular boundaries.

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