Product-Form Distribution and Reversibility of Inhomogeneous Symmetric Simple Exclusion Process with Open Boundaries

TL;DR

This work analyzes a continuous-time inhomogeneous symmetric exclusion process on a finite 1D lattice with open boundaries, where particles of $K$ types interact via type-dependent arrival, departure, and interior hop rates. The authors derive a product-form stationary distribution $p(x_1,\dots,x_N) = C \prod_{i:\, x_i\neq 0} (\alpha_{x_i}/\beta_{x_i})$ with normalization $C = \big(1+\sum_{k=1}^K \alpha_k/\beta_k\big)^{-N}$ and obtain site marginals matching an $M/G/1/0$ loss system, along with explicit arrival rates $J_k$ and mean sojourn times $U_k$. They prove ergodicity of the finite CTMC and, under $\alpha_k=\beta_k$, show equiprobable states and reversibility, with the uniform distribution $p(x_1,\dots,x_N)=(K+1)^{-N}$. These results provide closed-form performance metrics and connect the model to loss-queueing theory and network reversibility. The findings have implications for exactly solvable transport and queueing networks with open boundaries.

Abstract

We consider an inhomogeneous symmetric simple exclusion process on a one-dimensional lattice with open boundary conditions. The time scale is continuous. Particles of different types arrive to the utmost left and the utmost right site. If a particle is in a site that is neither the utmost left site nor the utmost right site, then the particle moves onto one site to the left or to the right. If a particle is either in the utmost left site or the utmost right site, then the particle leaves the system or moves onto one cell to the right or to the left, respectively. An arrival or a transfer of particle is possible only to a vacant site. The rate of arrival, exit or movement of a particle depends on its type and does not depend on the site from that the particle arrives or exits and on the direction the movement. The stationary distribution of the system states probabilities has been found. This distribution turns out to be multiplicative in the sense that the probability of the site state does not depend on the states of the other sites in the stationary mode, and the steady probability of any state of the system is equal to the product of the site states steady probabilities, and the probability of site ocupancy is the same as the server occupancy probability for M/G/1/0 loss system. We have found the arrival rate and the average sojourn time have been found. We have proved the reversibility of the process in time under the additional condition that the rate of arrival of a particle of prescribed type is equal to the rate of this type particle departure.