The Grand-Canonical Asymmetric Exclusion Process and the One-Transit   Walk
  The one-dimensional Asymmetric Exclusion Process (ASEP) is a paradigm for nonequilibrium dynamics, in particular driven diffusive processes. It is usually considered in a canonical ensemble in which the number of sites is fixed. We observe that the grand-canonical partition function for the ASEP is remarkably simple. It allows a simple direct derivation of the asymptotics of the canonical normalization in various phases and of the correspondence with One-Transit Walks recently observed by Brak et.al. 
R. A. Blythe, W. Janke, D. A. Johnston and R. Kenna