was defined for periodic boundary conditions). It would be interesting to investigate what is the physical interpretation, in terms of transition probabilities on the lattice, of all these models. In [@CrampeMRV15inhomogeneous] the transfer matrix was used, without specifying any inhomogeneity parameters, to define a discrete time process in the specific case of the totally asymmetric exclusion process for both periodic and open boundary conditions. A physical interpretation in terms of a sequential update was provided. Note that a similar result has been obtained earlier in the periodic homogeneous case in [@GM1].

In fact the use of the transfer matrix approach in the definition of discrete time exclusion processes follows the pioneering work of Baxter [@Baxter]. It was soon realized that the inhomogeneous transfer matrix of the six-vertex model can be used to define an asymmetric exclusion process with discrete time parallel update on the periodic lattice [@KandelDN; @SchutzS; @Schutz2; @Schutz]. Our construction is inspired by these works and aims to provide a general framework (independent of the $R$ matrix considered) to define an integrable discrete time Markov matrix from the transfer matrix machinery of integrable systems. Moreover we generalize it also for open boundary conditions using the Sklyanin transfer matrix 