Relaxation time in a non-conserving driven-diffusive system with parallel dynamics

We introduce a two-state non-conserving driven-diffusive system in one dimension under a discrete-time updating scheme. We show that the steady state of the system can be obtained using a matrix product approach. On the other hand, the steady state of the system can be expressed in terms of a linear superposition of Bernoulli shock measures with random walk dynamics. The dynamics of a shock position is studied in detail. The spectrum of the transfer matrix and the relaxation times to the steady state have also been studied in the limit of large system size.