What can one say on convergence to stationarity of a finite state Markov chain that behaves "locally" like a nearest neighbor random walk on the integer lattice ? The model we consider is a version of nearest neighbor lazy random walk on the state space 0,1,...,N: the probability for staying put at each site is 1/2, the transition to the nearest neighbors, one on the right and one on the left, occurs with probability 1/4 each, where we identify two sites, a and b as, respectively, the neighbor of 0 from the left and the neighbor of N from the right (but 0 is not a neighbor of a and N is not neighbor of b). This model is a discrete version of diffusion with redistribution on an interval studied by several authors in the recent past, and for which the exponential rates of convergence to stationarity was computed analytically, but had no intuitive or probabilistic interpretation, except for case where the jumps from the endpoints are identical (or more generally have the same distribution). We study convergence to stationarity probabilistically, by finding an efficient coupling. The coupling identifies the "bottlenecks" responsible for the rates of convergence and also gives tight computable bounds on the total variation norm of the process between two starting points. The adaptation to the diffusion setting is straightforward. Based on joint work with Hugo Panzo and Elizabeth Tripp.