The symmetric birth and death process in the integers $\{1, \ldots, N \}$ with linear rates is studied. The process moves slowly and spends more time in the neighborhood of the state 1. It represents our attempt at explaining the asymmetry between amounts of matter and antimatter by the inhomogeneity of the sojourn time. The state of the process reflects a relative frequency of an antimatter amount. The observed matter-antimatter imbalance in the Universe we consider a result of stochastic competition between them. The amount of matter significantly exceeds the amount of antimatter, which corresponds to the lower states of the process, and the path toward matter-antimatter equilibrium can be very long.