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We propose a generalized Ehrenfest urn model of many urns arrangedperiodically along a circle. The evolution of the urn model system is governedby a directed stochastic operation. Method for solving an $N$-ball, $M$-urnproblem of this model is presented. The evolution of the system is studied indetail. We find that the average number of balls in a certain urn oscillatesseveral times before it reaches a stationary value. This behavior seems to be apeculiar feature of this directed urn model. We also calculate the Poincar\'{e}cycle, i.e., the average time interval required for the system to return to itsinitial configuration. The result can be easily understood by counting thetotal number of all possible microstates of the system.Comment: 10 pages revtex file with 7 eps figure

Poincaré cycle of a multibox Ehrenfest urn model with directed transport