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The paper presents an analysis of output queued cell switches which are introduced with generic non-uniformly distributed traffic. Random arbitration is employed whereby non-empty queues compete equally for service within each switching interval. In particular, we study the case of two-state Markov-modulated arrivals in which input ports generate bursty streams that are non-uniformly distributed. Under the assumption of a memoryless server, the probability generating function of the interarrival process is utilized to derive closed-form expressions for the queue size distribution. The methodology established forms a flexible tool in determining bounds on the behavior and expected performance of output queued switches under a range of traffic scenarios. The validity of the analytical inference is established through simulation results.