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A novel performance analysis of output queued cell switches that are introduced with general independent heterogeneous traffic is presented. Random arbitration is employed whereby non-empty queues compete equally for service within each switching interval. In particular, the case of bursty two-state Markov-modulated arrivals is studied 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 utilised to derive an approximation for the queue size distribution. The methodology established forms a flexible tool in obtaining bounds on the behaviour and expected performance characteristics of output queued switches under a wide range of correlated traffic scenarios. The validity of the analytical inference is established through simulation results.