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In this paper, our investigation focuses on the performance evaluation of omega multistage interconnection networks which can natively support multi-class priority traffic. For this reason, a multi- buffered, single server switching element is presented and used in the construction of those networks, providing thus better quality of services. The performance evaluation model is based on steady state queuing analysis that leads to a novel utilization's formula. Based on the proposed analysis, other major performance indicators can also be accurately calculated. The analytical method was meticulously validated by classic simulation and a part of obtained results was also compared by existing related work. Moreover, the proposed analytical method was found to be very accurate on performance estimations for various network sizes under different offered loads supporting multi-priority traffic. The rationale behind investigating a multi-priority traffic system is to provide efficient networks supporting optimum quality of services (i.e. streaming media vs. file transfer).