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This paper analyses a multiple class single server M/M/1/K queue with finite capacity under weighted fair queuing (WFQ) discipline. The Poisson process has been used to model the multiple classes of arrival streams. The service times have exponential distribution. Analytical expressions for the flow balanced equations have been derived using Markov chain. This paper presents an analytical and numerical solution to the M/M/1/K queue with finite buffer under (WFQ ) service and the derivation of a general expression for the steady state probabilities for any buffer K. Numerical experiments corroborate the theoretical results are also offered.