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Abstract
We consider a general arrival and Markovian service time queueing system with one server under first come first served discipline, where the element of transition probability is given as matrix and the system can accommodate finite number of arrival, . Transition probability matrix and steady states probabilities were obtained. Numerical illustration is considered on D/M/1/5 queue, where the customers arrive at a rate of one per unit
time and for which the mean of the exponential service time equals to so as to reflect its usefulness in solving the real life problem.
Keywords: G/M/1/K Queueing System, Markovian arrival, Markovian service, Transition Probability,