keywords: Transition probability matrix, Markovian service, general arrival, PASTA property
A general arrival and Markovian service time queueing system with one server under first come first served discipline was considered, where the ij element of transition probability is given as matrix F and the system can accommodates infinite number of arrival. The steady state transition probability were obtained and compared with the result from Markovian arrival and Markovian service times queueing system (M/M/1). The formulae for waiting time distribution, probability distribution and density functions for the response time (total system time) in a G/M/1 system were obtained. Illustrative numerical examples were demonstrated to shown its usefulness in solving the real life problem. We arrived at the following values for the root of equation of Z-transform of the number of service completions that occur during an interarrival period, ξ^j= 0.600, 0.6115, 0.6153, 0.6206, 0.6223, 0.6267, ⋯, Which is eventually converges to ξ=0.645705. Therefore, the probabilities that it contains zero, one or two customers are given, respectively by 0.4, 0.12 and 0.182. The probability that arrival find the system empty is 0.5, the mean number of customers seen by an arrival is 1.0 and the mean time spent waiting in this system is 0.2.
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