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S. O. Agboola and G. B. Karokatose2

**keywords:** General distribution, performance measure, Pollaczek-Khintchine equation residual service

This present study, the residual service time on the M/G/1 queueing system where the arrival process is Poisson with rate λ and service times of customers are independent and identically distributed and obey an unspecified arbitrary or general distribution function is analysed. Our analysis is based on the fact that, an arrival is more likely to occur during a large service time than a small service interval since the service is a random variable having a general distribution. Using the imbedded Markov chain technique, the stochastic transition probability matrix f_ij and average residual service time in [0, t] are obtained by considered the area under the curve R(t) divided by t. Finally, we obtained the probability distribution function of the residual service time conditioned on the server being busy and expected residual service time. The numerical illustration is considered to show its applications in solving real life problem on M/D/1queue for which λ=1/2 and μ=1. The elements of the transition probability matrix are obtained as α_1= 0.303265; α_2= 0.075816;α_3= 0.075816; α_4= 0.075816; α_5= 0.075816;α_6= 0.075816; α_7= 0.075816. Also, by the use of the recursive procedure and settingρ_0=0.5, We obtain the followingP_1=0.32436: ∑_(i=0)^1▒〖p_i=0.824361〗, P_2=0.1226: ∑_(i=0)^2▒〖p_i=0.94696〗, P_3=0.037788: ∑_(i=0)^3▒〖p_i=0.98475〗, P_4=0.01091: ∑_(i=0)^4▒〖p_i=0.995658〗, P_5=0.003107: ∑_(i=0)^5▒〖p_i=0.998764〗, P_6=0.000884: ∑_(i=0)^6▒〖p_i=0.999648〗.