ZOR - Mathematical Methods of Operations Research (1996) 43:97-106 A Simple Method for Computing State Probabilities of the M[G[1 and GI/M[1 Finite Waiting Space Queues T. S. S. SRINIVASARAO and U. C. GUPTA Department of Mathematics, Indian Institute of Technology, Kharagpur-721302, India Abstract: This paper presents a simple method for computing steady state probabilities at arbitrary and departure epochs of the M/G/1/K queue. The method is recursive and works efficiently for all service time distributions. The only input required for exact evaluation of state probabilities is the Laplace transform of the probability density function of service time. Results for the GI/M/1/K -- 1 queue have also been obtained from those of M/G/1/K queue. Key Words: Computational, finite waiting space, queues Introduction Queues with finite waiting space have wide application in many areas, such as computers, communication systems, queueing networks, and manufacturing systems, etc. In recent years there has been a considerable interest in obtaining numerically state probabilities of finite queues using various techniques. The simple model M/G~1 finite queue had been under investigation for a long time and is discussed by Keilson (1966), Cooper (1981), Neuts (1981), Cohen (1982), Franken et al. (1982), Lavenberg (1983), Gross and Harris (1985), Baiocchi (1992) and many others. Recently Chaudhry et al. (1991) have obtained the probability distribution of the number in system for the M/G/1/K queue in terms of the roots of the associated characteristic equation, which can be evalu- ated using Chaudhry's (1993) QROOT software package. They also obtain results for the GI/M/1/K -- 1 queue from those of M/G/1/K queue. Their method works under the assumption that the probability density function (p.d.f.) of service time distribution has a rational Laplace transform (L.T.). In those cases where L.T, is not rational, one has to use an approximation. For example, the L.T. of the p.d.f, of deterministic distribution is not rational and hence the exact numerical results for M/D/1/K queue camaot be obtained using their method. However, they obtain approximate results for M/D/1/K queue from M/Eh/1/K 0340 9422/96/43 : 1/97-106 $2.50 9 1996 Physica-Verlag, Heidelberg