Research Abstract 
It is well known that many of social and natural phenomena can be described by queueing models atirl the theory pf(itieues has contributed much to analyze such systems. In fact, in past few decades. many theoretically and practically important results have been obtained. However, most of these results concern the steady state results as the time goes to intifinity. flence. rigorously speaking. these results can not be applied to real systems by the forms especially for such complex systems as communication had information networks, because we do not know how much the steady state approximation matters. As systems become complex and important. therefore. the need to have exact time dependent solutions increases. In this project, in order to reflect such social needs, we investigate the transient behavior of single server Markovian queueing systems and develop a computer system to evalutate the behavior. The solution is known which, however, includes an infinite sum of mo(lifie(i Bessel functions of second kind, so that the closed form solution is not terribly useful as far as computational tractability is concerned. Therefore, some approximation are iequireci from practical points of view. During the two years of this project, we have focused an some mathematical aspect and derived many properties that the time dependent solutions possess through joint works with Dr. Ward Whitt, et al., which sheds new light on the understanding of the transient behavior. The theory of Laguerre transform is applied for the numerical purposes. Some theory regarding the transient behavior of stochastic processes is also investigated, which is of independent interest.
