The statistics of queueing systems, such as the mean number in each queue and the variance, are often computed using steady state assumptions. In many systems, however, the queue parameters change with time and steady state assumptions lead to erroneous mean and variance quantities. It is therefore desirable to solve the transient system. Unfortunately, solutions to transient queueing systems are difficult to obtain. Although an analytic solution exists for the single M/M/1 queue, a network of two such queues remains an open problem. The approximation methods are used reduce the computational complexity of existing transient solutions and to provide insight into the behavior of systems for which no analytic solution exists. It is hoped that this research will serve to improve present methods of modelling computer networks [#!AAJA:1!#].