In this paper we propose a new method of determining the stability of queueing systems. We attain it using the absorbing process and introduce the untraceable events method to show the existence of the absorbing process. The advantage of our method is that we are able to discuss the stability of various variables for both discrete and continuous parameters in a general framework with nonstationary input. An untraceable event has the property that the state loses the memory of its origin. In a concrete model, we use the boundedness of the state at an epoch in time with respect to the initial condition and choose the form of the untraceable event corresponding to the input distribution.

We introduce the concept of the absorbing process for analysing a state process. Our aim is to show the existence of the absorbing process with probability one. This process is shown to be stationary, asymptotically stationary, periodic or a.m.s., if the input distribution has such properties. The real process is absorbed into this process so that its stability and some other properties are easily derived.

Consider a semi-Markov process X(t) defined on a subset of the non-negative integers with zero as an absorbing state and the non-zero states forming an irreducible class with exit to zero being possible. Conditions are given for the existence of the limits: where Xj(t) is the amount of time prior to time t spent in state j.

The limits (which are independent of the initial state) are evaluated when the sufficient conditions are satisfied.