This paper introduce a new approach to analysing the queueing process, namely lattice paths (LPs). Starting with the \(M/G/1\) queueing system with \(i_{0}\) customers, the authors analyze a busy period with the LP approach. The general distribution in the \(M/G/1\) queueing system is approximated by 3-phase Cox distribution, \(C_{3}\). When the LP approach is applied to the discretised queueing model, the limiting case \(h \rightarrow 0\) yields results for the continuous model. Thus, a wide range of \(M/G/1\) queues with non-rational Laplace-Stieltjes transforms can be approximated by \(C_{3}\), which has Markovian properties. These results are very valuable. They are applicable to almost any real queueing systems.