Analysis of a two-layered network by means of the power-series algorithm

We consider an extension of the classical machine-repair model, also known as the computer-terminal model or time-sharing model. As opposed to the classical model, we assume that the machines, apart from receiving service from the repairman, supply service themselves to queues of products. The extended model can be viewed as a two-layered queueing network, of which the first layer consists of two separate queues of products. Each of these queues is served by its own machine. The marginal and joint queue length distributions of the first-layer queues are hard to analyse in an exact fashion. Therefore, we apply the power-series algorithm to this model to obtain the light-traffic behaviour of the queue lengths symbolically. This leads to two accurate approximations for the marginal mean queue length. The first approximation, based on the light-traffic behaviour, is in closed form. The second approximation is based on an interpolation between the light-traffic behaviour and heavy-traffic results for the mean queue length. The obtained approximations are shown to work well for arbitrary loaded systems. The proposed numerical algorithm and approximations may prove to be very useful for system design and optimisation purposes in application areas such as manufacturing, computer systems and telecommunications.