Stationary analysis of the "Shortest Queue First" service policy: the asymmetric case

As a follow-up to a recent paper considering two symmetric queues, the \textit{Shortest Queue First} service discipline is presently analysed for two general asymmetric queues. Using the results previously established and assuming exponentially distributed service times, the bivariate Laplace transform of workloads in each queue is shown to depend on the solution $\mathbf{M}$ to a two-dimensional functional equation $$\mathbf{M} = Q_1 \cdot \mathbf{M}\circ h_1 + Q_2 \cdot \mathbf{M}\circ h_2 + \mathbf{L}$$ with given matrices $Q_1$, $Q_2$ and vector $\mathbf{L}$ and where functions $h_1$ and $h_2$ are defined each on some rational curve; solution $\mathbf{M}$ can then represented by a series expansion involving the semi-group $< h_1, h_2 >$ generated by these two functions. The empty queue probabilities along with the tail behaviour of the workload distribution at each queue are characterised.