On the stability of a class of non-monotonic systems of parallel queues

We investigate, under general stationary ergodic assumptions, the stability of systems of $S$ parallel queues in which any incoming customer joins the queue of the server having the $p+1$-th shortest workload ($p<S$), or a free server if any. This change in the allocation policy makes the analysis much more challenging with respect to the classical FCFS model with $S$ servers, as it leads to the non-monotonicity of the underlying stochastic recursion. We provide sufficient conditions of existence of a stationary workload, which indicate a “splitting” of the system in heavy traffic, into a loss system of $p$ servers plus a FCFS system of $S-p$ servers. To prove this result, we show {\em en route} an original sufficient condition for existence and uniqueness of a stationary workload for a multiple-server loss system.