On parallel queuing with random server connectivity and routing constraints

We study systems of parallel queues with finite buffers, a single server with random connectivity to each queue, and arriving job flows with random or class-dependent accessibility to the queues. Only currently connected queues may receive (preemptive) service at any given time, whereas an arriving job can only join one of its accessible queues. Using the coupling method, we study three key models, progressively building from simpler to more complicated structures. In the first model, there are only random server connectivities. It is shown that allocating the server to the Connected queue with the Fewest Empty Spaces (C-FES) stochastically minimizes the number of lost jobs due to buffer overflows, under conditions of independence and symmetry. In the second model, we additionally consider random accessibility of queues by arriving jobs. It is shown that allocating the server to the C-FES and routing each arriving job to the currently Accessible queue with the Most Empty Spaces (C-FES/A-MES) minimizes the loss flow stochastically, under similar assumptions. In the third model (addressing a target application), we consider multiple classes of arriving job flows, each allowed access to a deterministic subset of the queues. Under analogous assumptions, it is again shown that the C-FES/A-MES policy minimizes the loss flow stochastically.