Geometrically batched networks

We develop approximate solutions for the equilibrium queue length probability distribution of queues in open Markovian networks by considering each queue independently, constructing its arrival process as the join of each contributing queue's departing traffic. Without modulation and non-unit batches, we would only need to consider mean internal traffic rates, modelling each queue as M/M/1 to give an exact result by Jackson's theorem. However, bursty traffic significantly affects steady state queue lengths; for given throughput, mean queue length varies linearly with mean batch size. All batch sizes are geometrically distributed, so each queue is Markovian and has known analytical solution. Our analysis is based on properties of the output processes of these queues, their superposition and splitting, to form the arrival processes at all queues. In general, this leads to a fixed-point problem for the network's equilibrium. The numerical results of our approach are compared with simulation and show promising accuracy.