Asymptotic bounds of throughput in series-parallel queueing networks

We propose a piece-wise linear upper bound on the throughput rate from a network of series-parallel queues where arrivals occur through a single infinite queue. This bound is tight and is observed to be extremely accurate in forecasting the actual throughput rate. We also describe the monotonicity of throughput as a function of the arrival rate and specify a condition under which the upper bound may be computed. We approximate analytically the throughput measured as a function of the arrival rate for two tandem exponential queues, where the first queue has an infinite buffer while the second queue has a finite buffer. We extend this analysis to elementary split and merge queueing networks. We demonstrate the generality and robustness of this asymptotic property, for larger series-parallel networks with general service times and specify the set up of a single simulation experiment which can be used to retrieve the throughput for any arrival rate, as well as other networks performance measures.