SAFE Merge: Merging Streams in Discrete Time Stochastic Flow Networks

Abstract

Merging stochastic flows is a central operation in discrete time queueing networks. Furmans' renewal-based merge provides an efficient analytical approximation, but can become inaccurate when the temporal structure of the input streams restricts the attainable arrival configurations and the merged process therefore deviates from a renewal process. This paper introduces the SAFE (Senger–Altmann–Furmans Extension) merge, an analytical extension for zero relative temporal offset that removes inadmissible forward recurrence time values before applying the original merge procedure. The method is evaluated against simulation and a numerical merge using the Wasserstein distance. The results show that SAFE remains close to the simulation benchmark for cases in which Furmans' approximation becomes inaccurate, while the numerical approach may require up to 10^8 samples to reach comparable accuracy.