Using infinitesimal perturbation analysis of stochastic flow models to recover performance sensitivity estimates of discrete event systems

Stochastic Flow Models (SFMs) form a class of hybrid systems used as abstractions of complex Discrete Event Systems (DES) for the purpose of deriving performance sensitivity estimates through Infinitesimal Perturbation Analysis (IPA) techniques when these cannot be applied to the original DES. In this paper, we establish explicit connections between gradient estimators obtained through a SFM and those obtained in the underlying DES, thus providing analytical evidence for the effectiveness of these estimators which has so far been limited to empirical observations. We consider DES for which analytical expressions of IPA (or finite difference) estimators are available, specifically G/G/1 and G/G/1/K queueing systems. In the case of the G/G/1 system, we show that, when evaluated on the same sample path of the underlying DES, the IPA gradient estimators of states, event times, and various performance metrics derived through SFMs are, under certain conditions, the same as those of the associated DES or their expected values are asymptotically the same under large traffic rates. For G/G/1/K systems without and with feedback, we show that SFM-based derivative estimates capture basic properties of finite difference estimates evaluated on a sample path of the underlying DES.