The Lagrangian specification is used to simulate the transient filament stretching rheometer. Simulations are performed for a dilute PIE-solution modeled as a four mode Oldroyd-B fluid and a semidilute PIE-solution modeled as a non-linear single integral equation. The simulations are used to investigate a sequence of filament stretching rheometers. The questions of spacial and temporal homogeneity of the experiment are investigated. The simulations are compared with data from a specific rheometer by Tirtaadmadja and Sridhar.