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• A unified and rigorous ISB mathematical model is derived through equations. • The components of the new ISB model are verified by experiments. • The new ISB model is used to explain whether it is necessary to add an ISB parameter in BDS-2 and BDS-3 combined PPP. • The time-varying characteristics of ISB in multi-GNSS PPP are evaluated. • The influence of ISB estimation method on multi-GNSS PPP is discussed. The inter-system bias (ISB) is an important parameter in multi-GNSS precise point positioning (PPP). However, on the one hand, the generation mechanism and error components of ISB are not clear. On the other hand, it is unclear whether the ISB parameter should be added to the BDS-2/BDS-3 combined PPP. First, in order to solve these problems, an extended ISB mathematical model is proposed, which unifies the common errors between receiver and satellite, and extends the original ISB model. Second, to demonstrate the correctness of the new model, the components of the new ISB model are verified, and then it is used to explain whether the ISB parameter should be added to BDS-2/BDS-3 combined PPP. Furthermore, 41 stations from the MGEX network and precise products from COD (Center for Orbit Determination in Europe), GBM (Deutsches GeoForschungsZentrum) and WUM (Wuhan University) are used to calculate and analyze the multi-GNSS PPP and ISB during day of year (DOY) 307–365, 2020. Finally, we propose to use a different estimation method of ISB with different precise products to improve the positioning accuracy and shorten the convergence time. The experimental results show that: (1) ISB parameter is composed of five parts: time system error, receiver hardware delay, signal distortion biases (SDB), MGEX-realized (multi-GNSS experiment) time scale and other unmodeled deviations. (2) Due to different receiver hardware delay, SDB and MGEX-realized time scale between BDS-2 and BDS-3, it is necessary to add an ISB parameter in BDS-2/BDS-3 PPP. (3) In multi-GNSS PPP, if the ISB changes greatly but the traditional constant method is used to estimate the ISB parameter, the impact on single station PPP coordinates can reach decimeters. The statistical results demonstrate that the RMS of GBM(CON(Constant)) in East (E), North (N) and Up (U) directions are 2.34 cm, 0.60 cm, and 1.59 cm, respectively, while the RMS of GBM(RWK(Random walk)) decreased by 59.8 %, 13.3 %, and 18.2 %, respectively. [ABSTRACT FROM AUTHOR]