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In a silicon photomultiplier (SiPM), also referred to as multi-pixel photon counter (MPPC), many Geiger-mode avalanche photodiodes (GM-APDs) are connected in parallel so as to combine the photon counting capabilities of each of these so-called microcells into a proportional light sensor. The discharge of a single microcell is relatively well understood and electronic models exist to simulate this process. In this paper we introduce an extended model that is able to simulate the simultaneous discharge of multiple cells. This model is used to predict the SiPM signal in response to fast light pulses as a function of the number of fired cells, taking into account the influence of the input impedance of the SiPM preamplifier. The model predicts that the electronic signal is not proportional to the number of fired cells if the preamplifier input impedance is not zero. This effect becomes more important for SiPMs with lower parasitic capacitance (which otherwise is a favorable property). The model is validated by comparing its predictions to experimental data obtained with two different SiPMs (Hamamatsu S10362-11-25u and Hamamatsu S10362-33-25c) illuminated with ps laser pulses. The experimental results are in good agreement with the model predictions.