This article presents an approximate convolution model of a multiservice queueing system with the continuous FIFO (cFIFO) service discipline. The model makes it possible to service calls sequentially with variable bit rate, determined by unoccupied (free) resources of the multiservice server. As compared to the FIFO discipline, the cFIFO queue utilizes the resources of a multiservice server more effectively. The assumption in the model is that the queueing system is offered a mixture of independent multiservice Bernoulli-Poisson-Pascal (BPP) call streams. The article also discusses the results of modelling a number of queueing systems to which different, non-Poissonian, call streams are offered. To verify the accuracy of the model, the results of the analytical calculations are compared with the results of simulation experiments for a number of selected queueing systems. The study has confirmed the accuracy of all adopted theoretical assumptions for the proposed analytical model.