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In this paper, we address the modeling issues of cell movement and division with a special focus on the phenomenon of volume exclusion in a lattice-based, exact stochastic simulation framework. We propose a new exact method, called Reduced Rate Method -- RRM, that is substantially quicker than the previously used exclusion method, for large number of cells. In addition, we introduce three novel reaction types: the contact-inhibited, the contact-promoted, and the spontaneous reactions. To the best of our knowledge, these reaction types have not been taken into account in lattice-based stochastic simulations of cell cultures. These new types of events may be easily applied to complicated systems, enabling the generation of biologically feasible stochastic cell culture simulations. Furthermore, we show that the exclusion algorithm and our RRM algorithm are mathematically equivalent in the sense that the next reaction to be realized and the corresponding sojourn time both belong to the same reaction and time distributions in the two approaches -- even with the newly introduced reaction types. Exact, agent-based, stochastic methods of cell culture simulations seem to be undervalued and are mostly used as benchmarking tools to validate deterministic approximations of the corresponding stochastic models. Our proposed methods are exact, they are easy to implement, have a high predictive value, and can be conveniently extended with new features. Therefore, these approaches promise a great potential.