In normal human somatic cells, telomeres become shorter with each cell cycle until their chromosome end protection function fails causing the cells to senesce. Previously telomere loss was thought to be gradual, and the approach to cellular senescence was likened to a mitotic clock in which the shortest telomere of a progenitor cell determined the maximum number of cell doublings. In addition to the basal loss, recent evidence indicates that there are occasional large gains, losses and exchanges of telomeric DNA collectively called telomere dynamics. We are using discrete Monte Carlo computer simulations to explore the dependence of cellular proliferation on telomere dynamics. This work clearly shows that, when telomere lengths vary stochastically, a deterministic mitotic clock is an inadequate model of cellular proliferation. Compared to predictions of the mitotic clock model, senescent cells appear much earlier in simulated colony growth, just as they do in real cultures, and the cause of senescence is not necessarily the shortest initial telomere. Interestingly because of the stochastic nature of telomere dynamics, two indistinguishable progenitor cells may go through quite different colony growth experiences. The methods used to construct these simulations, and insights gained from them, will be presented in detail.