We explore some stochastic considerations regarding accumulation of mutations in relation to carcinogenesis. In particular, we consider the effect of exposure to specific agents, especially ionizing radiation, that may increase mutation rates. The formulation and consequences are a further development of the Armitage-Doll model; both in terms of background cancer where assumptions are substantially weakened, and in terms of the effect of specific mutagenic exposures through generally increasing mutation rates. Under our model the effect of exposure is equivalent to a change in age scale, adding to age a parametric multiple of cumulative dose to the mutagen, which leads to useful formulae for the relative risk. In particular, the excess relative risk at age a behaves approximately as a parametric multiple of the mean dose over ages prior to a. These results do not require assuming that some fixed number of mutations are required for malignancy. The implications are particularly useful in providing guidance for descriptive analyses since they have characteristics largely independent of parameter values. It is indicated that the model consequences conform remarkably well to observations from cohort studies of the A-bomb survivors, miners with prolonged exposure to radon, and cigarette smokers who stopped smoking at various ages.