The simplest and the most important distribution in survival analysis is the exponential distribution. In this paper, we investigate the influence of the random censorship model on the estimation of the scale parameter of the exponential distribution. The considered random censorship models are Koziol-Green model and the generalized exponential distribution model. Two models have different meanings. Through the simulation study, the averages of the estimated values of the parameter do not show big differences, however the MSE of the estimator tends to be bigger when the supposed model is significantly different from the true model.