A three-parameter generalized Poisson Lomax distribution, which contains two-parameter Lomax distribution as a special case, has to be obtained by a countable mixture with Poisson distribution. The mathematical attributes, as behavior of density and failure rate functions, compounding, moments, moment generating functions, and mean residual life, of this model and the strength-stress reliability function R=P(X>Y) are studied. Additionally, the maximum likelihood estimation of the new model parameters is derived. Lastly, the practical performance of the new model is demonstrated using censored and complete data sets with simulation study.