<p>A new class of counting processes generated by the mixture of the extended generalized Pólya process is defined and its properties are studied. The general form of the corresponding stochastic intensity is derived. Specifying geometric, negative binomial, Poisson, and binomial distributions as the mixing distributions, four parametric classes of counting processes are defined and stochastically characterized. It is shown that relevant monotonicity properties of the corresponding stochastic intensities do not follow ‘direct intuition’ and can dramatically change depending on the mixing distribution. The practical meaning of the considered parametric models is also interpreted from the reliability point of view.</p>