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In this work, we propose a simplification of the Pollaczek-Khinchine formulafor the ultimate time survival (or ruin) probability calculation in exchangefor a few assumptions on the random variables which generate the renewal riskmodel. More precisely, we show the expressibility of the distribution function$$\mathbb{P}\left(\sup_{n\geqslant1}\sum_{i=1}^{n}(X_i-c\theta_i)<u\right),\,u\in\mathbb{N}_0$$ via the roots of the probability generating function $G_{X-c\theta}(s)=1$,the expectation $\mathbb{E}(X-c\theta)$, and the probability mass function of$X-c\theta$. We assume that the random variables $X_1,\,X_2,\,\ldots$ and$c\theta_1,\,c\theta_2,\,\ldots$ are independent copies of $X$ and $c\theta$respectively, $c>0$, $X$ and $c\theta$ are independent non-negative andinteger-valued, and the support of $\theta$ is finite. We give few numericaloutputs of the proven theoretical statements when the mentioned randomvariables admit some particular distributions.

On the exact survival probability by setting discrete random variables in E. Sparre Andersen's model