Semi-Markov processes established a rich framework for many real-world problems. Semi-Markov processes include Markov processes, Markov chains, renewal processes, Markov renewal processes, Poisson processes, birth and death processes, and etc. Since semi-Markov processes describe more complex mathematical models of various objects, in many cases the obtained mathematical models are investigated numerically. But in some cases it is possible to study such mathematical models using analytical methods. In the presented paper, the authors chose to do the second way. In this paper we study the semi-Markov random walk processes with negative drift and positive jumps. The random variable or the moment, at which the process for the time reaches in zero level is introduced. An integral equation for the Laplace transform of conditional distribution of this random variable is obtained. In this paper length of jump is given by the gamma distribution with parameters $\alpha$ and $\beta$ resulting in a fractional order integral equation. In the class of gamma distributions, the resulting general integral equation of convolution type is reduced to a fractional order differential equation with constant coefficients. And also, the exact solution of the resulting fractional differential equation with constant coefficients has been found. Finally, using form of Laplace transform the expectation and variance of the random variable are found.

半马尔可夫过程为许多现实世界的问题建立了丰富的框架。半马尔可夫过程包括马尔可夫过程、马尔可夫链、更新过程、马尔可夫更新过程、泊松过程、生死过程等。 由于半马尔可夫过程描述了更复杂的各种对象的数学模型，在很多情况下得到的数学模型进行了数值研究。但在某些情况下，可以使用分析方法研究此类数学模型。在本文中，作者选择了第二种方式。在本文中，我们研究了具有负漂移和正跳跃的半马尔可夫随机游走过程。引入时间过程达到零水平的随机变量或时刻。得到了该随机变量条件分布的拉普拉斯变换积分方程。在本文中，跳跃长度由带参数的伽马分布给出$\alpha$ 和 $\beta$得到分数阶积分方程。在伽马分布类中，所得卷积类型的一般积分方程被简化为具有常数系数的分数阶微分方程。并且，已经找到了所得的具有常系数的分数阶微分方程的精确解。最后，利用拉普拉斯变换的形式求出随机变量的期望和方差。