A Poisson distribution is a count distribution in the homogeneous Poisson process. A truncated Poisson distribution can be thought as a count distribution, thus the truncated Poisson processes can be defined if we know that upper limit number of occurrences in homogeneous Poisson process. For this reason, right truncated homogenous Poisson process has been defined and obtained distribution of waiting times. Then numerical examples are given to illustrate results.