© 2022, The Author(s), under exclusive license to Springer Nature Switzerland AG.The approximate stochastic simulation algorithms are the alternative methods to simulate the complex biological systems with a loss in accuracy by acquiring from computational demand. These methods depend on the leap condition. Here, the study aims to construct an actual and close confidence interval for the parameter denoting the number of simultaneously reaction in the system, by expanding the leap condition and the hazard function by second and third order Taylor expansion in the same time. To reach the goal, we use the poisson τ -leap and approximate Gillespie algorithm. Moreover, we derive the maximum likelihood estimators (MLE) and the method of moment estimators (MME) of the simulation parameters and construct confidence interval estimators at a given significance level α for these extended version of algorithms. Finally, we theoretically present that the obtained k can generate more narrower results [1–5, 7, 10].