  A new method of the stochastic simulation algorithm (SSA), named the
Hashing-Leaping method (HLM), for exact simulations of a class of Markov jump
processes, is presented in this paper. The HLM has a conditional constant
computational cost per event, which is independent of the number of exponential
clocks in the Markov process. The main idea of the HLM is to repeatedly
implement a hash-table-like bucket sort algorithm for all times of occurrence
covered by a time step with length $\tau$. This paper serves as an introduction
to this new SSA method. We introduce the method, demonstrate its
implementation, analyze its properties, and compare its performance with three
other commonly used SSA methods in four examples. Our performance tests and CPU
operation statistics show certain advantage of the HLM for large scale
problems.
