  The Gillespie algorithm provides statistically exact methods for simulating
stochastic dynamics modelled as interacting sequences of discrete events
including systems of biochemical reactions or earthquake occurrences, networks
of queuing processes or spiking neurons, and epidemic and opinion formation
processes on social networks. Empirically, the inter-event times of various
phenomena obey long-tailed distributions. The Gillespie algorithm and its
variants either assume Poisson processes (i.e., exponentially distributed
inter-event times), use particular functions for time courses of the event
rate, or work for non-Poissonian renewal processes, including the case of
long-tailed distributions of inter-event times, but at a high computational
cost. In the present study, we propose an innovative Gillespie algorithm for
renewal processes on the basis of the Laplace transform. The algorithm makes
use of the fact that a class of point processes is represented as a mixture of
Poisson processes with different event rates. The method is applicable to
multivariate renewal processes whose survival function of inter-event times is
completely monotone. It is an exact algorithm and works faster than a recently
proposed Gillespie algorithm for general renewal processes, which is exact only
in the limit of infinitely many processes. We also propose a method to generate
sequences of event times with a tunable amount of positive correlation between
inter-event times. We demonstrate our algorithm with exact simulations of
epidemic processes on networks, finding that a realistic amount of positive
correlation in inter-event times only slightly affects the epidemic dynamics.
