  We explore the use of Array-RQMC, a randomized quasi-Monte Carlo method
designed for the simulation of Markov chains, to reduce the variance when
simulating stochastic biological or chemical reaction networks with
$\tau$-leaping. The task is to estimate the expectation of a function of
molecule copy numbers at a given future time $T$ by the sample average over $n$
sample paths, and the goal is to reduce the variance of this sample-average
estimator. We find that when the method is properly applied, variance
reductions by factors in the thousands can be obtained. These factors are much
larger than those observed previously by other authors who tried RQMC methods
for the same examples. Array-RQMC simulates an array of realizations of the
Markov chain and requires a sorting function to reorder these chains according
to their states, after each step. The choice of sorting function is a key
ingredient for the efficiency of the method, although in our experiments,
Array-RQMC was never worse than ordinary Monte Carlo, regardless of the sorting
method. The expected number of reactions of each type per step also has an
impact on the efficiency gain.
