  Max-infinitely divisible (max-id) processes play a central role in
extreme-value theory and include the subclass of all max-stable processes. They
allow for a constructive representation based on the pointwise maximum of
random functions drawn from a Poisson point process defined on a suitable
function space. Simulating from a max-id process is often difficult due to its
complex stochastic structure, while calculating its joint density in high
dimensions is often numerically infeasible. Therefore, exact and efficient
simulation techniques for max-id processes are useful tools for studying the
characteristics of the process and for drawing statistical inferences. Inspired
by the simulation algorithms for max-stable processes, theory and algorithms to
generalize simulation approaches tailored for certain flexible (existing or
new) classes of max-id processes are presented. Efficient simulation for a
large class of models can be achieved by implementing an adaptive rejection
sampling scheme to sidestep a numerical integration step in the algorithm. The
results of a simulation study highlight that our simulation algorithm works as
expected and is highly accurate and efficient, such that it clearly outperforms
customary approximate sampling schemes. As a by-product, new max-id models,
which can be represented as pointwise maxima of general location-scale mixtures
and possess flexible tail dependence structures capturing a wide range of
asymptotic dependence scenarios, are also developed.
