  We develop exact simulation (also known as perfect sampling) algorithms for a
family of assemble-to-order systems. Due to the finite capacity, and coupling
in demands and replenishments, known solving techniques are inefficient for
larger problem instances. We first consider the case with individual
replenishments of items, and derive an event based representation of the Markov
chain that allows applying existing exact simulation techniques, using the
monotonicity properties or bounding chains. In the case of joint
replenishments, the state space becomes intractable for the existing methods.
We propose new exact simulation algorithms, based on aggregation and bounding
chains, that allow a significant reduction of the state space of the Markov
chain. We also discuss the coupling times of considered models and provide
sufficient conditions for linear (in the single server replenishment case) or
quadratic (many server case) complexity of our algorithms in terms of the total
capacity in the system.
