  We consider multi-component matching systems in heavy traffic consisting of
$K\geq 2$ distinct perishable components which arrive randomly over time at
high speed at the assemble-to-order station, and they wait in their respective
queues according to their categories until matched or their ``patience" runs
out. An instantaneous match occurs if all categories are available, and the
matched components leave immediately thereafter. For a sequence of such systems
parameterized by $n$, we establish an explicit definition for the matching
completion process, and when all the arrival rates tend to infinity in concert
as $n\to\infty$, we obtain a heavy traffic limit of the appropriately scaled
queue lengths under mild assumptions, which is characterized by a coupled
stochastic integral equation with a scalar-valued non-linear term. We
demonstrate some crucial properties for certain coupled equations and exhibit
numerical case studies. Moreover, we establish an asymptotic Little's law,
which reveals the asymptotic relationship between the queue length and its
virtual waiting time. Motivated by the cost structure of blood bank drives, we
formulate an infinite-horizon discounted cost functional and show that the
expected value of this cost functional for the nth system converges to that of
the heavy traffic limiting process as n tends to infinity.
