  Clearing functions (CFs), which express a mathematical relationship between
the expected throughput of a production facility in a planning period and its
workload (or work-in-progress, WIP) in that period have shown considerable
promise for modeling WIP-dependent cycle times in production planning. While
steady-state queueing models are commonly used to derive analytic expressions
for CFs, the finite length of planning periods calls their validity into
question. We apply a different approach to propose a mechanistic model for
one-resource, one-product factory shop based on the analogy between the
operation of machine and enzyme molecule. The model is reduced to a singularly
perturbed system of two differential equations for slow (WIP) and fast (busy
machines) variables, respectively. The analysis of this slow-fast system finds
that CF is nothing but a result of the asymptotic expansion of the slow
invariant manifold. The validity of CF is ultimately determined by how small is
the parameter multiplying the derivative of the fast variable. It is shown that
sufficiently small characteristic ratio 'working machines : WIP' guarantees the
applicability of CF approximation in unsteady-state operation.
