Modeling and stability of production lines based on arrival-to-departure delays

A general production system with L input lines and one output line is studied in terms of the arrival-to-departure delay times of parts being processed by the system. In particular, the degree of stability of the system is defined in terms of the D-stability set which consists of all part interarrival times for which the resulting delays are bounded. Assumptions are then given which result in an equivalence between bounded delays and boundedness of the number of parts in the system. Bounds resembling Little´s law are generated for the delay times and the number of parts in, the system. Then a special class of production systems is considered which can be modeled by dynamical equations defined over the max-plus algebra. This leads to the concept of the Γ-stability set (corresponding to a type of internal stability) which turns out to be contained in the D-stability set