 
Summary: KINETIC AND FLUID MODEL HIERARCHIES
FOR SUPPLY CHAINS
D. ARMBRUSTER, D. MARTHALER, AND C. RINGHOFER
MULTISCALE MODEL. SIMUL. c 2003 Society for Industrial and Applied Mathematics
Vol. 2, No. 1, pp. 4361
Abstract. We present a model hierarchy for queuing networks and supply chains, analogous to
the hierarchy leading from the many body problem to the equations of gas dynamics. Various possible
mean field models for the interaction of individual parts in the chain are presented. For the case of
linearly ordered queues the mean field models and fluid approximations are verified numerically.
Key words. supply chains, gas dynamics, kinetic equations
AMS subject classifications. 65N35, 65N05
DOI. 10.1137/S1540345902419616
1. Introduction. Prevalent models for complex manufacturing systems and
supply chains are based either on discrete event simulations or on fluid models. They
all describe the following generic situation: A sequence of products or parts arrives
in the system at arbitrary arrival times an, n = 1, 2, . . . , N. They pass from one node
(supplier) in the system to the next at rates determined by the orders from the down
stream supplier and leave the system at exit times en, n = 1, 2, . . . , N. The object of
modeling this system is to compute the dependence of the exit times on the arrival
times and consequently the load of the system or "work in progress," at any time t,
