 
Summary: SIAM J. APPL. MATH. c 2006 Society for Industrial and Applied Mathematics
Vol. 66, No. 3, pp. 896920
A MODEL FOR THE DYNAMICS OF LARGE QUEUING
NETWORKS AND SUPPLY CHAINS
D. ARMBRUSTER, P. DEGOND, AND C. RINGHOFER
Abstract. We consider a supply chain consisting of a sequence of buffer queues and processors
with certain throughput times and capacities. Based on a simple rule for releasing parts, i.e., batches
of product or individual product items, from the buffers into the processors, we derive a hyperbolic
conservation law for the part density and flux in the supply chain. The conservation law will be
asymptotically valid in regimes with a large number of parts in the supply chain. Solutions of this
conservation law will in general develop concentrations corresponding to bottlenecks in the supply
chain.
Key words. supply chains, conservation laws, asymptotics.
AMS subject classifications. 65N35, 65N05
DOI. 10.1137/040604625
1. Introduction. This paper is concerned with the development and analysis of
continuum models for supply chains. We consider a chain of M suppliers or processors
S0, . . . , SM1. In the generic picture of a supply chain (see cf. [12] for an overview)
each supplier processes a certain good (measured in units of parts) and passes it on
to the next supplier in the chain. Labeling the parts by the index n, we denote
