Theoretical approach to congestion relaxation and saturated congestion in TASEP (totally asymmetric simple exclusion process) network with multi-chain section with flow/density-dependent rules of junctions

In this paper, we consider a TASEP (Totally Asymmetric Simple Exclusion Process) network with 1-in-n-out branching and n-in-1-out aggregation rules with density- and/or flow-dependent rules indirectly-dependent on time. From our extension of mean-field analysis to density- and/or flow- dependent rules, we can prove that the segments in chains do not have co-existence phase under the assumption that all segments in chains have the same phase in our rules. We show that the co-existence parabola of Down is independent of n, while the co-existence curve of Up is dependent on n, where n is the number of the chains. Also, the asymptotic curve of the co-existence curve of Up as n→∞ is a hyperbola in β−α plane, which is called 'Critical Hyperbola'. (α and β are the input and output probabilities, respectively.) The critical hyperbola is a boundary of the condition that all segments in the network are in HD phases, i.e., congested. Obviously, below the critical hyperbola, saturated congestion appears. We confirm the Critical Hyperbola is above of the Critical Ellipse we introduced in our previous papers. Moreover, these multiple-chain segments enable the network less congested than the single TASEP segment, which is called 'congestion relaxation'.