Non-existence of queues for system optimal departure patterns in tree networks

TL;DR

This paper proves that in tree networks with a single destination, optimal departure patterns can be achieved without queues, simplifying the problem to linear programming and eliminating deadweight losses.

Contribution

It establishes the non-existence of queues in system optimal departure patterns for tree networks, enabling LP reformulation of the problem.

Findings

Queues can be eliminated without increasing total delay costs

The MPCC can be transformed into a linear programming problem

Queues are deadweight losses in system optimal patterns

Abstract

This study proves the non-existence of queues for a dynamic system optimal (DSO) departure pattern in a directed rooted tree network with a single destination. First, considering queueing conditions explicitly, we formulate the DSO problem as mathematical programming with complementarity constraints (MPCC) that minimizes the total system cost which consists of the schedule and queueing delay costs. Next, for an arbitrary feasible solution to the MPCC, we prove the existence of another feasible solution where the departure flow pattern on every link is the same but no queue exists. This means that the queues can be eliminated without changing the total schedule delay cost. Queues are deadweight losses, and thus the non-existence theorem of queues in the DSO solution is established. Moreover, as an application of the non-existence theorem, we show that the MPCC can be transformed into a linear programming (LP) problem by eliminating the queueing conditions.