  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.
