The Downs-Thomson Paradox: Existence, Uniqueness and Stability of User Equilibria

Abstract

Consider a network where two routes are available for users wishing to travel from a source to a destination. On one route (which could be viewed as private transport) service slows as traffic increases. On the other (which could be viewed as public transport) the service frequency increases with demand. The Downs-Thomson paradox occurs when improvements in service produce an overall decline in performance as user equilibria adjust. Using the model proposed by Calvert [10], with a ·|M|1 queue corresponding to the private transport route, and a bulk-service infinite server queue modelling the public transport route, we give a complete analysis of this system in the setting of probabilistic routing. We obtain the user equilibria (which are not always unique), and determine their stability.

User equilibria for a parallel queueing system with state dependent routing

“…We prove Lemma 3.1 by considering a joint process, (X (1) , X (2) ), with both X (1) and X (2) obeying the law of X D , and transition rates such that if one of the conditions I-II is satisfied at time 0 with (X (1) , X (2) …”

“…Define the joint process (X (1) , X (2) ) with state space S × S and transition rates as follows: 1 , (a 2 + 1) mod N), (b 1 , (b 2 + 1) mod N))…”

“…Then both X (1) and X (2) follow the law of X D , and the coupling here is such that the processes X (1) and X (2) make the same transition together whenever possible.…”

“…We need to show that for any pair of states ((a, b), H 0 ; 0) = inf t≥0 {t : X (i) (t) ∈ H 0 |(X (1) , X (2) )(0) = (a, b)} be the marginal reaching times to the set H 0 . We will usually write T D (n) for T D (n, H 0 ; 0) and ((a, b), H 0 ; 0), since we are interested in reaching times only to the set H 0 .…”

“…Both Calvert [11] and Ho [22] gave numerical examples showing that the Downs-Thomson effect can occur for this system. A later study, Afimeimounga, Solomon and Ziedins [2] gave a complete analysis of this system in the setting of probabilistic routing; see also that paper for additional references on probabilistic routing.…”