User equilibria for a parallel queueing system with state dependent routing

Abstract

Consider a system of two queues in parallel, one of which is a ·|M|1 single-server infinite capacity queue, and the other a ·|G (N ) |∞ batch service queue. A stream of general arrivals choose which queue to join, after observing the current state of the system, and so as to minimize their own expected delay. We show that a unique user equilibrium (user optimal policy) exists and that it possesses various monotonicity properties, using sample path and coupling arguments. This is a very simplified model of a transportation network with a choice of private and public modes of transport. Under probabilistic routing (which is equivalent to the assumption that users have knowledge only of the mean delays on routes), the network may exhibit the Downs-Thomson paradox observed in transportation networks with expected delay increasing as the capacity of the ·|M|1 queue (private transport) is increased. We give examples where state-dependent routing mitigates the DownsThomson effect observed under probabilistic routing, and providing additional information on the state of the system to users reduces delay considerably.Mathematics Subject Classification (2010) 90B15 · 60K25 · 90B20 · 91A25 · 91A10 · 91A13

Monotonicity properties of user equilibrium policies for parallel batch systems

“…Numerical examples show that for our network, increasing service rates may lead to greater delays under the user equilibrium. As in [2], in our examples this non-monotonicity effect is considerably reduced under state-dependent routing as opposed to probabilistic routing-increasing information given to the users may improve performance in this respect.…”

“…Thus (n 1 , N 2 − 1) ∈ D * 2 by Definition 2.5, and D * is 1-level monotone for queue 2. Moreover b (2) …”

“…This was first observed with Braess's paradox [9,10] with the addition of an extra route to the system, and later authors have shown similar paradoxical effects can occur when service rates are increased (see, for instance, Calvert [12], Downs [16], Thomson [28], and Afimeimounga et al [1,2] for a discussion of the Downs- Thomson paradox). Numerical examples show that for our network, increasing service rates may lead to greater delays under the user equilibrium.…”

“…Figure 1 shows the structure of the system. Applications of batch-service queues include examples of public transportation, such as airport shuttle buses that leave when full (Afimeimounga et al [1,2] and Jia [20]). A modified model where the batch service may commence before the batch is full is more generally applicable to bus and shared taxi services.…”

“…For their models a one-dimensional state representation of the system suffices to calculate expected delays, since delays at the infinite server queue do not depend on the length of the queue. Afimeimounga et al [1,2] studied a system that is perhaps most closely related to the one we consider here, consisting of a batch-service queue and an M|M|1 queue, under both probabilistic and state-dependent routing, which did require a two-dimensional state representation. For the state-dependent case they showed, as we will do here, the existence and uniqueness of the user equilibrium.…”