Uniqueness of stochastic user equilibrium with asymmetric volume-delay functions for merging and diversion

Abstract

The main aim of this paper is to show how a new type of sufficient conditions can be used to prove uniqueness of a SUE model with non-separable arc cost-flow functions, even when their Jacobian is asymmetric and non-positive semi-definite. This apparently unusual setup for an assignment model permits to improve the representation of congestion in urban networks. Indeed, the supply models allowed by the standard uniqueness conditions, such as the monotonicity of separable cost-flow functions, can lack realism and thus may lead to wrong decision in the planning process. Actually, the main source of delay suffered by drivers when links are short is intersections, where vehicle flows conflict, competing to use the capacity of links ahead (merging), or are held back by other vehicles that are queuing (diversion). These traffic phenomena do not either lead to separable functions, or to symmetric Jacobians. A suitable supply model is then proposed to which the extended sufficient conditions are applied, showing that the uniqueness of the stochastic equilibrium can be proved also for more realistic volume-delay functions derived from traffic flow theory.

Appendix 1: Notation table

	Fixed point formulation of SUE
N	Set of network nodes
A ⊆ N × N	Set of network arcs
|A|	Number of arcs in the network
f a	Flow (volume) of arc a∈A; f a  = ∑ od∈D ∑ k∈Kod h k ×b ak
c a	Cost (generalized) of arc a∈A
f	(|A| × 1) vector of arc flows; f = B×h
c	(|A| × 1) vector of arc costs; c = c(f)
n	Vector of network parameters (e.g. arc capacities, free flow travel times, …)
ca(f; n), c = c(f)	Non-separable arc cost function
D ⊆ N × N	Set of demand od couples
d od	(Positive) demand flow from origin o∈N to destination d∈N, od∈D
K od	(Non-empty) set of elementary paths connecting o∈N to d∈N, od∈D
U k	Utility of path k∈K od (random variable)
p k	Probability of path k∈K od ; Pr[U k  ≥ U j ,∀ j∈K od ]
v k	Systematic utility of path k∈K od ; v k  = E[U k ]
ε k	Random residual of path k∈K od ; ε k  = U k  − v k
p k (v j , ∀ j∈K od )	Choice probability function of the generic path k∈K od
θ od	Parameter of the Logit model for the origin–destination couple od∈D
w k	Cost (generalized) of path k∈K od ; w k  = ∑ a∈A ca×b ak
b ak	b ak  = 1, if path k∈K od includes arc a, while b ak  = 0, otherwise
K = ∪ od∈D K od	Set of od paths
B	(|A| × |K|) arc-path incidence matrix, with generic element b ak
w	(|K| × 1) vector of the generalized costs; w = B T×c
h k	Flow of path k∈K od ; h k  = d od ×p k
h	(|K| × 1) vector of path flows
h = h(w)	Demand function; h k  = d od ×p k (−w j , ∀ j∈K od ), ∀ k∈K od and od∈D
d	(|D| × 1) vector of demand flows
Δ	(|D| × |K|) od-path incidence matrix, with generic element β(k∈K od )
S h	Set of feasible path flows; S h  = {h∈ℜ|K|: Δ×h = d, h ≥ 0}
f = f(c)	Network loading map; f(c) = B×h(B T×c)
S f	Set of feasible arc flows; S f  = {f∈ℜ|A|: f = B×h, h∈S h }
w = w(h)	Supply function; w(h) = B T×c(B×h)
S c	Set of feasible arc cost vectors; S c  = {c∈ℜ|A|: c = c(f), f∈S f }
β(x)	β(x) = 1, if x is true, while β(x) = 0, otherwise
	Uniqueness conditions of SUE
≽ (≼) 0	Applied to a square matrix, means that it is positive (negative) semi-definite
≻ (≺) 0	Applied to a square matrix, means that it is positive (negative) definite
||M||	Norm of square matrix M
λmax(M)	Largest eigenvalue of square matrix M
n = |K|-|D|	(Number of) reduced travel alternatives (or independent routes)
I −	Obtained from the identity matrix I |K| by removing the rows corresponding to the last path of each od pair
I°	Obtained from the identity matrix I |K| by removing the rows corresponding to all paths but the last path of each od pair
L	Space reduction matrix; L T = I −×(I |K| − Δ T×I°)
A(x)	Opposite of the demand function Jacobian in the reduced space of travel alternatives; A(x) = −I −×Jac[h(I −T×x)]×I −T
m	Network dimension parameter; m = λ max((B×L)T×(B×L))
α	Positive scalar that added to the diagonal of the arc cost function Jacobian makes it positive definite
	Static queue model
T	Duration of the demand peak (demand over capacity)
T 2	Duration of the congested period
e a	Effective capacity of arc a∈A
q a	Capacity of arc a∈A
g a	Green split of arc a∈A
π a	Priority of arc a∈A
ψ a	Reserved capacity to arc a∈A
s a	Sending flow of arc a∈A
	Volume delay function
l a	Length of arc a∈A
σ a	Free flow speed of arc a∈A
j a	Jam density of arc a∈A
ω a	Jam wave speed of arc a∈A
γ a	Shape factor of the fundamental diagram of arc a∈A
δ a	Fixed intersection delay of arc a∈A
χ a	Signal cycle (0, if there is no signal) of arc a∈A