DYNAMIC MODEL OF NETWORK WITH
REAL TIME TRAFFIC INFORMATION:
QUEUE EQUILIBRIUM AND STABILITY ANALYSIS
LEONID ENGELSON
Centre for Traffic Simulation Research
Royal Institute of Technology, Stockholm
and
Inregia AB
Abstract
This paper proposes an analytical method for stability analysis of traffic flows in a traffic
network with an advanced traveller information system. The presented model describes
within-day development of queues when drivers affected by real-time traffic information
choose their paths en route. The model reduces to a system of differential equations with
delays and discontinuous right hand sides. Equilibrium points of the system correspond to
constant queue lengths and coincide with solutions to a variational inequality problem.
Sufficient conditions for existence of an equilibrium are obtained. In purpose to investigate
qualitative properties of the model, we extend some results of the theory of Projected
Dynamical System (PDS) to the delay equations. The problem of Lyapunov stability of PDS
with delay reduces to the stability of minimal face flow which is a standard dynamical system
with delay in a lower dimension linear space. This allows to analytically investigate stability
of queue equilibria. A traffic network example which illustrates that possibility is provided.
This research is financially supported by The Swedish Transport & Communications
Research Board (KFB) and The Swedish Agency for Innovation Systems (VINNOVA) under
grants 1995-0382 and 1999-0262.
1
1. Introduction
The problem setting
Traffic information systems make use of the advances in computer and communication
technologies to improve performance of the transportation network by providing the traveller
with real-time information on travel conditions. Using the information, drivers can en route
adjust their choices and make better routing decisions for the remaining part of journey, i.e.
make an adaptive route choice.
However, instantaneous traffic information becomes obsolete if the majority of drivers
switch to other routes in response to the information. Overreaction occurs when drivers'
reactions to the information cause congestion to transfer from one road to another. This may
result in oscillation of traffic flows between alternative routes. Possibility of such oscillations
under shortest-time path guidance was discussed by Ben-Akiva et al. (1991) and repeatedly
supported by results of simulations, e.g. by Hall (1996). Based on result of a series of
experiments with the HUTSIM simulation tool, The 'INCOME' Project (1998) demonstrated
that dynamic route guidance may cause unstable situations where the traffic is assigned in
turns to two routes and that percentage of equipped vehicles, drivers' compliance and the
travel time feedback delay are important factors which affect the strength of oscillations.
To completely avoid the oscillations, the traffic information system should produce a
message that is consistent with the actual traffic conditions experienced by the drivers, i.e. it
should perform a recursive, or self-fulfilling forecast that takes into account changes in traffic
situation caused by dissemination of the message (see more detailed discussion in Engelson,
1997). Attempts to produce such a forecast meets however computational problems connected
with convergence of algorithms, see Bottom et al. (1999).
The source of the possible instability lies in the inevitable delay between the time when
the traffic conditions at a site are observed and the time when drivers, affected by the
information on the traffic conditions, arrive at the site and themselves influence traffic
conditions. Indeed, without delays the system would stabilise at some queue length values
because of the negative feedback between the queue lengths and drivers' route choices. In
fact, delays consists of time for information collection, processing, dissemination and the time
it takes for the drivers to go from the location where the information is received to the site the
information deals with. As a result, the flow arriving to a congestion site may still increase
during a considerable time after an extraordinary queue is detected, and still decrease after
dissolution of the queue. This eventually causes flow oscillations of increasing amplitude.
Up to now, investigation of flow stability under adaptive route choice was performed
mainly using simulation experiments. An analytical approach is preferable because it allows
general and trustworthy conclusions on qualitative properties of the system. Moreover, it
saves extensive time and human resources required for simulation experiments.
This paper analytically addresses the stability issue by means of a continuous-time
dynamic network model taking into account route updating during a trip. Our specific interest
is existence and stability of steady states which correspond to fixed queue lengths and travel
times.
Overview
Smith (1984) was probably the first to consider the problem of stability of traffic
assignment. Recently, models of day-to-day adjustment process of network flows were
2
presented and stability of these processes were investigated by Friesz et al. (1994), Nagurney
and Zhang (1998) and Cantarella and Cascetta (1995). However, none of those models
assumed adaptive route choice.
The time of driving from the place the information is received to the site of issue,
depends on drivers' location at the time of decision. Therefore the drivers' ability to perform
adaptive route choice should be treated explicitly when stability of the flows is investigated.
To consider the adaptive route choice, one has to model the drivers' routing decisions at each
node. Ran and Boyce (1996) presented a family of dynamic equilibrium models based on this
approach, gave various formulations of dynamic equilibria in the form of variational
inequalities and optimal control problems and proposed algorithms for calculation of the
equilibria. The complicated form of the model prevents analytic investigation of asymptotic
behaviour of flows.
Hoogendoorn (1997) discussed optimal control of a variable message sign (VMS) located
before a road bifurcation with two alternative paths and a bottleneck on each path.
Development of queues and time lags corresponding to travel times from the VMS until the
bottlenecks were taken into account. The optimal control problem was considered on a
bounded time interval and a method for approximate solution based on discrete time
approximation was proposed.
Methodology
We present a generalisation of Hoogendoorn's model to a more general network and
driver decision frame. The path travel times are determined by lengths of traffic queues at a
fixed set of bottlenecks. Drivers equipped with in-vehicle devices make an adaptive route
choice based on continually-received real-time information on current queues. The flow
arriving to a bottleneck is determined by the repetitive routing decisions. Following Weymann
et al. (1994) and Kuwahara and Akamatsu (1997), we consider the reactive dynamic
assignment with the link travel times explicitly taking into account the effects of queues under
the deterministic point queue concept.
The model reduces to a system of differential equations with time delays and with queue
lengths as state variables. The equilibrium states of the system, called queue equilibria,
correspond to constant queue lengths at all bottlenecks. The right-hand sides of the equations
are discontinuous due to the very nature of a queue. This prevents direct application of the
well developed theory of delay equations. In order to investigate the stability, we extend to the
delay case some results of the recent theory of Projected Dynamical Systems (PDS) presented
by Nagurney and Zang (1996).
The starting point for the theory was a Variational Inequality Problem (VIP). Dupuis and
Nagurney (1993) proposed an ordinary differential equation whose stationary points coincide
with solutions to the VIP. Nagurney and Zhang (1996) defined PDS as a family of solutions
to the equation and, in the case of VIP corresponding to a traffic network equilibrium problem
with fixed demand, interpreted the PDS as a continuous time route choice adjustment process
whereby the travellers between a OD pair switch from more costly routes to less costly routes
connecting this OD pair.
In this article, a new application of PDSs for transportation problems is presented. In the
case, when OD demands are constant, we describe the queue development by a Projected
Dynamical System with Delay (PDSD). The time delays make behaviour of PDSD more
complicated than that of PDS. For instance, monotonicity of the relevant function in the right
hand side of the equation does not imply stability. The problem of stability of PDSD can
3
however be reduced to the stability of minimal face flow which is a standard dynamical
system with delay in a linear subspace. This allows an analytical investigation of the stability
of queue equilibria.
Outline of the paper
In section 2, the dynamic model of traffic network with a driver information system is
constructed and formulated as a system of differential equations with delays and
discontinuous right hand sides. Existence and uniqueness of a solution to the system is
proved. Section 3 introduces and discusses the concept of queue equilibrium (QE) which is
obtained as a steady state of the system. It is shown that QEs coincide with solutions to a
variational inequality problem. Existence of the equilibrium is proved under certain
consistency conditions. Section 4 suggests a method of investigation stability of PDSD which
consists in reducing the problem to the stability of an ordinary delay equation on a subspace.
In section 5, the dynamic model, the queue equilibrium concept and the method of stability
investigation are illustrated by a simple network example. Section 6 formulates conclusion
and proposes directions for future research.
Notation
Sets
J
Set of all nodes in the network
L
Set of all links in the network
D
Set of all regular nodes in the network
B
Set of all bottlenecks in the network
Lm
Set of links leaving node m
E mbj
Set of efficient paths starting at node m, ending at node j and passing through
bottleneck b
E m* j
Set of efficient paths starting at node m and ending at node j
l
E mj
Set of efficient paths starting at node m with link l and ending at node j
Gmj
Set of simple paths starting at node m and ending at node j
Cipmj
Path choice set for those equipped drivers with destination j which have come
from origin i along path p to node m
l
Cipmj
Set of all paths from the path choice set Cipnj which leave the node m by link l
p(r , k , m)
Sub-path of cycle-free path r contained between nodes k and m
N
Set of all positive integers
R
Set of all real numbers
R+
Set of all non-negative real numbers
Rn
Euclid space of n-dimensional vectors with real components
R +n
Set of all n-dimensional vectors with non-negative components
4
Set of all continuous vector functions ϕ:[ − TM ,0] → R+n
C+
Constants
n
The total number of bottlenecks in the network
cb
Capacity of bottleneck b
τl
Travel time along link l
s( l )
Starting node of link l
τ0
Time period for collection and processing of travel time information
A
Total number of elements in set A
The maximal travel time along a simple path from a decision node to a
bottleneck
TM
Traffic Variables
Ab ( t )
Traffic flow arriving to bottleneck b
dij (t )
Flow of vehicles led by equipped drivers aiming to destination j and leaving
origin i at time t
qijbr (t )
Flow of vehicles led by equipped drivers with origin i and destination j
arriving at node b at time t along the path r from i to b
λ (t ; i , r , m, j , l ) Splitting rate at regular node m: the share of equipped drivers arriving at time t
at node m along path r from origin node i and aiming to destination j which
exit node m through link l
vb ( t )
Flow of vehicles led by unequipped drivers and arriving at bottleneck b
xb (t )
Queue length at bottleneck b
Tr (t )
The instantaneous travel time along the whole path r
5
2. Model of network
This section presents a dynamic model of queue development in a traffic network where
drivers choose their paths en route on the base of real-time information about travel times
along the paths. The model is summarised in the form of system of differential equations with
delays and discontinuities in the right hand sides.
Network elements and assumptions
Consider a traffic network represented by a directed graph with a finite set of nodes J and
finite set of links L ⊂ J × J where never (i , i ) ∈ L . Each node is either a regular node or a
bottleneck. The sets of regular nodes and bottlenecks are denoted by D and B respectively.
The regular nodes represent road intersections and activity centres and can serve as trip
origins and trip destinations. The bottlenecks represent the places on roads where capacity is
restricted and queues may develop. Each bottleneck has just two adjacent links - one entering
the bottleneck and one leaving the bottleneck1. A bottleneck can be neither origin nor
destination of any trip.
A sequence of contiguous links starting at a regular node is called a path if it does not
contain any cycles, i.e. subsequences that start and end at the same node. A path traverses a
node if it contains a link entering the node and a link exiting the node. A path is efficient if it
does not traverse more than one bottleneck. A simple path is a path that does not traverse any
bottleneck.
Each link has a constant travel time (cruise time). Passing through a node, except a
bottleneck, does not take any time. Hence any simple path has a fixed travel time equal to sum
of the link travel times. A passing time through a bottleneck, i.e. queuing time, is a dynamic
variable which depends on queue length and the bottleneck capacity. The concept of vertical
queue is adopted for the movement of vehicles through the bottleneck. Effects of queue
propagation to the upstream intersection is not considered in this article.
In the rest of the paper, the geometry of network and location of the bottleneck are
assumed to satisfy the following
Assumption 2.1. There exists an efficient path between any distinct pair of regular
nodes.2
Two types of drivers are considered - equipped with an information system terminal and
unequipped ones. The arriving flows of unequipped drivers at the bottlenecks are treated as
exogenous traffic variables.
Each equipped driver has a fixed origin and a fixed destination. Starting from the origin,
the driver chooses an efficient path to the destination he intends to follow and starts the trip.
1
It is well known that network capacity problems usually appear at intersections and a
queue develops on a link entering such an intersection. In that case, the capacity problem can
be represented by splitting the link and introducing a bottleneck node close to the intersection.
2
Although assumption on constant travel times except the restricted set of bottlenecks is
not fully realistic, it can be adopted for a study of route choice oscillations e.g. during the
morning peak hour if there are bottlenecks just at entrances to a city.
6
The choice of path is made upon the instantaneous travel times along efficient paths toward
the destination and other path characteristics which are constant. The instantaneous travel
times depend on current queue conditions and are reported to the equipped drivers by an
information system with some delay due to information collection and processing.
When the next node is reached, the driver revises his path and chooses a new intended
path to the destination. To make the choice, the driver uses new information about
instantaneous path travel times. The choice of intended path is repeated at each node the
driver passes through until the destination is reached.
It is well known that the real drivers do not consider all possible alternatives when the
path choice is made. At a decision node, the driver chooses a path from a choice set which is a
subset of all possible paths from the current node to the destination. In our model, the choice
sets are specified in the following two assumptions.
Assumption 2.2. For an equipped driver who has not passed through any bottleneck yet,
the choice set consists of such efficient paths from the current node to the driver’s destination
that do not have any common nodes, apart from the current one, with the path already passed
from the driver’s origin to the current node.
Assumption 2.3. For an equipped driver who has already passed through a bottleneck,
the choice set consists of such simple paths from the current node to the driver’s destination
that do not have any common nodes, apart from the current one, with the path already passed
from the driver’s origin to the current node.
It is easy to show that any path which can be realised by consequent choices from the
appropriate choice path sets specified by Assumptions 2.2 and 2.3 is efficient.3
Splitting rates
Denote Cipmj the path choice set from regular node m to destination j for an equipped
driver that has come from origin i of his trip to node m along efficient path p. Let k be the
number of paths in the path choice set.
Behaviour of equipped drivers is described by the family of path splitting functions
r
(Τ ) is the share of equipped
[ ] which are given in the model. The value Sipmj
r
Sipmj
: R+k → 0,1
drivers who choose path r ∈ Cipmj , where the argument vector Τ ∈ R+k consists of the last
reported travel times along the paths in Cipmj .
Note that the drivers are not generally supposed to minimise any time or cost criteria.
Besides the travel time, the path choice may depend on constant path specific attributes like
3
The assumption on avoidance of paths which go through nodes that the driver has
passed earlier has been imposed to guarantee that the set of all realised paths from an origin to
a bottleneck is finite. The assumption can be relaxed without affecting truthfulness of the
subsequent theory. For example, one can assume instead that the drivers avoid paths from the
current node to the destination which have more than one (or two) common links with the
path already passed and/or relate that condition only to the drivers which have not passed
through any bottleneck yet. It also possible to reformulate the whole theory in the framework
of path enumeration.
7
travel distance, number of turns or scenery. Those attributes can be incorporated in definitions
of the path splitting functions.
Let l be a link starting from regular node m. The share of equipped drivers arriving at
time t at node m along efficient path p from origin node i and aiming to destination j which
exit node m through link l is equal to
λ (t ; i , p, m, j , l ) =
∑l
({
r
S ipmj
Tu (t − τ 0 ); u ∈ Cipmj
r ∈Cipmj
})
(2.1)
l
where Cipmj
consists of paths in the choice set starting with link l, Tu (t − τ 0 ) is the
instantaneous travel time for path u at time t − τ 0 reported by the information system at time t
and τ 0 is the constant time period for information collection and processing.
Path travel time
Let εrb = 1 if path r goes through bottleneck b, εrb = 0 otherwise. Due to the
assumption on constant link travel times, the travel time by efficient path r is
Tr =
∑ τl
l ∈r
+
∑ εrb xb
b ∈B
cb
(2.2)
where xb and cb are queue length and capacity of bottleneck b, respectively.
Equation for queue length
Denote Ab (t ) the traffic flow arriving to bottleneck b. We consider the bottleneck as a
restriction with constant service rate (capacity) cb and use the deterministic fluid
approximation of traffic. It follows from (Daganzo, 1997, p. 31), that
x b (t ) = π + ( xb (t ), Ab (t ) − cb )
(2.3)
where overdot denotes the derivative with respect to time and the function π + of two real
arguments is defined as
δ ,
π + ( x, δ ) = 
 0,
if x > 0 or δ > 0
.
if x ≤ 0 and δ ≤ 0
The queue length changes with the rate equal to the difference between the total arriving
flow and the bottleneck capacity, until the queue vanishes. When the queue length is zero, it
will remain so until the arriving flow exceeds the capacity.
8
Equation for flow arriving at the bottleneck
The flow of vehicles arriving at the bottleneck b is obtained as a sum of two flows,
namely the flow of vehicles led by unequipped drivers v b ( t ) and the flow of vehicles led by
equipped drivers. Consequently,
Ab ( t ) =
∑ ∑ ∑ qijbr (t ) + vb (t )
(2.4)
i ∈D j ∈D r ∈Gib
where Gib is the set of all simple paths starting at node i and ending at bottleneck b
and qijbr (t ) is the flow of vehicles led by equipped drivers with origin i and destination j,
arriving at the bottleneck b at time t by path r from i to b.
To calculate the flow arriving by path r with the origin i and the destination j, one has to
multiply the total travel demand for the corresponding OD pair by splitting rates at the first
and all intermediate nodes along the path to ensure that only flow leaving a node by the link
on path r is included.
The travel demand and the splitting rates at nodes on path r should be calculated at
previous time instants, when the flow arriving at bottleneck b at time t leave the
corresponding nodes. To find such an instant, one should reduce t by travel time by the
corresponding sub-path of path r. The resulting equation is
(
q ijbr ( t ) = d ij ( t − T p ( r , i , b ) ) ∏ λ t − T p ( r , s ( l ), b ) ; i , p (r , i , s ( l ) ), s ( l ), j , l
l ∈r
)
(2.5)
where
d ij (t ) is the flow of vehicles led by equipped drivers aiming to destination j and leaving origin
i at time t,
s(l ) is the starting node of link l,
p(r , i , m) is the sub-path of path r between nodes i and m,
Tp is the travel time along the whole path p.
Network model in the form of a differential equation with delay.
Denote n the total number of bottlenecks in the network. Substituting (2.2) into (2.1) and
(2.5); (2.1) into (2.5); (2.5) into (2.4); (2.4) into (2.3), one obtains the system of equations
(
(
)
)
xDb ( t ) = π + xb (t ),− f b t , X ( t − T1 ) ,..., X ( t − TM ) − cb ,
b ∈B ,
(2.6)
where X (t ) = ( x1 (t ),..., x n (t )) , and the delays T1 ,..., TM are travel times from decision
nodes to bottlenecks plus the constant time τ 0 for collection, processing and dissemination of
the traffic information. Note that there is a finite number of delays because there is just a
finite number of realised paths from origins to the bottlenecks (see the note at the end of
T
9
subsection 2.1). Moreover, those delays are constant as the paths are simple. Without lost of
generality, we can assume that 0 < T1 < T2 <... < TM .
Specific form of functions f b depends on structure of the network, path splitting
functions, OD-demand and arriving flows of unequipped drivers to the bottlenecks (see
example in section 5). All these elements are given in the model. The minus before fb is
inserted for the convenience of subsequent exposition.
The whole model can be summarised as one equation in R n :
(
) )
(
X (t ) = Π X ( t ),− f t , X ( t − T1 ),..., X ( t − TM ) − c
where
(
f = ( f 1 ,..., f n ) , Π ( x1 ,..., x n ) , (δ1 ,..., δn )
T
c = (c1 ,..., cn ) .
T
(2.7)
T
) = (π ( x ,δ ),...,π ( x ,δ ))
+
1 1
+
n
n
T
and
T
Existence and uniqueness of a solution
The equation (2.7) is a differential equation with time delays, belonging to a wider class
of functional differential equations (cf. Hale and Verduyn Lunel, 1993). However, the
conventional assumption that the right hand side of the equation is continuous by the
unknown function, is not satisfied in the case of equation (2.7). On the other hand, due to the
operator Π in the right hand side, the equation (2.7) resembles the class of ordinary
differential equations studied by Nagurney and Zhang (1996).
As a delay equation, the equation (2.7) requires an initial function to be given on the
[
]
maximal delay interval − TM ,0 to determine a solution. On the other hand, discontinuity of
the right hand side allows only to define a solution in the Caratheodory sense.
[
]
Let ϕ: − TM ,0 → R+n be a fixed continuous function.
[
[
We say that function X : − TM ,+∞ → R+n is a solution to the equation (2.7) with initial
condition
[
]
X (t ) = ϕ (t ), t ∈ − TM ,0
(2.8)
if (2.8) holds, X (⋅) is absolutely continuous and satisfies (2.7) on [ −TM ,+∞[ , save on a set of
Lebesgue measure 0.
Existence and uniqueness of a solution to the initial value problem (2.7), (2.8) can be
proved under very weak conditions on the function f.
Theorem 2.1. Assume that the function f is measurable and bounded on any bonded set.
[
[
Then there is a unique solution X : − TM ,+∞ → R+n to the equation (2.7) with initial
condition (2.8).
10
[
]
Proof. Define X (t ) = ϕ (t ) for t ∈ − TM ,0 . The solution is constructed further step by
[
]
step. Namely, for t ∈ 0, T1 , define
(
)
g( t ) = f t , X (t − T1 ),..., X (t − T M ) + c
(2.9)
Since function ϕ is continuous, premise of the theorem implies that function g is
measurable and bounded. By Theorems 2 and 3 in Dupuis and Nagurney (1993; see also
[
]
Remark on pp. 25-26), there exists a unique solution Y (⋅) on 0,T1 to equation
Y ( t ) = Π K (Y ( t ) ,− g (t ))
(2.10)
]
]
satisfying the conditionY (0) = X (0) . Define X ( t ) = Y (t ), t ∈ 0, T1 . Obviously, X (t )
[
]
solves (2.7), (2.8) on − TM , T1 .
[
]
Now define g( t ) by (2.9) on T1 ,2T1 and solve (2.10) with initial value Y (T1 ) = X (T1 )
obtaining unique solution to (2.7), (2.8) on − TM ,2T1 etc.
[
]
Such steps define by induction a solution to the problem (2.7), (2.8) on interval
[ −TM ,+∞[ . The solution is unique since it necessarily satisfies the equation (2.10) with
function g defined by (2.9).
This proves the theorem.
r
are measurable and the functions dij and vb are
Note that, if the functions Sipnj
measurable and bounded on each bounded interval, then the function f satisfies the required
conditions,
and
(2.7),
(2.8)
admits
a
unique
solution.
11
3. The Projected Dynamical System with Delays and Queue equilibrium
In this section, the concepts of a Projected Dynamical System with Delays (PDSD) and
Queue Equilibrium (QE) are defined and the characterisation of QE as solutions to a
variational inequality problem is given. Next, the QE is compared with other network
equilibrium concepts. Then sufficient conditions for existence of QE are established.
Projected Dynamical System with Delays
Consider the autonomous case of equation (2.7) corresponding to constant OD demands
d ij and constant flows of unequipped drivers vb :
(
) )
(
X ( t ) = Π X ( t ) ,− f X (t − T1 ),..., X (t − T M ) − c .
(3.1)
The equation (3.1) defines a dynamic system in the space C+ of all continuous vector
functions ϕ:[ − TM ,0] → R+n . Indeed, let Φ (ϕ , t )(⋅) be a function in C+ defined by
Φ (ϕ , t )(θ ) = X (t + θ )
[
]
where X (⋅) is a solution to (3.1) with the initial condition X (t ) = ϕ (t ), t ∈ − TM ,0 .
In analogy with Nagurney and Zhang (1996), we call the mapping Φ: C+ × R+ → C+ a
Projected Dynamical System with Delays (PDSD) generated by equation (3.1).
Note that the definition of PDSD can be extended to the much more general case of
equation with distributed delay.
Definition and characterisation of queue equilibria
Consider an fixed point of the PDSD generated by equation (3.1), i.e. a function ϕ ∈C+
such that Φ (ϕ , t ) = ϕ for all t ≥ 0 . It is easy to see that ϕ is necessary a constant function.
Since components of vector X are queue lengths, the following definition is motivated.
A vector X ∗ ∈ R+n is a queue equilibrium (QE) if the constant vector function
X (θ ) ≡ X ∗ is a fixed point of the PDSD generated by (3.1).
(
)
In other words, X ∗ = x1∗ ,..., x n∗ is a QE if the following condition is satisfied: if during
a TM long time interval the queues are constant and equal to x1∗ ,..., x n∗ then it will remain so
forever.
In the following, we give a characterisation of QE via a variational inequality. In order to
do that, introduce a vector function h: R+n → R n by
h( X ) = f ( X ,..., X ) .
(3.2)
12
Theorem 3.1. Let X ∗ ∈ R+n . The following conditions are equivalent:
(i)
X ∗ is a QE;
(
)
(ii) Π X ∗ ,− h( X ∗ ) − c = 0 ;
(iii) X ∗ is a solution to the variational inequality
h( X *) + c, X − X * ≥ 0 ∀X ∈ R+n .
(3.3)
Proof. Equivalence of (i) and (ii) follows directly from the definition of QE. Equivalence
of (ii) and (iii) is due to Lemma 1 in Dupuis and Nagurney (1993).
Comparison with other concepts of network equilibrium
It is useful to compare the concept of QE with conventional equilibrium concepts. First,
note that since travel demand, link characteristics and queue lengths are constant, the link and
path flows determined by those factors are constant as well. Hence QE is a kind of static
equilibrium.
The main difference between QE and static user/system equilibria is that the former is the
state of queues while the latter are the states of flow rates. In our model, queue lengths
unequivocally determines flow rates by the path travel times and the path splitting functions.
However, the flow rates do not generally determine the queue lengths. Indeed, the queue
length can take any fixed value when the arriving flow rate is equal to the bottleneck capacity.
Thus it is not possible to judge if the network is in QE state when only the flow rates are
given. One has also to know the queue lengths.
In the contrary, to formulate the definition of deterministic static user equilibrium
(DSUE) assignment, one has first to suppose that the path flow rates unequivocally determine
path travel times by a family of static volume-delay functions. Here the path travel time do
not generally determine the flow rates. In DSUE, the drivers distribute themselves by routes
in such a way that all used routes between a specific OD pair have the same cost (travel time)
which is not higher that the cost (travel time) of any unused route between the same OD pair.
Thus it is not possible to judge if the network is in DSUE state when only the travel times are
given. One has also to know the flow rates.
Our approach does not suggest equal cost or travel time for used routes between the same
OD pair, but distribution of drivers by routes is governed by path splitting functions which are
defined at all decision nodes.
When the travel times are constant, one can aggregate the path splitting functions at
decision nodes to obtain path splitting functions for the whole paths from origin to
destination. For each OD pair, the aggregated function will determine path shares depending
on path travel times. The stochastic static user equilibrium (SSUE) can also be interpreted in
such a way, with an aggregated path splitting function defined as probability that perceived
travel time along a specific path is less than along other paths. Definition of SSUE requires
also volume-delay functions while QE is formulated with the help of capacity restrictions.
Both DSUE and QE can be considered as limit cases of the generalised SSUE where the
aggregated path splitting functions are defined for all routes depending on path travel times
13
but not necessary minimising the perceived travel time. Indeed, DSUE is obtained at
extremely high sensitivity of path choice on difference in path travel time while QE is result
of extra sensitive volume-delay functions.
Existence of QE
A sufficient condition of existence of QE will be proved using the variational inequality
formulation (3.3).
To express the vector function h( ⋅) in a convenient way, remind that hb ( X ) = − Ab
where Ab should be expressed via X from (2.4), (2.5), (2.1) and (2.2) assuming constant
queue lengths xb , OD demands dij and flows of unequipped drivers vb . Consequently, the bth component of vector function h is
hb ( X ) = − ∑
∑ dij Zijb ( X ) − vb
(3.4)
j ∈D i ∈D
where the bottleneck split functions Zijb : R+n → R , i , j ∈ D, b ∈ B , are defined by equation
Zijb ( x1 ,..., x n ) =
∑ ∏ λ 0 (i , p (r , i , s ( l ) ), s ( l ), j , l )
(3.5)
r ∈Gib l ∈r
where the path shares λ0 depend on x1 ,..., x n by
λ 0 (i , p, m, j , l ) =
∑
({
r
Sipmj
Tu ; u ∈ Cipnj
l
r ∈Cipmj
})
(3.6)
and
Tu =
∑ τ l + ∑ εub xb
l ∈u
b∈B
cb
(3.7)
Obviously the value Zijb ( X ) is the bottleneck share, i.e. the share of equipped drivers going
from origin i to destination j which pass through the bottleneck b. Note that
∑ Zijb ( X ) ≤ 1
b∈B
∀i , j ∈ D, X ∈ R+n .
(3.8)
The following notations and assumptions are necessary for formulation of the theorem on
sufficient conditions of existence of QE. For W ⊂ D × D , denote β (W ) the set of all
bottlenecks on efficient paths connecting OD pairs belonging to W. For I ⊂ B , denote W(I)
the set of such OD pairs that any efficient path connecting the pair goes through a bottleneck
belonging to I.
Assumption 3.1. For any non-empty set W ⊂ W ( B) ,
14
∑
(i , j ) ∈W
d ij <
∑(
cb − vb
b ∈β (W )
).
Assumption 3.2. If I ⊂ B and X m ∈ R+n , m ∈ N is such a sequence of vectors that
( )
xbm → +∞ for b ∈ I and xbm converges for b ∈ B − I , then Zijb X m → 0 for b ∈ I ,
(i , j ) ∈ D × D − W ( I ) .
Define residual capacity of a bottleneck as its capacity minus the arriving flow of
unequipped drivers to the bottleneck.
Assumption 3.1 means that the total travel demand of equipped drivers for a set of OD
pairs is less than the total residual capacity of all bottlenecks the efficient paths between these
OD pairs go through. This condition concerns only sets consisting of OD pairs between which
no path without bottlenecks exist. Assumption 3.2 concerns the bottleneck split functions and
can be interpreted in the following way. If queue on an efficient path connecting an OD pair
grows infinitely while queue on another efficient path is bounded (or there is no bottleneck on
the latter path) then the share of equipped drivers choosing the former path vanishes.
Theorem 3.2. Let the functions Zijb , i , j ∈ D, b ∈ B be continuous on R+n and residual
capacities of all bottlenecks be positive. Assume Assumptions 3.1 and 3.2. Then a QE exists.
Proof of the theorem is presented in Appendix A.
The Assumptions 3.1, and 3.2 might be difficult to test directly. They have however
easily testable sufficient criteria that are presented below.
Proposition 3.1. Suppose there exists a distribution by paths of the total demand of
equipped drivers for each OD pair, such that flow arriving to any bottleneck is less than the
residual capacity. Then Assumption 3.1 holds.
Proof. Denote Ei * j the set of all efficient routes from i to j. Premise of the proposition
means that there exist flows zr such that
∑
∀ i, j ∈ D
zr = dij
r ∈Ei * j
(3.9)
and
∑ ∑ ∑ zr < cb − vb ∀b ∈ B .
i ∈D j ∈D r ∈E
(3.10)
ibj
Let W ⊂ W0 . Then
Ei * j =
t Eibj = t Eibj ∀(i , j ) ∈W .
b ∈B
b ∈β (W )
Hence
15
∑
(i , j ) ∈W
d ij =
∑
∑ zr = ∑
(i , j ) ∈W r ∈Ei* j
∑
∑ zr ≤ ∑
(i , j ) ∈W b ∈β (W ) r ∈Eibj
∑
∑ zr < ∑ (cb − vb ) ,
b ∈β (W ) (i , j ) ∈D × D r ∈E ibj
b ∈β (W )
Q.E.D.
Thus, to establish Assumption 3.1 it is enough to check that the system of linear
equations and inequalities (3.9), (3.10) has a solution.
Note that the premise of Proposition 3.1 with the words "less than" replaced by the words
"no more than" becomes a necessary condition of QE. The required distribution is provided
by network flows under QE.
To formulate a sufficient condition for Assumption 3.2, note that properties of the
r
,
bottleneck split functions Zijb are determined by properties of the path splitting functions Sipnj
see equations (3.5) and (3.6).
Recall from section 2 that Cipmj is the path choice set from regular node m to destination
j for an equipped driver that has come from origin i to node m by the simple path p. A
r
is now formulated.
property of path splitting functions Sipmj
Assumption 3.3. Let i , m, j ∈ D, p ∈ Gim , U ⊂ Cipmj , U ≠ Cipmj and k =| Cipmj | . Let
(
µ
µ
Τ µ = T1 ,..., Tk
) ∈R
k
+
, µ ∈ N , be such a sequence of travel time vectors that Trµ → +∞
( )
r
for all r ∈U and Trµ converges for all r ∈ Cipmj − U . Then Sipmj
Τ µ → 0 for all r ∈U .
This property means that, at each decision node, when travel time on some paths from the
choice set tend to some constants, the share of drivers which choose paths with infinitely
increasing travel times will tend to zero.
In particular, the random utility model of path choice, with observed path utilities defined
as unbounded increasing functions of path travel time, delivers path splitting functions that
conform with Assumption 3.3. For instance, path choice behaviour specified by the logit
model
r
Sipnj
=
exp( −γTr )
∑ exp(−γTu )
u ∈Cipnj
satisfies Assumption 3.3.
We will now formulate the proposition which provides sufficient conditions for required
properties of bottleneck split functions.
r
Proposition 3.2. Assume that the path splitting functions Sipnj
satisfy Assumption 3.3.
Then the bottleneck splitting functions defined by (3.5) and (3.6) satisfy Assumption 3.2.
The formal proof of this proposition, though straightforward, requires cumbersome
manipulations with indexes and is therefore omitted.
Finally, Theorem 3.1 and Propositions 3.1 and 3.2 imply the following
16
r
Corollary 3.1. Let the functions Sipnj
( i , n, j ∈ D, p ∈ Gin , r ∈ Cipnj ) be continuous and
satisfy Assumption 3.3. Assume that there exists a distribution by paths of the total demand of
equipped drivers for each OD pair, such that flow through any bottleneck is less than the
residual capacity. Then a QE exists.
17
4. Stability of the Projected Dynamical System with Delays
Section 3 presented the concept of queue equilibrium which is the equilibrium state of
the differential equation (4.1) describing queue development in a road network. In this
section, we initialise the study of dynamic stability of queue equilibria. Information on
stability is necessary for discrimination between those stable equilibrium states that are
anticipated to occur realistically and those unstable to be eliminated for consideration in
applications. When asymptotic stability of an equilibrium is shown, one can deduce that
oscillations around the equilibrium, if any, will vanish.
An extensive literature exists on methods of stability investigation of equations with
delays. The theory assumes that the right hand side of the equation continuously depends on
the state. Zhang and Nagurney (1995) (ZN) studied stability of Projected Dynamical Systems
(PDS) without delays. They have shown that the problem of stability of PDS can be reduced
to the one of stability of a classical dynamical system on a lower dimension linear space. On
the other hand, ZN have proved that stability of PDS follows from monotonicity of the
relevant vector function.
Some results from ZN are transferable to the PDSD. In particular, we show that the
properties of stability and asymptotic stability of PDSD at a regular solution of corresponding
VIP are inherited from minimal face flow induced by the PDSD. Consequently, stability of
PDSD can be exploited through the classical stability theory of differential equations with
delays.
In contrast with PDS, monotonicity of the relevant vector function does not imply
stability of PDSD. That is shown by the example in section 5.
Definitions of Stability
Consider the autonomous delay equation
(
) )
(
XD ( t ) = Π X ( t ) ,− f X (t − T1),..., X (t − TM ) − c ,
Assume
{
that
function
C+ = ϕ ϕ:[− TM ,0] →
R+n , ϕ
f
satisfies
}
is continuous
ϕ ∈ C+ . For any ϕ ∈ C+ , denote X
ϕ
(4.1)
conditions
and
of
Theorem
2.1.
Denote
ϕ = sup{|ϕ (θ )|:−TM ≤ θ ≤ 0}
for
the unique solution of (4.1) satisfying the initial
condition
[
]
X ϕ (t ) = ϕ (t ), t ∈ − TM ,0 .
(4.2)
(
)
Let X ∗ ∈ R+n be a QE, i.e. Π X ∗ ,− h( X ∗ ) − c = 0 where h is defined by (3.2).
For convenience, we recall the following definitions concerning stability study of
retarded functional differential equations (cf. Hale and Lunel, 1993).
18
The equilibrium state X ∗ of equation (4.1) is said to be stable if for any ξ > 0 there is a
δ > 0 such that ϕ ∈ C+ , ϕ − X ∗ < δ
implies
ϕ
X ( t ) − X ∗ < ξ for all t ≥ 0 . The
equilibrium state X* of equation (4.1) is said to be asymptotically stable if it is stable and
ϕ
there exists a δ 0 > 0 such that ϕ ∈ C+ , ϕ − X ∗ < δ0 implies X ( t ) → X ∗ as t → +∞ . The
equilibrium state X* of equation (4.1) is a finite time attractor if there exists a δ 0 > 0 such
ϕ
that if ϕ ∈ C+ , ϕ − X ∗ < δ 0 then there is a T ∈ R+ such that X ( t ) = X ∗ when t ≥ T .
Local Properties under Regularity
In the following, we assume the subsequent assumptions.
Assumption 4.1. X* is a regular solution to variational inequality (3.2) in the sense of
ZN. In the actual case with R+n as the feasible region this condition means that, at QE X*,
arriving flows to bottlenecks without queues are strictly less than the bottleneck capacities.
r
Assumption 4.2. The path splitting functions Sipmj
are continuous.
Note that the last assumption implies continuity of the vector function f.
For
{
X = ( x1 ,..., x n ) ∈ R+n ,
any
I 0 ( X ) = {b: xb = 0} ,
denote
}
and
S ( X ) = Y = ( y1 ,..., y n ): yb = 0 ∀b ∈ I 0 ( X ) . Denote z1( X ) and z2 ( X ) orthogonal
projections of the vector X-X* on the linear subspace S(X*) and its orthogonal complement
S ⊥ ( X *) respectively: z1( X ) = PS ( X *) ( X − X *) , z2 ( X ) = PS ⊥ ( X *) ( X − X *) .
The following lemma is obtained from Lemma 3.1 in ZN.
Lemma 4.1. There exists an α>0 such that for all X ∈ R+n
h( X *) + c, z2 ( X ) ≥ α | z2 ( X )| .
Let B( x , ε ) denote the open ball with radius ε centred in x.
The next lemma follows from the proof of Theorem 3.3 in ZN.
Lemma
4.2.
There
exists
a
λ>0
such
that
B( X *, λ ) ∩ S ( X *) ⊂ R+n
and
Π ( X , Y ) = PS ( X *) Y for any X ∈ B( X *, λ ) ∩ S ( X *) and any Y ∈ B( − h( X *) − c, α 2) where
α is the constant provided by Lemma 4.1.
As projection PS ( X *) is continuous, it is easy to show by the method of steps that, for
any continuous initial function ϕ:[ − TM ,0] → B(0, ε ) ∩ S ( X *) , there is a unique solution to
the equation
( (
) )
zD(t ) = − PS ( X *) f z( t − T1 ) + X *,..., z( t − TM ) + X * + c
which is a delay equation in the subspace S(X*).
19
(4.3)
The family of solutions to equation (4.3) is called minimum face flow (MFF) induced by
equation
(4.1)
at
the
equilibrium
point
X*.
Due
to
Lemma
4.2,
PS ( X *) ( − h( X *) − c) = Π( X *,−h( X *) − c) = 0 , i.e. 0 is an equilibrium state of equation (4.3).
Moreover, a solution to (4.3) starting with an initial function whose values lie in the
set B( 0, ε ) ∩ S ( X *) with ε small enough, is identical to a solution of (4.1) by a translation
from origin to X*.
It is shown below that stability of equation (4.1) is determined by stability of equation
(4.3). To prove that, one needs to show that solutions to (4.1) starting in a neighbourhood of
X* arrive to B( X *, ε ) ∩ S ( X *) and stay there for at least time TM . That is confirmed by the
following two lemmas.
Lemma 4.3. For any ε > 0 , there exists a δ > 0 such that for any continuous function
[
]
ϕ:[ − TM ,0] → B ( X *, δ ) ∩ K there is a T0 ∈ 0, 2 z 2 (ϕ (0)) α such that
X ϕ (T0 ) ∈ B( X *, ε ) ∩ S ( X *) and
[
X ϕ ( t ) ∈ B( X *, ε ) for t ∈ 0, T0
(4.4)
]
(4.5)
where α is the constant provided by Lemma 4.1.
Proof. Proof of the part i) is conducted with the similar arguments as that of Lemma 3.2
in ZN. The part ii) can be proved the same way as the similar assertion in the proof of
Theorem 3.4 in ZN.
Lemma 4.4. For any ς > 0 and ∆ > 0 , there exists an ε > 0 such that if a solution x( ⋅)
to the equation (4.1) satisfies
[
x( t ) ∈ B( X *, ε ) for t ∈ − TM , T0
]
(4.6)
and
x(T0 ) ∈ B( X *, ε ) ∩ S ( X *)
(4.7)
with some T0 ≥ 0 , then
[
]
x( t ) ∈ B ( X *, ς ) ∩ S ( X *) for t ∈ T0 , T0 + ∆ .
(4.8)
Proof of this lemma is presented in Appendix B.
Stability of PDSD and the Corresponding MFF
The major results of this section are now ready for presentation. First, it is pointed out by
the following theorem that a PDSD has the best stability around regular solutions to the
corresponding VIP when they are extreme points of the polyhedron K.
20
Theorem 4.1. If X*=0 is a regular solution to the VIP(H, R+n ), then it is a finite time
attractor for the PDSD (4.1) with a continuous function F.
Finally, we establish theorems which allows to investigate stability of PDSD using the
corresponding MFF.
Theorem 4.2. Assume that X* is a regular solution to the VIP(H, R+n ). Then stability of
equilibrium point X* of the equation (4.1) is equivalent to stability of equilibrium point 0 of
the delay equation (4.3) in the subspace S(X*).
Theorem 4.3. Assume that 0 is a regular solution to the VIP(H, R+n ). Then asymptotic
stability of equilibrium point X* of the equation (4.1) is equivalent to asymptotic stability of
equilibrium point X* of the equation (4.3).
Theorems 4.1, 4.2 and 4.3 are easily proved using Lemmas 4.3 and 4.4 and uniqueness of
solution to (4.1). The proofs are given in Appendix C.
The meaning of results of this section is following. Suppose that a QE is found. If, at
equilibrium, flow arriving at each bottlenecks is strictly less than the residual bottleneck
capacity, then the equilibrium is a finite time attractor. Consequently, when all the queues are
small enough on the whole initial time interval, the queues disappear in a finite time.
If, at equilibrium, queues at some bottlenecks are positive while at all other bottlenecks
arriving flow is strictly less than the residual bottleneck capacity (the regularity condition),
then stability of the equilibrium can be studied by fixing queues to 0 at the latter set of
bottlenecks and investigation stability of the resulting delay equation with continuous right
hand side.
21
5. Example
The dynamic network model presented in section 2 can be illustrated using the example
network with two bottlenecks in Fig. 5.1. Assume that OD travel demand of equipped drivers
consists of d 35 (t ) and d 45 (t ) .
l1
3
1
l5
l7
l4
l3
5
l8
4
l6
2
l2
Fig. 5.1. Example network with two bottlenecks.
There are two efficient paths r1 = {l1, l5} , r 2 = {l4 , l2 , l6} between OD pair (3,5) and two
efficient paths r 3 = {l3 , l1, l5} , r 4 = {l2 , l6} between OD pair (4,5). Suppose the path splitting
functions be of logit type:
(
(
))
1
S3r∅
35 = 1 / 1 + exp γ ( T1 − T2 ) ,
(
(
))
2
S3r∅
35 = 1 / 1 + exp γ ( T2 − T1 ) ,
(5.1)
(
(
))
3
S4r∅
45 = 1 / 1 + exp γ (T3 − T4 ) ,
(
(
4
S4r∅
45 = 1 / 1 + exp γ ( T4 − T3 )
))
where T1 , T2 , T3 , T4 are travel times along respective paths and γ is the common
sensitivity parameter. Then equations (2.14) for the bottlenecks 1 and 2 take the following
form:





,
d35 ( t − τ1 )
d45 ( t − τ1 − τ 3 )
xD1 (t ) = π +  x1 (t ),
+
+ v1 (t ) − c1
  x1( t − τ1 ) x2 ( t − τ1 )
  x1( t − τ 1 − τ 3 ) x2 ( t − τ1 − τ 3 )




1 + expγ 
−
− θ1  1 + expγ 
−
− θ2  


c1
c2
c1
c2
 
 
 
 


22






d35 ( t − τ 2 − τ 3 )
d45 ( t − τ 2 )
xD2 (t ) = π +  x2 (t ),
+
+ v2 (t ) − c2 
  x2 ( t − τ 2 − τ 3 ) x1( t − τ 2 − τ 3 )
  x2 ( t − τ 2 ) x1( t − τ 2 )




1 + expγ 
−
+ θ1  1 + expγ 
−
+ θ2  


c
c
c
c








2
1
2
1


where θ1 = τ 4 + τ 2 + τ 6 − τ1 − τ 5 , θ2 = τ 2 + τ 6 − τ 3 − τ1 − τ 5 , and τ k is the fixed travel time
over link lk , k = 1,...,6 .
Now let OD demand d 35 and d 45 and flows of unequipped drivers v1 , v 2 through the
bottlenecks be constant. The path splitting functions (5.1) obviously satisfy the assumption
3.3. By Proposition 3.2 and Theorem 3.2, a queue equilibrium state exists if c1 > v1 , c2 > v 2
and
d 35 + d 45 < c1 + c2 − v1 − v 2 .
(5.2)
To find it, one has to assume x1 , x 2 constant and to solve the system of equations
 x1 = 0

or
 x > 0
1

 x2 = 0
or

 x2 > 0
and
E1( x1, x2 ) ≤ 0
and
E1( x1, x2 ) = 0;
and
E2 ( x1, x2 ) ≤ 0
and
E2 ( x1, x2 ) = 0
(5.3)
where
E1 ( x1, x2 ) =
[
d35
d45
]
+
1 + exp γ ( x1 c1 − x2 c2 − θ2 )
]
+
d45
1 + exp γ ( x1 c1 − x2 c2 − θ1)
[
+ v1 − c1
(5.4)
+ v2 − c2 .
(5.5)
]
and
E2 ( x1, x2 ) =
[
d35
1 + exp γ ( x2 c2 − x1 c1 + θ1)
[
1 + exp γ ( x2 c2 − x1 c1 + θ2 )
]
It follows from (5.2) that
E1 ( x1 , x 2 ) + E 2 ( x1 , x 2 ) = d 35 + d 45 + v1 − c1 + v 2 − c2 < 0 ,
hence E2 ( x1 , x2 ) < 0 when E1 ( x1 , x2 ) = 0 and E1 ( x1 , x2 ) < 0 when E2 ( x1 , x2 ) = 0 . Since
E1 ( x1 , x 2 ) decreases by x1 and increases by x 2 , so the equilibrium state X * = ( x *1 , x *2 ) is
unique and its location is determined by the following conditions:
if E1 (0,0) ≤ 0 and E 2 (0,0) ≤ 0 then X * = ( 0,0) ;
23
(5.6)
If E1 (0,0) > 0 then X * = ( x *1 ,0) where x *1 solves E1 ( x *1 ,0) = 0 ;
(5.7)
If E 2 (0,0) > 0 then X * = (0, x *2 ) where x *2 solves E 2 (0, x *2 ) = 0.
(5.8)
This can be interpreted in the following way.
Call normal the flows which establish in the network when the information on absence of
queues in continually reported to the equipped drivers. If the residual capacity of each
bottleneck admits the normal arriving flow, then the queue equilibrium is the state of absence
of queues. If the residual capacity of the first (second) bottleneck is not enough to let freely
pass the normal arriving flow, then the queue equilibrium suggests a positive constant queue
at the first (second) bottleneck and no queue at another one.
Now
turn
to
(
the
problem
)
H ( x1 , x 2 ) = − E1 ( x1 , x 2 ), E 2 ( x1 , x 2 )
N (0) =
{( x , x ) : x
T
1
2
1
}
of
T
.
stability.
In
the
The
function
case
(5.6),
is
H
equal
S ( 0) = { 0}
to
and
≤ 0, x2 ≤ 0 . The solution 0 to the corresponding VIP is regular if
E1 (0,0) > 0 and E 2 (0,0) > 0 . In this case, the Theorem 4.1 ensures that the queue
equilibrium is a finite time attractor, i.e. queues, if any, dissolve in a finite time.
In the case (5.7), the solution to equation E1 ( x1 ,0) = 0 is
x1∗
=
c1
γ
where
ln
(
)
w + w 2 − 4(c1 − v1 − d 35 − d 45 )(c1 − v1 ) exp − γ (θ1 + θ 2 )
(
)
2(c1 − v1 ) exp − γ (θ1 + θ 2 )
(
)
w = d 35 exp( − γθ 2 ) + d 45 exp( − γθ1 ) − (c1 − v1 ) exp( − γθ1 ) + exp( − γθ 2 ) . Here
we have S ( X *) =
{( x ,0) : x ∈ R} . The solution
T
1
1
( )
X * = x1∗ ,0
to the VIP is regular if
E 2 ( x1∗ ,0) > 0 . In this case, the Theorems 4.2 and 4.3 reduce the problem of stability to the
investigation of stability of null solution to the one-dimensional delay equation
zD(t ) =
[
1 + exp γ
((
d 35
)
z t − τ1 c1 + x1∗ c1 − θ1
)]
+
[(
d 45
1 + exp γ z( t − τ1 − τ 3 ) c1 + x1∗ c1 − θ2
)]
+ v1 − c1 (5.9)
where z(t ) = x1 (t ) − x1∗ .
The first-order approximation equation for (5.9) is
zD(t ) = −az( t − τ1 ) − bz( t − τ1 − τ 3 )
(5.10)
with the characteristic equation
p + a exp( − pτ1 ) + b exp( − pτ1 − pτ 3 ) = 0
(5.11)
24
where
a=
(
γ d 35 exp γ x1∗ c1 − γθ1
[
(
c1 1 + exp γ
x1∗
c1 − γθ1
)
)]
2
, b=
(
γ d 45 exp γ x1∗ c1 − γθ2
[
(
c1 1 + exp γ
x1∗
c1 − γθ2
)
)]
2
.
By Krasovskii (1963), the null solution to the equation (5.9) is asymptotically stable if all
solutions to the equation (5.11) have negative real parts, and unstable if there is a solution to
(5.11) with positive real part. A method presented by Pinney (1958) can be used to identify
the stability area in the (a,b)-plane. For instance, in the case τ1 = τ 3 = 01
. , the stability area is
shown in Fig. 5.2 where the separating curve Γ is given by
a=−
y cos( 0.2 y )
sin( 01
. y)
, b=
y
tan( 01
. y)
, 0 < y < 10π .
b
20
15
Unstable
10
5
Stable
0
-20
-15
-10
-5
a
0
5
10
15
20
-5
Unstable
-10
-15
-20
. ) − bz(t − 0.2) .
Fig. 5.2. Stability area of the equation zD(t ) = −az(t − 01
Varying γ for fixed d 35 = d 45 = 1000 , v1 = v2 = 1000 , c1 = 1700, c2 = 2400 ,
τ 1 =... = τ 6 = 01
. , it is easy to calculate that (a,b) leaves the stability area at γ 0 = 127.16 ,
That is, the queue equilibrium is stable for γ < γ 0 and unstable for γ > γ 0 . As expected,
increasing sensitivity worsens stability of the equilibrium.
Note that the vector function H is strictly monotone and locally strongly monotone at x*
for any combination of parameter values. Hence stability of PDSD does not follow from the
monotonicity of H.
25
6. Conclusion
In this paper, a new dynamic network model is proposed. The model explicitly takes into
account a driver information system which supports route switching decisions en-route. It has
been shown that the model is suitable for analytic investigation of stability of route flow
oscillations eventually arising due to time delays in the system Traffic Conditions →
Information → Driver Behaviour → Traffic Conditions.
The proposed methodology presents a new application of the recent theory of PDS for
dynamic transportation models. That theory has been applied to the transportation problems
by Nagurney and Zhang (1996). Note that, beside PDS, there are other dynamical systems
whose steady states coincide with solutions to the VIP. For example, Friesz et al. (1994)
developed a "global projective dynamics" approach based on a tatonnement adjustment
process of both path flows and perceived OD costs. Their differential equations with
continuous right hand sides describe anticipatory behavior on the part of drivers in avoiding
any constraint boundary (Friesz et al., 1998). Furthermore, Cantarella and Cascetta (1995)
proposed a family of deterministic and stochastic discrete processes which, due to travellers'
learning, converge to traffic equilibrium.
It is not possible to deduce a correct dynamic model based on the information about the
steady states only. The knowledge about the route choice adjustment process has to be taken
from the real world. Mahmassani (1990) addressed the issue experimentally by investigation
of real commuters' route choice in a simulated environment with various information supply
strategies and found that the system converged to a steady state, albeit at different rates.
All those approaches describe the day-to-day route adjustment process. On the other
hand, equation (2.7) describes within-day development of queues when drivers choose their
paths to destinations on the base of current queue lengths.
As pointed by Friesz et al. (1996), the traffic flows result from people decisions which
are made with very different frequencies. Therefore dynamic of the flows should be modelled
using a hierarchy of processes of various 'speeds'. So the changes of OD-demand are slow as
compared with the dynamics of queues considered in this paper and therefore it is appropriate
to consider a 'moving queue equilibrium' corresponding to consecutive states of the demand
adjustment process.
There are several directions in which the initiated research can be developed. It is useful
to analyse robust stability and stability under stochastic perturbation of the queue equilibrium.
This would allow the equation for queue length based on the point queue approach to be
replaced by more realistic link volume conservation equation. The stability study can be
continued in the non-autonomous case with variable OD travel demand.
The presented model can be formulated in more general framework that allows multiple
bottlenecks on a route. This will result in equations with delay size dependent on the unknown
function. Development of a mathematical theory of stability of such equations is an
interesting and useful task.
References
Ben-Akiva M., de Palma A. and Kaysi I. (1991) Dynamic network models and driver
information systems. Transp. Research, 25A(5), 251-266.
26
D. E. Boyce (1988) Contributions of transportation network modelling to the
development of a real-time route guidance system. Paper presented at the First International
Conference on Transportation for the Future, Sweden, May 1988.
P. H. L. Bovy and E. Stern (1990). Route Choice: Wayfinding in Transportation
Networks, Kluwer Academic Publishers, Dordrecht.
G. E. Cantarella and E. Cascetta (1995). Dynamic Processes and Equilibrium in
Transportation Networks: Towards a Unifying Theory, Transportation Science, Vol. 29, pp.
305-329.
C. F. Daganzo (1997). Fundamentals of Transportation and Traffic Operations,
Pergamon, Oxford.
P. Dupuis and A. Nagurney (1993). Dynamical Systems and Variational Inequalities.
Annals of Operarions Research, Vol. 44, pp. 9-42.
T. L. Friesz, D. Bernstein, N. J. Mehta, R. L. Tobin and S. Ganjazadeh (1994). Day-today Dynamic Network Disequilibria and Idealized Traveler Information Systems, Operations
Research, Vol. 42, pp. 1120-1136.
T. L. Friesz, D. Bernstein and R. Stough (1996). Dynamic Systems, Variational
Inequalities and Contril Theoretic Models for Predicting Urban Network Flows,
Transportation Science, Vol. 30, pp. 14-31.
T. L. Friesz, S. Shah and D. Bernstein (1998). Disequilibrium Network Design: A New
Paradigm for Transportation Planning and Control, in: L. Lundqvist, L.-G. Mattsson, T. J. Kim
(Eds.) Network Infrastructure and the Urban Environment, Springer-Verlag, Berlin, pp. 99111.
J. K. Hale and S. M. Verduyn Lunel (1993). Introduction to Functional Differential
Equations, Springer-Verlag, New York.
R. W. Hall (1996). Route Choice and Advanced Traveller Information Systems on a
capacitated and Dynamic Network. Transp. Research, 4C(5), 289-306.
S. P. Hoogendoorn (1997). Optimal control of dynamic path information panels. Proc. 8th
IFAC Symposium on Transportation Systems, Chania, Greece.
The 'INCOME' Project (1998). Strategies for Driver Information Systems and Urban
Traffic Control in Gothenburg (Deliverable 16). European Commission, Directorate General
for Transport (DGVII).
D. E. Kaufman, R. L. Smith, K.E. Wunderlich (1998). User-equilibrium properties of
fixed points in dynamic traffic assignment, Transp. Res., 6C(1-2), 1-16.
N. N. Krasovskii (1963). Stability of Motion, Stanford University Press, Stanford,
California.
M. Kuwahara and T. Akamatsu (1997). Decomposition of the reactive dynamic
assignments with queues for a many-to-many origin-destination pattern, Transp. Res., 31B(1),
1-10.
H. Mahmassani (1990). Dynamic Models of Commuter Behavior: Experimental
Investigation and Application to the Analysis of Planned Traffic Disruptions. Trans. Res.,
Vol. 24A, pp. 465-484.
A. Nagurney (1998). Network Economics: A Variational Inequality Approach, Kluwer
Academic Publishers, Boston.
27
A. Nagurney and D. Zhang (1996). Projected Dynamnical Systems and Variational
Inequalities with Applications, Kluwer Academic Publishers, Boston.
A. Nagurney and D. Zhang (1998). Introduction to Projected Dynamnical Systems for
Traffic Network Equilibrium Problems, in: L. Lundqvist, L.-G. Mattsson, T. J. Kim (Eds.)
Network Infrastructure and the Urban Environment, Springer-Verlag, Berlin, pp. 125-156.
E. Pinney (1958). Ordinary difference-differential equations, Univ. Californ. Press,
Berkeley-Los Angeles.
B. Ran and D. Boyce (1996). Modeling Dynamic Transportation Networks. SpringerVerlag, Berlin.
T. E. Smith, T. L. Friesz, D. H. Bernstein and Z.-G. Suo (1997) A comparative analysis
of two minimum-norm projective dynamics and their relationship to variational inequalities,
in: Complementarity and variational problems, SIAM, Philadelphia, pp. 405-424.
J. Weymann, J.-L. Farges and J.-J. Henry (1994). Dynamic Path Guidance with Queueand Flow-Dependent Travel Time, Transp. Res., 2C(3), 165-183.
D. Zhang and A. Nagurney (1995). On the stability of Projected Dynamical Systems, J.
Optimization Theory and Appl., 85(1), 97-124.
28
Appendix A.
{
}
Proof of Theorem 3.2. Let Km = X ∈ R+n : X ≤ m , m ∈ N . Then Km is non-empty, convex
and compact. It follows from continuity of functions Zijb and equality (3.4) that function h is
continuous. By Theorem 1.4 in Nagurney (1993), there exists a solution Y ∈ Km to the
variational inequality
h(Y ) + c, X − Y ≥ 0 ∀X ∈ Km .
(A.1)
Denote the solution X m . We claim that X m < m for some m.
Assume the contrary, i.e. X m ≥ m ∀m ∈ N . Then one can select by the diagonal process
a sub-sequence of X m with some components positive and tending to infinity while all other
components converge to some numbers. Formally, denoting the sub-sequence as X m , there is
a set of indexes I ⊂ B , I ≠ ∅ such that xbm → +∞ and xbm > 0 for b ∈ I , while xbm
converges for b ∈ B − I .
Substituting X = X m − xbmℑb into (A.1), where ℑb ∈ R n denotes the vector with 1 in the
( ( ) )(
)
b-th location and 0 elsewhere, one concludes that hb X m + cb ⋅ − xbm ≥ 0 , which implies
( )
that hb X m + cb ≤ 0 for b ∈ I , i.e. (by (3.4))
∑ ∑ dij Zijb ( X m ) ≥ cb − vb
∀m ∈ N , b ∈ I .
(A.2)
i ∈D j ∈D
On the other hand, it follows from (3.8) that
( )
∑
dij ∑ Zijb X m ≤
∑ dij .
( i , j ) ∈W ( I ) b∈I
( i , j ) ∈W ( I )
(A.3)
If W ( I ) ≠ ∅ then Assumption 3.2 is applicable and
∑
( i , j ) ∈W ( I )
dij <
∑ ( cb − vb ) ≤ ∑ ( cb − vb )
b ∈β ( W ( I ) )
(A.4)
b ∈I
where the last inequality follows from the fact that β (W ( I ) ) ⊂ I . If W ( I ) = ∅ then inequality
( )
(A.4) is true too because residual capacities are positive. Assumption 3.2 yields Zijb X m → 0
for b ∈ I and (i , j ) ∈ D × D − W ( I ) , hence for m large enough
∑
( i , j ) ∈D × D −W ( I )
dij
∑ Zijb ( X m ) < ∑ ( cb − vb ) −
b ∈I
b ∈I
29
∑
( i , j ) ∈W ( I )
dij
(A.5)
because the right hand side is positive due to (A.4). Adding up (A.3) and (A.5), one obtains
∑ ∑ dij ∑ Zijb ( X m ) < ∑ (cb − vb )
i ∈D j ∈D
b ∈I
b∈I
that contradicts (A.2). Hence, the claim is correct.
By Theorem 1.5 in Nagurney (1993), there exists a solution X * ∈ R+n to the variational
inequality (3.2). By Theorem 3.1, this yields a queue equilibrium state, Q.E.D.
30
Appendix B.
Proof of Lemma 4.4. Let ε 0 > 0 be less than ς , less than λ provided by Lemma 4.2 and
so small that I ( X ) ⊂ I ( X *) when | X − X *|< ε0 , and f ( X1,..., X M ) − h( X *) < α 2 when
| X i − X *| < ε0 for i=1,…,M, where α is the constant provided by Lemma 4.1.
Denote µ an integer larger than ∆ T1 . For k=1,…,µ, choose ε k < ε k −1 2 and such that
f ( X1,..., X M ) − h( X *) < ε k −1 2 τ
(B.1)
when | X i − X *|≤ ε k for i=1,…,M.
We claim that, for any k=1,…,µ, if T ≥ 0 ,
x (t ) ∈ B( X *, ε k ) for t ∈[ − TM , T ]
(B.2)
x( T ) ∈ S ( X *)
(B.3)
and
then
x( t ) ∈ B( X *, ε k −1) ∩ S ( X *) for t ∈ T , T + T1 .
[
]
(B.4)
{
}
Indeed, if it is not true, then v := inf t ∈ [T , T + τ ] : x(t ) ∉ B( X *, ε k −1 ) ∩ S ( X *)
∈]T , T + τ [ due to (B.3) and continuity of x( ⋅) . As ε k −1 ≤ ε 0 < λ , we have
x( t ) ∈ B( X *, λ ) ∩ S ( X *) for t ∈[T , v ]
(B.5)
where λ is provided by Lemma 4.2. It follows from (B.2) that
| x (t − Ti ) − X *| ≤ ε k < ε0
(B.6)
for i=1,…,M and t ∈[T , v ] and therefore
f ( x(t − T1),..., x( t − TM )) − h( X *) < α 2
(B.7)
by the choice of ε0 . Applying Lemma 4.2, it follows from (B.5) and (B.7) that
Π K x (t ),− f ( x (t − T1),..., x (t − TM )) = − PS (0) f ( x (t − T1),..., x (t − TM )) for t ∈[T , v ] . Hence
(
)
31
v
(
)
x (v ) = x (T ) + ∫ Π K x (t ),− f ( x (t − T1 ),..., x (t − TM )) − c dt
T
v
(
)
= x (T ) − ∫ PS (0) f ( x (t − T1 ),..., x (t − TM )) − c dt
(B.8)
T
Due to (B.6) and (B.1), we have f ( x (t − T1),..., x (t − TM )) − h( X *) < ε k −1 2 T1. Since
− PS ( X *) (h( X *) + c) = Π( X *,− h( X *) − c) = 0 , (B.2) and (B.8) yield
v
x (v ) ≤ x (T ) + ∫ PS ( X *) f ( x (t − T1 ),..., x (t − TM )) − PS ( X *) h( X *) dt
T
< ε k + ( v − T ) ⋅ ε k −1 2T1 < ε k −1 .
Let η > 0 be so small that
x (t ) < ε k −1 for t ∈[ v , v + η ] .
(B.9)
By the definition of v, there is w ∈[v , v + η] such that x ( w) ∉ B( X *, ε k −1) ∩ S ( X *) .
Hence, by (B.9),
x ( w) ∉ S ( X *) .
(B.10)
Assumption (B.2), the definition of v and (B.9) imply that x (t ) ∈ B( X *, ε k −1) for
t ∈[ − TM , w] . Hence x (t ) < ε0 and x (t − Ti ) < ε0 for t ∈[T , w] , i=1,…,M. By the choice of
ε0 , both
I ( x (t )) ⊂ I ( X *)
(B.11)
and (B.7) are true for t ∈[ T , w] . It follows from the proof of Lemma 3.2 in ZN that (B.11)
and (B.7) together imply
(
)
2
d
z2 ( x (t )) 2 ≤ 0 .
dt
(B.13)
In view of (B.3), we have z2 ( x ( T )) = 0 and from (B.13) z2 ( x ( w)) = 0 which contradicts
(B.10). Hence the claim is correct.
Set ε = ε µ and let x( ⋅) be a solution to (4.1) satisfying (4.6) and (4.7). Then (B.2) and
(B.3) are satisfied for k=µ and T = T0 . Using the claim, we obtain
32
[
]
x( t ) ∈ B ( X *, ε µ −1) ∩ S ( X *) for t ∈ T0 , T0 + T1 .
(B.14)
Now (B.2) and (B.3) are satisfied for k=µ−1 and T = T0 + T1 , and the claim together with
(B.14) yields x( t ) ∈ B( X *, ε µ − 2 ) ∩ S ( X *) for t ∈ T0 , T0 + 2T1 . After µ steps, one has
[
[
]
]
x( t ) ∈ B( X *, ε0 ) ∩ S ( X *) for t ∈ T0 , T0 + µT1 which provides (4.8) due to the choice of ε0
and µ, Q.E.D.
33
Appendix C.
Proof of Theorem 4.1. Using continuity of f, choose ε > 0 less than λ provided by
Lemma 4.2 and so small that f (u1 ,..., u M ) ∈ B(h(0),α 2 ) when u i < ε , i = 1,..., M . By
Lemma 4.3, there exists a δ 0 > 0 such that if ϕ ∈ C + , ϕ < δ 0 then there is a T ∈ R+ such
that X ϕ (T ) = 0 and X ϕ (t ) < ε for 0 ≤ t ≤ T .
Let Y (t ) be a vector function coinciding with X ϕ (t ) for − T M ≤ t ≤ T and equal to 0
for
t>T.
Then
for
t≥T
we
Y (t − T i ) < ε ,
have
f (Y (t − T1 ),..., Y (t − TM )) ∈ B(h(0),α 2 )
and,
by
Π (Y (t ),− f (Y (t − T1 ),..., Y (t − TM ))) = − PS (0) f (Y (t − T1 ),..., Y (t − TM
i = 1,..., M ,
hence
Lemma
4.2,
)) = 0 because S (0) = {0}.
Thus Y (t ) is a solution to the equation (4.1). Due to uniqueness of solution, X ≡ Y and the
theorem is proved.
Proof of Theorem 4.2. Suppose the steady point X* of the equation (4.1) is stable and let
ξ > 0 . Choose so small η > 0 that f (u1 ,..., u M ) − h( X *) < α 2 when u i < ε , i = 1,..., M
where α is the constant provided by Lemma 4.1. Let ε0 = min( ξ , η, λ ) where λ is provided by
Lemma 4.2. There exists a δ > 0 such that for each initial function ϕ ∈ C + , ϕ − X * < δ , the
solution X ϕ (t ) to (4.1) never leaves B( X *, ε 0 ) .
Assume
[
]
ψ : − T M ,0 → S ( X *)
is
continuous
and
ψ <δ .
Let
v := inf {t ≥ 0 : X * + z (t ) ≠ X (t )} where z is a solution to (4.3) with initial function ψ and X is
a solution to (4.1) with initial function ϕ (t ) = X * +ψ (t ) . Suppose that x(t ) and X * + z (t ) do
not coincide, i.e. v < +∞ . By continuity, X (v ) = X * + z (v ) , hence X * + z (v ) ∈ B( X *, ε 0 ) .
There exists η > 0 so small that X * + z (t ) ∈ B( X *, ε 0 ) for t ∈[v , v + η] . Then
((
)
(
)
)
f z t − T 1 + X *,..., z t − T M + X * − h( X *) < α 2 and, by Lemma 4.2,
(
(( )
(
) ))
= − PS ( x*) f (z (t − T 1 ) + X *,..., z (t − T M ) + X *) = zD (t )
Π K X * + z (t ),− f z t − T 1 + X *,..., z t − T M + X *
(C.1)
which implies that X * +z (⋅) is a solution to (4.1) on [v , v + η] . Due to uniqueness of
solution to (4.1), X (t ) = X * + z (t ) for t ∈[v , v + η] , which contradicts the definition of v.
Thus the supposition is not correct, and z (t ) ≡ x(t ) − X * never leaves B(0, ε 0 ) ⊂ B(0, ξ )
which proves stability of steady point 0 of (4.3).
Now assume that (4.3) is stable at 0 and set ξ > 0 . Choose ε0 as in the first part of the
proof. There exists a ς > 0 such that for any continuous ψ : [−T M ,0] → B(0, ς ) ∩ S ( X *) one
has zψ (t ) ∈ B(0, ε0 ) for all t. Let ε > 0 be the number specified for ς and ∆ = T M in
Lemma 4.4, and let δ > 0 be the number provided by Lemma 4.3. For any
34
ϕ ∈ C + , ϕ − X * < δ , the Lemmas 4.3 and 4.4 ensure that X ϕ (t ) ∈ X * + B(0, ε ) for
[
]
[
]
t ∈ − T M , T0 and X ϕ (t ) ∈ X * + B (0, ς ) ∩ S ( X *) for t ∈ T0 , T0 + T M .
By Lemma 4.2, solution z to equation (4.3) starting with the initial function
(
[
)
]
ψ (θ ) = X ϕ θ + T0 + T M − X * , θ ∈ − T M ,0 , satisfies equation (C.1) and therefore
z (t ) + X * is a solution to the equation (4.1). By the uniqueness of solution to (4.1),
(
)
X ϕ (t ) = X * + z t − T0 − T M for t ≥ T0 , hence X ϕ (t ) − X * < ε 0 < ξ for all t, Q.E.D.
The proof of Theorem 4.3 is straightforward and uses arguments similar to those used
to prove Theorem 4.
35