In section 3, we propose a methodology to model traf-
fic flow as a min-plus linear system. A particular 2
inputs-2 outputs system is presented as a generic rep-
resentation for a wide-variety of elementary roadway
stretches. Then we show how such elementary mod-
els can be composed to obtain a model for a succes-
sion of roadway sections.
In section 4, two examples are presented. The first
one concerns a basic road section (without any inter-
section). The proposed model is simple but rough in
the sense that it leads to considered that an infinity of
vehicles can simultaneously run on the section. The
second example is a refinement (taking into account
the limited capacity), and the composition of two el-
ementary models is experimented. In both cases, we
derive from simulation the fundamental diagram that
links the flow to the density of vehicles on the road.
In section 5, we discuss characteristics of the pro-
posed model comparatively with traffic flow models
in the literature.
2 PRELIMINARIES
In this section, we first explain why vehicular traffic
flow on a road section can be studied as a linear sys-
tem over min-plus algebra. Then we recall that min-
plus linear systems, and in particular traffic flow, can
be represented by their impulse responses.
2.1 Traffic on a Roadway Stretch is a
Min-plus Linear System
As usual when studying complex systems, we con-
sider a roadway network as an assembly of road sec-
tions. Each roadway stretch will be seen as a min-
plus linear system, and so, a complex infrastructure
will be studied as the system resulting from the as-
sembly of corresponding elementary subsystems. In
other words, each road section is considered to be a
simple min-plus linear system whose input and out-
put correspond to the flows of vehicles respectively
entering and leaving the section.
Let us recall that min-plus linear systems are sys-
tems for which the property of linearity, also called
”principle of superposition”, can be applied to the two
binary operations min and + of the min-plus algebra
(see for example (Gaubert, 1992)).
Definition 2.1 (Signal, Min-plus Linear System) A
signal u is defined as a map from Z to R ∪ {−∞}.
A system S is called min-plus linear if for all signal
u, v, and ∀t ∈ Z,
[S (min(u, v))](t) = min([S (u)](t), [S (v)](t)), (1)
and ∀u, ∀a ∈ R, ∀t ∈ Z,
[S (a+ u)](t) = a+ [S (u)](t). (2)
in which S (u) is the system output signal in response
to input u.
To study a road section as a min-plus linear sys-
tem, let us give the following meanings for its input
and output :
• the input u is a counter of vehicles entering the
road section: u(t) denotes the cumulated number
of vehicles having entered the road section up to
time t,
• the output y is a counter of vehicles leaving the
road section: y(t) denotes the cumulated number
of vehicles having left the road section up to time
t.
It is assumed that vehicles are conserved along a
road section
1
, i.e., u(t) is always equal to the sum of
y(t) and the number of vehicles on the road section at
time t.
Under this assumption, the amount of vehicles out
of the road section with min(u,v) as input flow, is
claimed to be equal to the minimum of quantity of
vehicles out of the road section obtained with u and v
considered separately.
On the one hand, since min(u, v) corresponds to a less
dense flow than the ones given separately by u and v,
vehicles of the flow given by min(u, v) cross the road
section at least as fast as the ones in flows u and v
considered separately. So we deduce the following
inequality:
∀t , [S (min(u, v))](t) ≥ min([S (u)](t), [S (v)](t)),
(3)
On the other hand, causality of the system induces
the converse inequality. More precisely, since
∀t, [min(u, v)](t) ≤ u(t),
we have
∀t, [S (min(u, v))](t) ≤ [S (u)](t),
that is the amount of vehicles out the road section
is at any time t greater with u than with min(u, v)
as input flow. With similar arguments we have
∀t, [S (min(u, v))](t) ≤ [S (v)](t), and we deduce that
[S (min(u, v))](t) ≤ min([S (u)](t), [S (v)](t)). (4)
Inequalities (3) and (4) satisfied by vehicular traf-
fic flow on a roadway stretch correspond to the first
condition (1) defining a min-plus linear system.
1
Additions or withdrawals of vehicles via entry/exit
lanes will be taken into account as additional inputs/outputs
in the model.
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