algebraic approach for performance bound calculus on transportation networks

10

Transportation Research Record: Journal of the Transportation Research Board, No. 2334, Transportation Research Board of the National Academies, Washington, D.C., 2013, pp. 10–20.DOI: 10.3141/2334-02

Université Paris-Est, Institut Français des Sciences et Technologies des Transports, de l’Aménagement, et des Réseaux, Génie des Réseaux de Transports Terrestres et Informatique Avancée, F-93166 Noisy-le-Grand, France. Corresponding author: N. Farhi, [email protected].

tribution of this paper is a traffic model that permits derivation of that bound with a theoretical formula. The upper bound of the travel time is derived as a function of the average car density on the road and is then compared with the formula that gives the average travel time on the basis of the same modeling. A short review of network calculus theory and min-plus algebra is given (10) to fix the notation and the language. The model is then presented. It is inspired from the cell transmission model (11, 12) and from the algebraic models of traffic flow proposed elsewhere (13–19). The model applies to a single-lane ring road. The cumulative flows are seen as time signals, and the car dynamics are written algebraically as a min-plus linear system (10). The impulse response of that system is then interpreted in terms of guaranteed minimum service on the road. An upper bound for the travel time of cars through the road is then derived from that guar-anteed minimum service, as done basically in deterministic network calculus (7, 8). Thus, a formula giving an upper bound for the travel time as a function of the average car density on the road is obtained. Finally, a procedure for extending this approach to intersections and large transportation networks is proposed.

Review of NetwoRk CalCulus

The procedure here is to consider a single-lane road as a server and apply the network calculus theory to determine an upper bound of the travel time of cars through the road. In this section some basic results of network calculus theory are described. A time function f (t) (written with a small letter) expresses a given flow at time t, while F(t) (written with a capital letter) denotes the cumulative flow ∫ t

0 f (s)ds. For a flow U arriving to a server that is empty of data at time zero, a maximum arrival curve α is associated to provide an upper bound on the arrivals: α is a (maximum) arrival curve for U if

U t U s t s s t( ) ( ) ( )− ≤ α − ∀ ≤ ≤0

In addition, a minimum service curve β is associated with the server to provide a lower bound on the service. If the output from the server is denoted Y, β is defined as follows: β is a (minimum) service curve for the server if

Y t U s t s ts t

( )( ) ( ) ( )≥ + β − ∀ ≥≤ ≤

min 00

Two indicators of service performance are considered: backlog and virtual delay. The backlog B(t) of data in the server at a given time t is defined by

B t U t Y t( ) ( ) ( )= −

Algebraic Approach for Performance Bound Calculus on Transportation NetworksRoad Network Calculus

Nadir Farhi, Habib Haj-Salem, and Jean-Patrick Lebacque

An application of the basic results of deterministic network calculus theory to road traffic flow theory is presented. Network calculus is a theory based on min-plus algebra. This calculus uses algebraic tech-niques to compute performance bounds in communication networks, such as maximum end-to-end delays and backlogs. The objective of this paper is to investigate the application of those techniques for determin-ing performance bounds on road networks, such as upper bounds for travel times. A traffic model is proposed for cars moving in a single-lane ring road without passing. The model is compatible with network calcu-lus theory and permits derivation of an upper bound of the travel time of cars on the road. An approach for extending the model to calculate upper bounds for the travel times of cars on paths passing through intersections in a whole network is also proposed.

See Table 1 for the mathematical notation used in this paper.Recent advances in information and communication technologies

permit the collection of valuable information on the traffic state by means of probe vehicles (1, 2). The information is either used to derive reliable traffic indicators or sent (after any required analysis, filtration, and reformulation) to connected vehicles via intelligent transportation equipment to improve traffic conditions. One of the most important traffic indicators that drivers need to optimize their trip is the travel time through the possible paths to their destinations. Even though the average value of the travel time estimation is determinative for drivers, its deviation may be important in some cases. To evaluate the deviation of travel time, either the probability distribution of the travel time or minimum and maximum bounds for travel time can be determined. The literature gives several methods for estimating travel times (3–6).

This paper presents a traffic model that permits the derivation of an upper bound for the travel time of cars passing through a single-lane ring road. The model is deterministic and uses algebraic techniques of the network calculus theory (a theory for performance-bound calculus in communication and computer networks) (7–9). The objective of this work is to adapt the algebraic approach of network calculus theory to transportation networks.

A first step in applying the deterministic network calculus to deter-mine an upper bound of the travel time is proposed. The main con-

Farhi, Haj-Salem, and Lebacque 11

The virtual delay d(t) caused by the server at time t is defined by

d t h Y t h U t{ }) ) )( ( (= ≥ + ≥inf 0,

If n0 is assumed in the server at time zero, the definition of the virtual delay is modified by replacing the signal Y with Y − n0.

Arrival and service curves are easily seen not to be unique. To obtain good bounds on the backlog and on the virtual delay on a given server, it is necessary to consider “good” arrival and service curves. A good arrival (respectively, service) curve is simply the minimal (respectively, maximal) one; more details are given elsewhere (7, 8).

Shown below is a basic result of the deterministic network calculus (7, 8) that gives three bounds for a unique server. If α is an arrival curve for an arrival flow U to a given server that offers a minimum service curve β, then the virtual delay is bounded as follows:

d t h t t h tt

{ }{ }) ) )( ( (≤ ≥ α ≤ β + ∀ ≥≥

sup inf 0, 00

The backlog is bounded as follows:

B t s s ts

( )( ) ( ) ( )≤ α − β ∀ ≥≥

sup 00

The curve t ° sups≥0(α(t + s) − β(s)) is an arrival curve for the departure flow Y from the server.

The maximum backlog and delay on a server are then given simply as the maximal vertical and horizontal distances between the arrival and the service curves; see Figure 1.

GuaRaNteed seRviCe oN siNGle-laNe RiNG Road: the Model

An elementary model for the calculus of minimum guaranteed ser-vice for a single-lane ring road seen as a server is presented in this section. The objective of the model is to derive an upper bound for the travel time of cars passing through the road from the guaranteed minimum service of the road. The model is based on the cumulative car flow variables Q (see notation below). It is inspired from the cell transmission model (11, 12); see also work by Lebacque (20). The min-plus linear system theory (10) is then used to derive the minimum guaranteed service.

Consider a single-lane road where cars move without passing. To be able to fix the car density and to simplify the model, a ring road

TABLE 1 Mathematical Notation

u(t) Inflow at time t

U(t) Cumulated inflow from time zero to time t

y(t) Outflow at time t

Y(t) Cumulated outflow from time zero to time t

α Maximum arrival curve (time function)

β Minimum service curve (time function)

F Set of time functions { f is non-decreasing and f (t) = 0, ∀t < 0}

⊕ Element-wise operation (min-plus addition in F): ( f ⊕ g)(t) := min( f (t), g(t))

* Min-plus convolution in F: ( f p g)(t) := inf0≤s≤t( f(s) + g(t − s))

Min-plus deconvolution in F: ( f g)(t) := sups≥0( f(t + s) − f(s))

ε Zero element of the dioid (F, ⊕, p): ε(t) = +∞, ∀t ≥ 0

e Unity element of the dioid (F, ⊕, p): e(t) = +∞, ∀t > 0, and e (0) = 0

f k Convolution power: f k = f p f p . . . p f (k times)

B(t) Backlog at time t: B(t) := U(t) − Y(t)

d(t) Virtual delay at time t: d(t) := inf{h ≥ 0, Y(t + h) ≥ U(t)}

γ p Particular function in F: γ p(t) = +∞ ∀t > 0, and γ p(0) = p

δT Particular function in F. δT(t) = 0, ∀t ≤ T, and δT(t) = +∞ ∀t > T

f + max (0, f )

q Car flow

qmax Maximum car flow

qi(t) Car outflow from the ith section at time t

Qi(t) Cumulated car outflow from the ith section from time zero to time t

ni Number of cars in the ith section at time zero

nmax Maximum number of cars that a section can contain

ρ Average car density in the road

ρc Critical car density

ρj Jam car density

v Free car speed

w Backward wave speed

Δx Section length

q(ρ) Car-flow function of the car density (fundamental traffic diagram)

τ(ρ) Travel time function of the car density

τmax(ρ) Upper bound of the travel time, function of the average car density

`mk=1 f(k) min1≤k≤m f(k)

(a)

Arrival flow

Arrival curve α

Service Departure flow

Service curve β

U Y

(b)

α(t )

β(t )

t

b

d

FIGURE 1 (a) Schema of server and (b) maximum delay d and maximum backlog b determined graphically as maximum horizontal and vertical distances between arrival and service curves, respectively.

12 Transportation Research Record 2334

is considered; see Figure 2. The road is divided into m sections of length Δx. The maximum number of cars on one section is denoted by nmax. The jam density ρj on the road is then given by

n

xjρ =

∆(1)max

The following notation is used:

U(t) = cumulated inflow of cars to the road from time zero up to time t;

Y(t) = cumulated outflow of cars from the road from time zero up to time t;

qi(t) = outflow of cars from the ith section [the section between positions iΔx and (i + 1)Δx] at time t;

Qi(t) = ∫t0 qi(s)ds, the cumulated outflow of cars from the ith section

up to time t; and ni = number of cars in the ith section [the section between

positions iΔx and (i + 1)Δx] at time zero.

The following fundamental traffic diagram for the road is assumed:

q v w j( )( )( )ρ = ρ ρ − ρmin , (2)

where q, ρ, v, and w denote, respectively, the car flow, the car density, the free car speed, and the backward wave speed on the road; see Figure 3. The maximum flow is then given by

q

v w

j=ρ

+1 1 (3)max

Recall the well-known case (15) of an autonomous single-lane ring road, with no entry U and no exit Y. In this case the dynamics of the system are written as follows:

Q t Q tx

vn Q t

x

wnm m{ }( ) = −

∆

+

∆

+min , –1 2 1

v

flow q

density ρ

w

qmax = ρj

+1v

1w

ρc = wρj

v+wρj0

FIGURE 3 Fundamental traffic diagram.

FIGURE 2 Single-lane ring road and corresponding event–graph model.

nm−1

nm n1

n2

n3

n4

(a)

(nm−1,∆t )

Qm−1

(n2,∆t )

(n3,∆t )

(n4,∆t )

(nm,∆t ) (n1,∆t )

Qm

Q1

Q2

Q3

Q4

Q5

U Y

(n1, w ∆x)(nm,w

∆x)(nm−1, w

∆x) (n2, w ∆x)

(n3, w ∆x)

(n4, w ∆x)

(b)


Q t Q tx

vn Q t

x

wn

i m

i i i i{ }{ }

( ) =∆

+

∆

+

∀ ∈ −

+min – , –

2, 3, . . . , 1

–1 –1 1 1

(4)

Q t Q tx

vn Q t

x

wnm m m m{ }( ) =

∆

+

∆

+min – , – (4)–1 –1 1

where

n x n n nk j k k= ρ ∆ − = −max

System 4 is a discrete time event system for which the dynamics are well understood. System 4 can be represented by an event graph (a class of Petri nets), and its dynamics can be written linearly in min-plus algebra (10). Because the event graph representing Sys-tem 4 is strongly connected, the asymptotic car flow on the road is the same at every section and is given by the minimum over the average weights of the graph circuits (15). Three circuits are distinguished in the event graph of Figure 2.

• Interior circuit with the average weight:

n

m x

v

n

m xv v

i

i

m

i

i

m

∑ ∑∆ =

∆= ρ= =1 1

• Exterior circuit with the average weight:

n

m x

w

wi

i

m

j

∑( )∆ = ρ − ρ=1

• Loops over each section with the average weight:

nx

v

x

w

xx

v

x

w

qj

∆ + ∆ =ρ ∆

∆ + ∆ =maxmax

The asymptotic car flow is then given by

q q v w ii j{ }( )( ) ( )ρ = ρ = ρ − ρ − ρ ∀min ,

and retrieves the fundamental diagram (see Equation 2) assumed in the model.

The average travel time through the road is then given by

qm x

v wm x

j( )( )( )

τ ρ =ρρ

∆ =ρ

ρ − ρ

∆max1

,1

In the following, the item of interest is the maximum travel time through the road, not the average travel time. For that, an inflow and an outflow of cars on the road are introduced. Cars are assumed to enter into the road from the entry of Section 1 and to exit it from the same point after passing through the road. The dynamics (System 4) are then rewritten as follows:

Q t U t Q tx

vn Q t

x

wnm m{ }( ) ( )= −

∆

+

∆

+min , , –1 2 1

Q t Q tx

vn Q t

x

wn

i m

i i i i i{ }{ }

( ) =∆

+

∆

+

∀ ∈ −

+min – , –

2, 3, . . . , 1

–1 –1 1

(5)Q t Q t

x

vn Q t

x

wnm m m m{ }( ) =

∆

+

∆

+min – , ––1 –1 1

Y t Q tx

vn Q t

x

wnm m{ }( ) = −

∆

+ −

∆

+min , (5)2 1

Notice the difference between the dynamics of Q1, which are conditioned by the inflow U, and the dynamics of Y, which are not conditioned by the inflow U.

The same modeling approach as above is considered, but here the variables are seen as signals; see Baccelli et al. (10). The dynamic System 5 is first shown to be linear in a certain algebraic structure, and then that linearity is used to derive a guaranteed service of the road seen as a car server.

Consider the set of time functions

F f f t t{ }( )= = ∀ <is nondecreasing and 0, 0

endowed with the following two operations:

• Addition (elementwise minimum): ( f ⊕ g)(t) := min{ f(t), g(t)}, ∀t ≥ 0, and

• Product (min-plus convolution): ( f p g)(t) := min0≤s≤t{ f (s) + g(t − s)}, ∀t ≥ 0.

A dioid structure (F, ⊕, p) is then obtained (8, 10). The zero element of that dioid is denoted by ε and defined by ε(t) = +∞, ∀t ≥ 0, while the unity element is denoted by e and is defined by

e tt

t( ) =

=

+∞ ≥

0 if 0

for 0

The following notation is defined:

• Power: f k denotes the product (min-plus convolution) of f with itself k times.

• Additive closure: f *(t) := e ⊕ f ⊕ f 2 ⊕ f 3 ⊕ . . .• Deconvolution: ( f ∅ g)(t) := sups≥0( f (t + s) − g(s)).

In addition, the two particular signals γ p (the gain signal) and δT (the shift signal) in F are defined as follows (see Appendix A for more details on those signals).

tp t

tp( )γ =

=

+∞ >

if 0

for 0

tt T

T( )δ =≤

+∞

0 if

otherwise

By using the notation of the min-plus convolution, the definitions of arrival and service curves can be rewritten as follows:

• α is an arrival curve for U if U ≤ α p U.• β is a (minimum) service curve for the server if Y ≥ β p U.


The cumulative flows U; Qi, 1 ≤ i ≤ m; and Y are time functions (or signals) in F. Therefore, by using the notation of addition and product in F, System 5 is linear according to those operations and is written as follows. [Note that, as in standard algebra, the product operation is sometimes not symbolized (that is to say, f p g may simply be written fg).]

Q Q Q Un x vm

n x wm= γ δ ⊕ γ δ ⊕∆ ∆1 2

1

Q Q Q i min x v

in x w

ii i { }= γ δ ⊕ γ δ ∀ ∈ −∆

−∆

+− 2, 3, . . . , 11 1

1

(6)

Q Q Qmn x v

mn x wm m= γ δ ⊕ γ δ∆

−∆−

1 11

Y Q Qn x vm

n x wm= γ δ ⊕ γ δ∆ ∆ (6)21

Notice that, in the case where all the signals U; Qi, 1 ≤ i ≤ m; and Y are initialized to zero (or to the same constant), the flow Z = [Y − Σm

i=1ni]+ is the outflow corresponding to the inflow U. In this case, under the assumption that cars move without passing, the travel time of a car arriving on the road at time t is inf{h ≥ 0, Z(t + h) ≥ U(t)}. Furthermore, because the item of interest here is determination of an upper bound (that may be realized at the initial time), the initialization of all the signals to zero is important. For that, it is sufficient to add (min-plus addition) to every signal of System 6 the signal unity e. Then, the dynamics (System 6) are written as follows:

Q Q Q U en x vm

n x wm= γ δ ⊕ γ δ ⊕ ⊕∆ ∆1 2

1

Q Q Q e i min x v

in x w

ii i { }= γ δ ⊕ γ δ ⊕ ∀ ∈ −∆

−∆

+− 2, 3, . . . , 11 1

1

(7)

Q Q Q emn x v

mn x wm m= γ δ ⊕ γ δ ⊕∆

−∆−

1 11

Y Q Q en x vm

n x wm= γ δ ⊕ γ δ ⊕∆ ∆ (7)21

System 7 is then written as follows:

Q A Q B U E= ⊕ ⊕p p (8)

Y C Q e= ⊕ (8)p

where

Q = (Q1 Q2 . . . Qm)′,

A

n x w n x v

n x v n x w

n x v n x w

n x v n x w

n x w n x v

m

m m

m m

,

/1

1 2

2 3

2 1

1

L

L

M

M O O O O

L

L

=

ε γ δ ε ε γ δ

γ δ ε γ δ ε ε

ε γ δ ε γ δ ε

ε

ε ε γ δ ε γ δ

γ δ ε ε γ δ ε

∆ ∆

∆ ∆

∆ ∆

∆ ∆

∆ ∆

− −

−

B

e

,M

M=

ε

ε

ε

C = (ε γ n–1δΔx/w ε . . . ε γnmδΔx/v), and E = (e e . . . e)′.

System 8 is a min-plus linear system. A basic result of the min-plus system theory (10) gives the impulse response of System 8:

Y CA BU E e CA BU CA E e( )= ⊕ ⊕ = ⊕ ⊕* * * (9)

Note that an impulse response Y = CA*BU would be obtained if System 6 is considered instead of System 7 (that is, if the initial conditions of System 6 are not taken into account). In that case, it would be concluded directly that CA*B is a minimum service curve of the road seen as a server [because Y ≥ (CA*B) p U], and an upper bound for the travel time through the road would be derived. But here, as mentioned above, the initial conditions of the dynamical system must be taken into account, because an upper bound of the travel time through the road is sought rather than the average travel time. To be able to derive upper bounds from Equation 9, as is done from a service curve, the following extension is proposed.

Minimum service couple: Considered here is the case in which the curve service is given with an additional affine term. More precisely, (β, λ) is a service couple for a server if

Y U≥ β ⊕ λp

As can easily be checked (see Appendix B), to obtain the three bounds given above, for that case, it is sufficient to replace β with β ⊕ λ. The following hold:

• The maximum backlog is bounded as follows:

B t s s ts

( )( )( ) ( )( ) ≤ α − β ⊕ λ ∀ ≥≥

sup 00

• The maximum delay is bounded as follows:

d t h t t h tt

sup inf 0, 00

{ }{ }( )( ) ( ) ( )≤ ≥ α ≤ β ⊕ λ + ∀ ≥≥

• An arrival curve for the departure flow is α ∅ (β ⊕ λ). That is,

Y Y( )( )≤ α ∅ β ⊕ λ p

The two first bounds are then given by the maximum vertical and horizontal distances between the curves α and β ⊕ λ. Note that the curve (β ⊕ λ) is not necessarily a minimum service curve for the server.

The introduction of the additional curve λ in the characterization of the service is related here to the initialization of the system. Indeed, as mentioned above, the output of System 6 (without initializing the sig-nals) is simply given by Y ≥ β p U, with β = CA*B, while the initializa-tion of the signals introduces additional terms and renders the output affine with respect to the input Y ≥ β p U ⊕ λ, with λ = CA*E ⊕ e.

Theorem 1. A minimum service couple for the single-lane ring road, seen as a server, is (β, λ), given by

amp x[ ]β = γ ∆ +*– p

emp x

k

mm k x m k x v

k

mk m x k x wj j` `( )λ = β ⊕ γ γ δ ⊕ γ δ ⊕

( )− ∆

=

−ρ− ρ ∆ − ∆

=

−ρ − ρ ∆ ∆

++ +

1

1

1

1


where

a mp x m x v x x v x w m x m x wj j= γ δ ⊕ γ δ ⊕ γ δ( )∆ ∆ ρ ∆ ∆ +∆ ρ −ρ ∆ ∆

Proof. System 8 is an affine min-plus system, and Y = CA*BU ⊕ CA*E ⊕ e. A* must be computed. Several methods can be used to compute A*. By definition of A*, one can simply compute A2, A3, . . . , then deduce A* = `k≥0 Ak by simplifying all the terms. It can easily be verified that A* is given as follows:

A

a i j

a

i j k m

ij

n

j x v

n

j i x wk

k j

i

k

k i

j

∑ ∑( )

{ }

∗ =

∗ =

∗ γ δ ⊕ γ δ

≠

( ) ( )− ∆ − ∆=

−

−

−

if

if , with cyclic in 1, 2, . . . ,

1

1 1

To compute CA*B, only the first column of A* is needed, because only the first entry of B is not null. Thus, A*B = (A*).1 and is given by

pA B ae n x v

nm x w

nm x v n x w

k

k

m

k

k

m

m

∑

∑∗ = ∗

γ δ ⊕ γ δ

γ δ ⊕ γ δ

( )

( )

∆ − ∆

− ∆ ∆

=

=

−

. . .1

1

1 2

1

1

Therefore, CA* B = a* a.Similarly, it can be verified that

CA E a ak

m nm k x v

k

m nk x w

i

i k

m

i

i

k

∑ ∑∗ = ∗ ⊕ γ δ ⊕ γ δ( )

=

−− ∆

=

−∆= + =

1

1

1

11 1` `p

Then it is not difficult to verify that

n m k xi

i k

m

j∑ [ ]≥ ρ − ρ ∆= +

+

1

n k m xi

i

k

j∑ [ ]≥ ρ − ρ ∆=

+

1

Now because the item of interest is the car outflow from the road that comes from the car inflow, without counting the cars in the road at time zero (and that stay there all the time because the road is cir-cular), the variable Z = [Y − Σm

i=1ni]+ = [γ−mρΔxY]+ must be expressed as a function of U. Thus, the couple of curves (γ−mρΔxCA*B, γ−mρΔx(CA*E ⊕ e)) is a minimum service couple for the ring road.

Finally, [γ−mρΔx p a p a*]+ = [γ−mρΔx p a*]+, since (a p a*)(t) = a*(t) ∀t > 0 and (a p a*)(0) = a(0) < mρΔx (see Properties P2 and P4 in Appendix A). ◾

To show the shape of the minimum service couple given in Theo-rem 1, an academic example is taken.

Example 1. Let m = 6, Δx =1, v = 1, w = 1⁄2, and ρj = 1. Then nmax = 1, ρc = 1⁄3, and qmax = 1⁄3. In addition, three cases are considered, in which the average car density on the road is varied. First notice that a* can also be written as follows (see Property P3 in Appendix A):

p pa m x m x v x x v x w m x m x vj j( )( )( )∗ = γ δ ∗ γ δ ∗ γ δ ∗( )ρ∆ ∆ ρ ∆ ∆ +∆ ρ −ρ ∆ ∆

There are three cases:

1. Σ mi=1ni = 1, that is, ρ = 1⁄6 < ρc. Then a = γ1δ6 ⊕ γ5δ12 and

a* = (γ1δ6)*. Hence the minimum service couple (β, λ) is given by

[ ]( ) ( )β = γ γ δ ∗ = γ γ δ ∗− +1 1 6 –1 1 6p

λ = β ⊕ γ δ ⊕ γ δ ⊕ γ δ ⊕ γ δ = β0 5 1 6 2 8 3 10

2. Σmi=1ni = 2, that is, ρ = 1⁄3 = ρc. Then a = γ1δ3 and a* = (γ1δ3)*.

Hence the minimum service couple (β, λ) is given by

[ ]( )β = γ γ δ ∗ +–2 1 3

λ = β ⊕ γ δ ⊕ γ δ0 8 1 10

3. Σmi=1ni = 3, that is, ρ = 1⁄2 > ρc. Then a = γ1δ3 ⊕ γ3δ12 and a* =

(γ1δ3)*(γ3δ12)*. Hence the minimum service couple (β, λ) is given by

[ ]( ) ( )β = γ γ δ ∗ γ δ ∗ +–3 1 3 3 12

λ = β ⊕ γ δ = β0 10

The minimum service couples (given in Theorem 1) corresponding to each of the three cases are shown in Figure 4. ◾

A corollary of Theorem 1 is given below, where practical formu-las are obtained by relaxing the minimum service couple given in Theorem 1.

Corollary 1. A minimum service couple (β, λ) for the road is given by

t q t[ ]( ) ( )( )β = ρ τ ρ +–

t t w tw

m xjj

( ) ( )λ = β ⊕ ρ −ρ

ρ∆

+2 1

where q(ρ) and τ(ρ) are the average flow and average travel time given by the fundamental diagram (see the section on review of network calculus).

Proof. From the curve β given by Theorem 1, the following is known:

vtn

m x vi

i

m

∑γ δ

≥ ρ∆−

*1

q tn x v x w( )γ δ ≥∆ +∆ *max

max

wtn

m x wj

i

i

m

∑( )γ δ

≥ ρ − ρ∆=

*1

Then

v q w t

n

m x v n x v x w

n

m x w

j

i

i

m

i

i

m

∑ ∑

{ }( )

( )γ δ

γ δ γ δ

≥ ρ ρ − ρ

∆ ∆ +∆ ∆= =

*

*

*

min , ,max

1 max 1p p


But because ∀ρ, min{ρv, (ρj − ρ)w} ≥ qmax,

v w t

n

m x v n x v x w

n

m x w

j

i

i

m

i

i

m

∑ ∑

{ }( )

( )γ δ

γ δ γ δ

≥ ρ ρ − ρ

∆ ∆ +∆ ∆= =

*

*

*

min ,

1 max 1p p

Hence

p p m x

v w tm x

v w

v w t mx

vm

x

w

n

m x v n x v x w

n

m x w

jj

jj

i

i

m

i

i

m

∑ ∑

{ } { }

{ }

( )

( ) ( )

( )

γ δ

γ δ γ δ

− ρ ∆

≥ ρ ρ − ρ −ρ ∆

ρ ρ − ρ

≥ ρ ρ − ρ −∆ ρ

ρ − ρ∆

∆ ∆ +∆ ∆

+

+

+

= =

*

*

*

min ,min ,

min , max ,

1 max 1

From the curve λ given in Theorem 1, the following is known:

v t mm x

vk

mm k x m k x v

jj

m x m x v

j

j

γ δ ≥ρ − −

ρρ

∆

γ δ

( )

( )

=

−ρ− ρ ∆ − ∆

+

ρ−ρ ∆ − ∆

+

+

max,

1

1

1

`

Then

v t mx

vmp x

k

mm k x m k x v

jjγ γ δ ≥ ρ −

∆

( )− ∆

=

−ρ− ρ ∆ − ∆

++

1

1

`

In addition,

w tm x

wk

mk m x k x w

jj

m m x m x wj j` γ δ ≥ ρ −ρ

ρ∆

γ δ

( ) ( )

=

−ρ − ρ ∆ ∆

+− ρ − ρ ∆ − ∆

+ +

max ,1

11 1

Low densityγ0 δ5 ⊕ γ1 δ6 ⊕ γ2 δ8 ⊕ γ3 δ10

γ0 δ8 ⊕ γ1 δ10

γ0 δ10

λ = β = β ⊕ λ

λ = β = β ⊕ λ

λ

t

β

Critical density

High density

FIGURE 4 Calculus of service couple for single-lane ring road in three cases of low, critical, and high car density.


Then

w tm x

wm x

k

mk m x k x w

jj

j`γ γ δ ≥ ρ −ρ

ρ∆

− ρ∆

=

−ρ − ρ ∆ ∆

++ 2

1

1

Finally, it is easy to verify that

t v t mx

vj( )β ≤ ρ −

∆

+

because q(ρ) ≤ vρj and τ(ρ) ≥ mΔx/v. ◾

Notice that the term wρj[t − (2mρ/ρj)(Δx/w)]+ is more important than the term β(t) in the service couple given in Corollary 1. The curve β(t) simply gives the average service of the road because q(ρ) is the average car flow and τ(ρ) is the average travel time on the road.

Corollary 2. If the arrival flow U is regular, with an arrival curve α(t) = rt, with r ≤ qmax, then an upper bound τmax(ρ) for the travel time through the road is given as a function of the average car density in the road as follows:

wm x

v w wm x

j j j

( ) ( )τ ρ = τ ρρ

ρ∆

=ρ

ρ − ρρ

ρ

∆max ,2 1

max1

,1

,2 1

(10)

max

Proof. τmax(ρ) is simply the horizontal distance between the curves α(t) = rt and β ⊕ λ given in Corollary 1. ◾

It is easy to see from Equation 10 that τmax(ρ) is independent of r as long as r ≤ qmax and that

w v

w

v

w

v

j j

jj j[ ]

( ) ( )

( ) ( )< ⇒

τ ρ > τ ρ ρ ∈ρ ρ

τ ρ = τ ρ ρ ∈ ρρ ρ

for2

,2

for 0, \2

,2

(11)

max

max

w v j[ ]( ) ( )≥ ⇒ τ ρ = τ ρ ∀ρ ∈ ρ0,max

Figure 5 shows τ(ρ) and τmax(ρ) in the case w < v.The choice of the traffic flow model (cell transmission model)

and the choice of the fundamental traffic diagram (triangular one) are important. These two choices permitted the writing of the car dynamics on the road linearly in the min-plus algebra and the deriva-tion of the service couple as an impulse response of a min-plus linear system. Nevertheless, the choice of the fundamental traffic diagram is not restrictive. In fact, it is easy to see that the triangular fundamental diagram can be replaced with a trapezoid diagram without affecting the linearity of the model or the derivation of an upper bound for the travel time through the road.

Road NetwoRk CalCulus

In this section the first ideas on extending the results obtained on a single-lane road to intersections and whole transportation networks are given. For multilane roads, one fundamental diagram may be used for all lanes and the same approach applied as in the case of a single-lane road. The same model and results are assumed to be used here for multilane roads. However, other models similar to the one given above can be developed for multilane traffic.

Before the procedure for extending the approach to intersections and networks is presented, an important result of the determinis-tic network calculus on the series composition of servers must be recalled and extended (it shows in particular the power of the alge-braic approach against other approaches). The result indicates that a minimum service curve of the series composition of two servers guaranteeing β1 and β2 as minimum service curves for each of them is simply the curve β1 β2. The result can easily be proved by using the associative property of the min-plus convolution; more details are given elsewhere (7, 8). Furthermore, since the min-plus convolution is also commutative, the order of the composition of the two servers is not important.

For this model, a similar result is needed for a composition of servers offering minimum service couples (rather than minimum

FIGURE 5 t(r) and tmax(r) in case w = v/2.

00

Average and maximum travel times

Average car density

τ

τmax

ρcρj

2

m∆xw

m∆xv

ρj

2wv


service curves). It is easy to see that the composition of two servers offering two service couples (β1, λ1) and (β2, λ2) guarantees a ser-vice couple (β1 p β2, β1 p λ2 ⊕ λ1). Indeed, from Y = β1 p Z ⊕ λ1 and Z = β2 p U ⊕ λ2, it is seen that Y = (β1 p β2) p U ⊕ (β1 p λ2 ⊕ λ1).

In communication and computer networks, the (residual) guaran-teed service for all (input, output) couples through every switch-ing router is determined first. Then it is sufficient to compose (with a min-plus convolution) all the guaranteed services through all the arcs of a given path to determine the service guaranteed through the whole path. Finally, the maximum end-to-end delays on a communication network are determined by calculating the maximum delay on each (origin–destination) path on the network simply by using its guaranteed service curve and considering the whole path as an elementary server. The residual guaranteed service calculus on a given input–output couple of a given router takes into account the control policy set in that router; more details are given elsewhere (7, 8).

A similar procedure is considered possible for transportation networks. The main difference between data traffic in communica-tion networks and car traffic in transportation networks is the inter-action between particles (drivers observe reaction times, contrary to data packets) expressed in the fundamental diagram of the road. This difference is already taken into account in the one road model (presented above). Thus, to extend the approach to complicated transportation networks, the residual guaranteed service calculus on input–output couples of routers must be adapted to apply in inter-sections, for transportation control policies. Then one only needs to compose elementary road services and residual guaranteed services to obtain guaranteed services on whole paths. Upper bounds for travel times can be derived similarly. One of the most difficult issues that must be resolved to extend the approach presented in this paper to big networks is the presence of cyclic dependencies of inflows arriving at one intersection, even though some elementary results exist to deal with that issue (7).

The calculus of upper bounds of travel times in a treelike net-work will be explained by considering the transportation network of Figure 6.

In Figure 6, the notation A+ indicates some point downstream of intersection A. The notation C −

A indicates some point upstream of intersection C in the direction of intersection A. All other symbols are interpreted similarly. The fundamental diagrams of the roads (A+, C −

A), (B+, C −B), C +, E −

C), (D+, E −D), and (E +, F −

E) are assumed

known. The approach for calculating an upper bound for the travel time from A to F in Figure 6 is the following.

1. Determine the guaranteed service on each of the roads (A+, C−A),

(B+, C −B), (C+, E −

C), (D+, E −D), and (E+, F −

E), independent of the inter-sections. The fundamental diagrams of the roads and the model presented above are used for this determination.

2. Calculate the residual services on the merge C:– The service guaranteed on C for the aggregate inflows coming

from C−A and C−

B is assumed to be simply the guaranteed service on the road (C +, E −

C).– The control policy on C is assumed to be known.– From the above two items of information, calculate the resid-

ual guaranteed services for the flows (C −A, C +) and (C −

B, C +). This will be done by adapting well-known results of network calculus theory on the calculus of residual guaranteed service. (This has not yet been done.)3. Calculate the residual services on the merge E. By the same

method as above (Step 2), calculate the residual guaranteed services for the flows (E −

C, E +) and (E −D, E +).

4. Finally, the guaranteed service for the flow (A+, F −E), or simply

(A, F), is given by the series composition of the services (A+, C −A),

(C−A, C+), (C+, E−

C), (E−C, E+), and (E+, F−

E), respectively. The guaranteed services for (B, F) and (D, F) are obtained similarly.

An upper bound for the travel times from A (respectively, B and D) to F are then derived from the guaranteed services (A, F), (B, F), and (D, F), respectively, as done on a single road.

CoNClusioN

A network calculus traffic model that permits the derivation of an upper bound for the travel time of cars passing through a single-lane ring road has been presented in this paper. An important advantage of the model is its algebraic formulation, which is powerful in comparison with other formulations (e.g., the series composition). Even though the model is basic and elementary, its developments and extensions may be promising. Future work will be on the realization of the model extension process. In addition, the effectiveness of the approach will be demonstrated by performing numerical investigations of effective data sets.

appeNdix a

details oN siGNals gp aNd dT

Details on some particular signals in F, as well as certain properties used in the section on guaranteed service on a single-lane ring road (the traffic model), are given here. The signals γ p and δT being defined in that section, the signals γ p p δT and (γ p p δT)* are easy to obtain. For the signal γ p p δT,

tp t T

t T

p T( )[ ]

( )γ δ =∈

+∞ >

if 0,

ifp

For the signal (γ p p δT)*,

tt

kp kT t k T k Np T( )

( )( )γ δ =

≤

≤ < + > ∈

*

0 if 0

for 1 ,p

AA+

C+

B+

D+

E+

B

D

EF

CCA

–

CB–

ED–

EC–

FE–

FIGURE 6 Treelike network.


Furthermore, it can be verified (see Figure A-1) that

p tpT

t tp T *( ) ( )γ δ ≥ ∀

The convolution of the signals γ p and δT with a given signal f is explained here.

f t f t f t p tp p( ) ( )( ) ( ) ( )γ ⋅⋅= γ = + ∀p

f t f t f t T p tT T( ) ( )( ) ( ) ( )δ ⋅⋅= δ = − + ∀p

f t f t f t T p tp T p T( ) ( )( ) ( ) ( )γ δ ⋅⋅= γ δ = − + ∀p p

The following additional properties are cited here (8, 10):

P1. ∀f ∈ F, f * ≤ f p f * ≤ fP2. ∀f ∈ F, f (0) = 0 ⇒ f p f * = f *P3. ∀f, g ∈ F, (f ⊕ g)* = f * p g*P4. ∀f ∈ F, e ⊕ f p f * = f *

APPENDIX B

MINIMUM SERVICE COUPLE

The three bounds of network calculus in the case of a minimum service couple (β, λ) instead of a minimum service curve β (that is, Y ≥ β * U ⊕ λ instead of Y ≥ β * U) are clarified here.

• Maximum backlog:

B t U t Y t U t U t s s t

s s t t

s s z z

s s s s

s s

s t

s t

z s z z

s s

s

{ }{ }

{ }

{ }( )

( )

( ) ( )

( ) ( )

{ }( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )

( )= − ≤ − − + β λ

≤ α − β α − λ

≤ α − β α − λ

≤ α − β α − λ

≤ α − β ⊕ λ

≤ ≤

≤ ≤

≥ ≤ ≤ ≥

≥ ≥

≥

min min ,

max max ,

max max max , max

max max , max

max

0

0

0 0 0

0 0

0

• Maximum delay: Let t ≥ 0. Let τ ≥ 0 such that τ < d(t) := inf{h ≥ 0, U(t) ≤ Y(t + h)}. Then U(t) > Y(t + τ) and

Y t U t s s ts t{ }{ }( ) ( ) ( ) ( )+ τ ≥ + τ − + β λ + τ

≤ ≤ +τmin min ,

0

That is,

s s t Y t U t s s t{ }( )( ) ( )( )∃ ≤ ≤ + τ + τ ≥ + τ − + β λ + τ, 0 min ,0 0 0 0

Then

s s t U t U t s s t{ }( )( ) ( )( )∃ ≤ ≤ + τ ≥ + τ − + β λ + τ, 0 min ,0 0 0 0

Therefore, if U(t) ≥ U(t + τ − s0) + β(s0), then s0 > τ, and thus

s U t U t s s( ) ( ) ( ) ( )α − τ ≥ + τ − > β–0 0 0

Hence

h t t h

h t t ht

t

{ }{ }( )

( ) ( )

( ) ( )

τ < ≥ α ≤ β +

≤ ≥ α ≤ β ⊕ λ +≥

≥

sup inf 0,

sup inf 0,0

0

If U(t) ≥ λ(t + τ), then because U(0) = 0, α(t) ≥ U(t) ≥ λ(t + τ). Hence

h t t h

h t t h

t

t

{ }

{ }( )

( ) ( )

( ) ( )

τ < ≥ α ≤ λ +

≤ ≥ α ≤ β ⊕ λ +

≥

≥

sup inf 0,

sup inf 0,

0

0

• Departure flow:

Y t Y s U t U s z z s

U t U s z z U t s

t s z z t s

t s z z t s z z

t s z z t s

z s

z s

z s

z s z s

z s

{ }{ }

{ }{ }

{ }

{ } { }

{ }

( )

{ }

( )

( ) ( )

( ) ( )( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

− ≤ − − + β λ

≤ − − − β − λ

≤ α − + − β α − λ

≤ α − + − β α − + − λ

≤ α − + − β λ ≤ α ∅ β ⊕ λ −

≤ ≤

≤ ≤

≤ ≤

≤ ≤ ≤ ≤

≤ ≤

max min ,

max max ,

max max ,

max max , max

max min ,

0

0

0

0 0

0

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The Transportation Network Modeling Committee peer-reviewed this paper.

algebraic approach for performance bound calculus on transportation networks

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