Algebraic Approach for Performance Bound Calculus on Transportation Networks

Transcription

1 Algebraic Approach for Perforance Bound Calculus on Transportation Networks (Road Network Calculus) Nadir Farhi, Habib Haj-Sale & Jean-Patrick Lebacque Université Paris-Est, IFSTTAR, GRETTIA, F Noisy-le-Grand, France. (Corresponding author).

2 N. Farhi, H. Haj-Sale and J.-P. Lebacque 2 Abstract We propose in this article an adaptation of the basic techniques of the deterinistic network calculus theory to the road traffic flow theory. Network calculus is a theory based on in-plus algebra. It uses algebraic techniques to copute perforance bounds in counication networks, such as axiu end-to-end delays and backlogs. The objective of this article is to investigate the application of such techniques for deterining perforance bounds on road networks, such as axiu bounds on travel ties. The ain difficulty to apply the network calculus theory on road networks is the odeling of interaction of cars inside one road, or ore precisely the congestion phase. We propose a traffic odel for a single-lane road without passing, which is copatible with the network calculus theory. The odel perits to derive a axiu bound of the travel tie of cars through the road. Then, basing on that odel, we explain how to extend the approach to odel intersections and large-scale networks. Keywords: traffic flow theory, axiu travel tie, network calculus, in-plus algebra.

3 N. Farhi, H. Haj-Sale and J.-P. Lebacque 3 Notations u(t) U(t) y(t) Y(t) α β inflow at tie t. cuulated inflow fro tie zero to tie t. outflow at tie t. cuulated outflow fro tie zero to tie t. axiu arrival curve (tie function). iniu service curve (tie function). F the set of tie functions f is non decreasing and f t = 0, t < 0. eleent-wise operation (in-plus addition in F). f g t in f t, g t. in-plus convolution in F. f g t inf 0 s t f s + g t s. in-plus de-convolution in F. (f g)(t) sup s 0 f t + s f s. ε the zero eleent of the dioid (F,,). ε t = +, t 0. e the unity eleent of the dioid (F,,). e t = +, t > 0, and e 0 = 0. f k convolution power. f k = f f f (k ties). B(t) the backlog at tie t. B t U t Y(t). d(t) the virtual delay at tie t. d t Inf 0, Y t + U t. γ p a particular function in F. γ p t = + t > 0, and γ p 0 = p. δ T a particular function in F. δ T t = 0 t T, and δ T t = + t > T. [expr] + ax 0, expr. q q ax q i (t) Q i (t) n i n ax car-flow. the axiu car-flow. the car outflow fro the i th section at tie t. the cuulated car outflow fro the i th section fro tie zero to tie t. the nuber of cars in the i th section at tie zero. the axiu nuber of cars that a section can contain.

4 N. Farhi, H. Haj-Sale and J.-P. Lebacque 4 ρ ρ c ρ j v w x q(ρ) τ(ρ) τ ax (ρ) the average car-density in the road. the critical car-density. the ja car-density. the free car speed. the backward wave speed. the section length. the car-flow function of the car-density (fundaental traffic diagra). the travel tie function of the car-density. a axiu bound of the travel tie, function of the car-density. k=1 f(k) in 1 k f(k). 1 Introduction The recent advances in inforation and counication technologies perit to obtain valuable inforation on the traffic state by eans of probe vehicles (Ain & al, 2008), (Herrera & Bayen, 2008). The inforation is then either used to derive reliable traffic indicators, or sent (after required analyses, filtrations and reforulations) to connected vehicles via intelligent transportation equipents, in order to iprove the traffic conditions. One of the ost iportant traffic indicators that drivers need to receive in order to optiize their trip, is the travel tie through the possible paths to their destinations. Even though the average value of the travel tie estiation is deterinant for drivers, its deviation ay be very iportant in soe cases. In order to evaluate the deviation of the travel tie, one can deterine either the probability distribution of the travel tie, or iniu and axiu bounds for it. Several ethods exist in the literature to estiate travel ties (Coifan, 2002), (Claudel & Bayen, 2008), (Claudel, Hofleitner, Mignerey, & Bayen, 2008), (Ng, Szeto, & Waller, 2011). We present in this article a traffic odel that perits to derive a axiu bound of the travel tie of cars passing through a single-lane road. The odel is deterinistic and uses algebraic techniques of the network calculus theory (a theory for perforance bound calculus in counication and coputer networks) (Chang, 2000), (Le Boudec & Thiran, 2001), (Jiang & Liu, 2008). The objective of our work is to adapt the algebraic approach of the Network Calculus theory to transportation networks.

5 N. Farhi, H. Haj-Sale and J.-P. Lebacque 5 We propose in this article a first step of applying the deterinistic network calculus to deterine the axiu bound of the travel tie. The ain contribution of this article is a traffic odel that perits the derivation of that bound with a theoretic forula. Moreover, the axiu bound of the travel tie is derived as a function of the average car-density on the road, and is then copared to the forula that gives the average travel tie, basing on the sae odeling. In section 2 we give a short review in the network calculus theory and in the in-plus algebra (Baccelli, Cohen, Olsder, & Quadrat, 1992), in order to fix the notations and the language. The odel is presented in section 3. It is inspired fro the cell transission odel (Daganzo, 1994), (Daganzo, 1995), and written on a single-lane road. The cuulative flows are seen as tie signals and the car-dynaics is written algebraically as a in-plus linear syste (Baccelli, Cohen, Olsder, & Quadrat, 1992). The ipulse response of that syste is then interpreted in ter of guaranteed iniu service on the road. A axiu travel tie of cars through the road is then derived fro that guaranteed iniu service, as done basically in deterinistic network calculus (Chang, 2000), (Le Boudec & Thiran, 2001). Thus, a forula giving a axiu bound for the travel tie as a function of the average cardensity in the road is obtained. In section 4, we propose a procedure for extending our approach to intersections and large transportation networks. 2 Review in network calculus The procedure here is to consider a single-lane road as a server and apply the network calculus theory to deterine a axiu bound of the travel tie of cars through the road. In this short section we recall soe basic results of the network calculus theory. In order to fix notations, a tie function f(t) (written with a sall letter) expresses a given flow at tie t, t while F(t) (written with a capital letter) denotes the cuulative flow f(s)ds. For an 0 arrival flow U arriving to a server that is epty of data at tie zero, a axiu arrival curve α is associated in order to upper-bound the arrivals. Arrival curve. α is a (axiu) arrival curve for U if U t U s α t s, 0 s t. In the other side, a iniu service curve β is associated to the server, in order to lowerbound the service. If the output fro the server is denoted Y, then β is defined as follows. Service curve. β is a (iniu) service curve for the server if Y t in U s + β t s, t 0. 0 s t Two indicators of the service perforance are considered. The backlog B(t) of data in the server at a given tie t is defined by B t = U t Y t. The virtual delay d(t) caused by the server at tie t is defined by d t = Inf 0, Y t + U t. In the case where initial data n 0 is assued in the server at tie zero, the definition of the virtual delay reains correct by replacing the signal Y by Y n 0. It is easy to see that arrival and service curves are not unique. In order to obtain good bounds

6 N. Farhi, H. Haj-Sale and J.-P. Lebacque 6 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 (resp. service) curve is siply the inial (resp. axial) one; see (Le Boudec & Thiran, 2001) and (Chang, 2000) for ore details. We recall below a basic result of the deterinistic network calculus (Le Boudec & Thiran, 2001) (Chang, 2000), 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 iniu service curve β, then we have The virtual delay is bounded as follows. d(t) sup t 0 Inf 0, α t β t +, t 0. The backlog is bounded as follows. B(t) sup s 0 α s β s, t 0. The curve t sup s 0 (α t + s β s ) is an arrival curve for the departure flow Y fro the server. The axiu backlog and delay on a server are then given siply as the axial vertical and horizontal distances between the arrival and the service curves; see Figure 1. Figure 1. On the left side: Schea of the server. On the right side: The axiu delay d and the axiu backlog b deterined graphically as the axiu horizontal and vertical distances between the arrival and the service curves, respectively. 3 Guaranteed service on a single-lane road (the odel) We present in this section an eleentary odel for the calculus of iniu guaranteed service for a single-lane road seen as a server. The objective of this odeling is to derive the axiu travel tie of cars passing through the road, fro the guaranteed iniu service of the road. The odel is written on the cuulative car-flow variables Q (see notations below). It is inspired fro the cell transission odel (Daganzo, 1994), (Daganzo, 1995); see also (Lebacque, 1996). The in-plus linear syste theory (Baccelli, Cohen, Olsder, & Quadrat, 1992) is then used to derive the iniu guaranteed service. Let us consider a single-lane road where cars ove without passing. In order to be able to fix the car-density, and to siplify the odel, we consider a ring road; see Figure 2. The road is divided into sections of length x. The axiu nuber of cars on one section is denoted by n ax. The ja density ρ j on the road is then given by

7 N. Farhi, H. Haj-Sale and J.-P. Lebacque 7 We use the following notations: ρ j = n ax x. (1) U t : The cuulated inflow of cars to the road fro tie zero up to tie t. Y t : The cuulated outflow of cars fro the road, fro tie zero up to tie t. q i (t) : The car outflow fro the i t section (the section between positions i x and i + 1 x) at tie t. t 0 Q i (t) = q i s ds : The cuulated car outflow fro the i t section up to tie t. n i : The nuber of cars in the i t section (the section between positions i x and i + 1 x) at tie zero. Figure 2. A single-lane ring road and the corresponding event-graph odel. Moreover, we assue the following fundaental traffic diagra for the road. q ρ = in vρ, w ρ j ρ, (2) Where q, ρ, v, 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 axiu flow is then given by q ax = ρ j 1 v + 1. w (3)

8 N. Farhi, H. Haj-Sale and J.-P. Lebacque 8 Figure 3. The fundaental traffic diagra. Let us first recall the well known case (Farhi, 2008) of an autonoous single-lane ring road, with no entry U and no exit Y. In this case the dynaics of the syste is written as follows. Q 1 t = in Q t x v + n, Q 2 t x w + n 1. Q i t = in Q i 1 t x v + n i 1, Q i+1 t x w + n i, i 2,3,, 1 Q t = in Q 1 t x v + n 1, Q 1 t x w + n, (4) where n k = ρ j x n k = n ax n k. The syste (4) is a discrete tie event syste for which the dynaics is well understood. Indeed, the syste (4) can be represented by an event graph (a class of Petri nets), and its dynaics can be written linearly in in-plus algebra (Baccelli, Cohen, Olsder, & Quadrat, 1992). Since the event graph representing the syste (4) is strongly connected, the asyptotic car flow on the road is then the sae at every section, and is given by the iniu over the average weights of the graph circuits (Farhi, 2008). Three circuits are distinguished in the event graph of Figure 2. The interior circuit with the average growth rate i=1 n i i=1 n i x/v = x The exterior circuit with the average growth rate i=1 n i v = vρ. x w = w ρ j ρ. The loops over each section, with the average growth rate n ax x/v + x w = ρ j x x/v + x w = q ax. The asyptotic car-flow is then given by q ρ = q i (ρ) = in{vρ, w (ρ ρ j ) }, i. and retrieves then the fundaental diagra (2) assued in the odel.

9 N. Farhi, H. Haj-Sale and J.-P. Lebacque 9 The average travel tie through the road is then given by τ(ρ) = x ρ q(ρ) x x = ax, v w ρ ρ j ρ. In the following, we are interested on the axiu travel tie through the road, rather than the average travel tie. For that, we introduce an inflow and outflow of cars on the road. We assue that cars enter into the road fro the entry of section 1, and exit it fro the sae point after passing through the road. The dynaics (4) are then rewritten as follows: Q 1 t = in U t, Q t x v + n, Q 2 t x w + n 1. Q i t = in Q i 1 t x v + n i 1, Q i+1 t x w + n i, i 2,3,, 1. (5) Q t = in Q 1 t x v + n 1, Q 1 t x w + n. Y t = in Q t x v + n, Q 2 t x w + n 1. Let us notice the difference between the dynaics of Q 1 which is conditioned by the inflow U, and the dynaics of Y which is not conditioned by the inflow U. We consider the sae odeling approach as above, but here we will see the variables as signals; see (Baccelli, Cohen, Olsder, & Quadrat, 1992). We first show that the dynaic syste (5) is linear in a certain algebraic structure, and then use this linearity to derive a guaranteed service of the road seen as a car server. Let us consider the set of tie functions F = f is non decreasing and f t = 0, t < 0, endowed with the following two operations. Addition (eleent-wise iniu): f g t in f t, g t, t 0. Product (in-plus convolution): f g t in 0 s t {f s + g(t s)}, t 0. We then obtain a dioid structure (F,, ) (Baccelli, Cohen, Olsder, & Quadrat, 1992) (Le Boudec & Thiran, 2001). The zero eleent of that dioid is denoted by ε and defined by ε t = +, t 0, while the unity eleent is denoted by e and defined by e t = 0 if t = 0. + for t 0. In addition, we consider the following notations: Power: f k denotes the product (in-plus convolution) of f with itself k ties. Additive closure: f t e f f 2 f 3 De-convolution: (f g)(t) = sup s 0 f t + s g s. and the two particular signals γ p (the gain signal) and δ T (the shift signal) in F (see Appendix A for ore details on those signals). γ p t = p if t = 0 + for t > 0 and δ T t = 0 if t T + otherwise

10 N. Farhi, H. Haj-Sale and J.-P. Lebacque 10 We notice that by using the notation of the in-plus convolution, the definitions of arrival and service curves can be rewritten as follows. α is an arrival curve for U if U α U. β is a (iniu) service curve for the server if Y β U. Now, since the cuulative flows U, Q i, 1 i, and Y are tie functions (or signals) in F, then by using the notations of addition and product in F, the syste (5) is linear according to those operations, and is written as follows 1. Q 1 = γ n δ x v Q γ n 1δ x/w Q 2 U. Q i = γ n i 1δ x v Q i 1 γ n iδ x/w Q i+1, i 2,3,, 1. Q = γ n 1δ x v Q 1 γ n δ x/w Q 1. Y = γ n δ x v Q γ n 1δ x/w Q 2. (6) Moreover, since we are not only interested in the average quantities, but on the axiu bounds, we include the initial conditions of the syste (5) (the signals are null at tie zero). To include those conditions, it is sufficient to add (in-plus addition) to each signal of syste (6) the signal unity e. Then, the dynaics (6) are written as follows. Q 1 = γ n δ x v Q γ n 1δ x/w Q 2 U e. Q i = γ n i 1δ x v Q i 1 γ n iδ x/w Q i+1 e, i 2,3,, 1. Q = γ n 1δ x v Q 1 γ n δ x/w Q 1 e. Y = γ n δ x v Q γ n 1δ x/w Q 2 e. (7) The syste (7) is then written as follows. Q = A Q B U E Y = C Q e (8) where Q = (Q 1 Q 2 Q ), A = ε γ n 1 δ x w ε ε γ n δ t γ n 1 δ t ε γ n 2 δ x w ε ε ε γ n 2 δ t ε γ n 3 δ x w ε ε ε ε γ n 2 δ t ε γ n 1 x w δ γ n δ x w ε ε γ n 1 δ t ε, B = e ε ε ε, C = ε γ n 1δ x/w ε ε γ n δ t and E = (e e e). The syste (8) is a in-plus linear syste. A basic result of the in-plus syste theory (Baccelli, Cohen, Olsder, & Quadrat, 1992) gives then the ipulse response of syste (8): 1 Note that, as in the standard algebra, the product operation is soeties just not sybolized (that is to say that f g can siply be written fg.)

11 N. Farhi, H. Haj-Sale and J.-P. Lebacque 11 Y = CA BU E e = CA BU CA E e. (9) Note that an ipulse response Y = CA BU would be obtained if syste (6) is considered instead of syste (7) (that is if the initial conditions of syste (6) are not taken into account). In that case, we would conclude directly that CA B is a iniu service curve of the road seen as a server (since we have Y CA B U), and derive a axiu bound for the travel tie through the road. But here, as entioned above, it is necessary to take into account the initial conditions of the dynaical syste, since we are interested in the axiu bound of the travel tie through the road, rather than the average travel tie. In order to be able to derive axiu bounds fro the forula (9), as done fro a service curve, we propose the following extension. Miniu service couple We consider here the case where the curve service is given with an additional affine ter. More precisely, we say that β, λ is a service couple for a server if Y β U λ. Then we can easily check (see Appendix B), that to obtain the three bounds given above, for that case, it is sufficient to replace β with β λ. That is to say that: The axiu backlog is bounded as follows. B(t) sup α s (β λ) s, t 0. s 0 The axiu delay is bounded as follows. d t sup Inf 0, α t (β λ) t +, t 0. t 0 An arrival curve for the departure flow is α (β λ). That is Y (α (β λ)) Y. Note that the curve (β λ) is not necessarily a iniu service curve for the server. The two first bounds are then given by the axiu vertical and horizontal distances between the curves α and β λ. Theore 1. A iniu service couple for the single-lane road, seen as a server, is β, λ given by β = γ ρ x a λ = β γ ρ x γ ρ kρ + j x δ ( k) x v γ kρ j ρ + x k x w δ k=1 k=1 e + where a = γ ρ x δ x/v γ ρ j x δ x v+ x w γ (ρ j ρ) x x w δ Proof. The syste (8) is an affine in-plus syste. We then have Y = CA BU CA E e. We need to copute A. Several ethods can be used to copute A. By definition of A, one can siply copute A 2, A 3, etc, then deduce A = k 0 A k by siplifying all the ters. We can easily check that A is given as follows.

12 N. Farhi, H. Haj-Sale and J.-P. Lebacque 12 (A ) ij = a if i = j a i 1 n γ k=j k δ i j t j 1 n γ k=i k δ j i x w if i j, with k cyclic in 1,2,,. In order to copute CA B, we need only the first colun of A, since only the first entry of B is not null. Thus, A B = (A ) 1 and is given by A B = a e γ n 1 δ t γ k=2 n k δ 1 x w 1 γ k=1 n k δ 1 t γ n δ x w. Therefore, CA B = a a. Siilarly, we can check that 1 1 CA E = a a γ i=k+1 n iδ ( k) x v γ i=1 n i k=1 Then, it is not difficult to check that i=k+1 k i=1 n i n i [ρ kρ j ] + x. [kρ j ρ] + x. k=1 k k x w δ Now, since we are interested in the car outflow fro the road that coes fro the car inflow, without counting the cars being in the road at tie zero (and stay there all tie because the + road is circular), we need to express the variable Z = [Y n i ] = γ ρ x Y + in function of U. We then conclude that the couple of curves γ ρ x CA B, γ ρ x CA E e is a iniu service couple for the ring road (see property (P4) in Appendix A). Finally, we have γ ρ x a a + = γ ρ x a +, since a a t = a t, t > 0 and a a 0 = a(0) < ρ x (see properties (P2) and (P5) in Appendix A). In order to show the shape of the iniu service couple given in Theore 1, let us take an acadeic exaple. Exaple 1. Let = 6, x = 1, v = 1, w = 1 2, ρ j = 1. Then we have n ax = 1, ρ c = 1 3 and q ax = 1 3. In addition, we take three cases, where we vary the average car-density on the road. Let us first notice that a can also be written as follows (see property (P3) in Appendix A). a = (γ ρ x δ x/v ) γ ρ j x δ x v+ x w (γ ρ j ρ x δ x w ) Then we have the three cases: n i = 1 i=1, that is ρ = 1 6 < ρ c. Then a = γ 1 δ 6 γ 5 δ 12 and a = (γ 1 δ 6 ). i=1

13 N. Farhi, H. Haj-Sale and J.-P. Lebacque 13 Hence the iniu service couple (β, λ) is given by β = γ 1 γ 1 δ 6 + = γ 1 γ 1 δ 6. λ = β γ 0 δ 5 γ 1 δ 6 γ 2 δ 8 γ 3 δ 10 = β. i=1 n i = 2, that is ρ = 1 3 = ρ c. Then a = γ 1 δ 3 and a = γ 1 δ 3. Hence the iniu service couple (β, λ) is given by β = γ 2 γ 1 δ 3 +. λ = β γ 0 δ 8 γ 1 δ 10. i=1 n i = 3, that is ρ = 1 2 > ρ c. Then a = γ 1 δ 3 γ 3 δ 12, a = γ 1 δ 3 γ 3 δ 12. Hence the iniu service couple (β, λ) is given by β = γ 3 γ 1 δ 3 γ 3 δ λ = β γ 0 δ 10 = β. The iniu service couples (given in Theore 1) corresponding to each of the three cases, are shown in Figure 4. We give below a corollary of Theore 1, where by relaxing the iniu service couple given in Theore 1, we obtain practical forulas. Corollary 1. A iniu service couple (β, λ) for the road is given by β t = q ρ t τ ρ + λ t = β(t) wρ j t 2ρ x ρ j w where q(ρ) and τ(ρ) are the average flow and average travel tie given by the fundaental diagra (given in section 2). +

14 N. Farhi, H. Haj-Sale and J.-P. Lebacque 14 Figure 4. Calculus of the service couple for the single-lane ring road in three cases of low, critical and high car-density. Proof. 1. Fro the curve β given by Theore 1, we have γ n x i=1 i δ v γ n ax δ x v + x w γ n x i=1 i δ w ρv t, q ax t, ρ j ρ wt.

15 N. Farhi, H. Haj-Sale and J.-P. Lebacque 15 Then γ n x i=1 i δ v γ n ax δ x v + x w γ n x i=1 i δ w in ρv, q ax, ρ j ρ w t. But since ρ, in ρv, ρ j ρ w q ax, then γ x i=1 n i δ v γ n ax δ x v + x w γ x i=1 n i δ w in ρv, ρ j ρ w t. Hence γ i=1 n iδ x v γ n ax δ x v + x w γ i=1 n iδ x w ρ x ρ x in ρv, ρ j ρ w t in ρv, ρ j ρ w in ρv, ρ j ρ w t ax x v, ρ ρ j ρ + + x w Fro the curve λ given in theore 1, we have then 1 k=1 γ ρ kρ + j x ( k) x v δ ρ x γ γ In the other hand, we have then 1 k=1 ax vρ j t ρ ρ j 1 k=1 γ kρ j ρ + x k x w δ ρ x γ γ 1 k=1 3. Finally, it is easy to check that x v ρ kρ + j x ( k) x v δ ax wρ j t ρ x ρ j w kρ j ρ + x k x w δ β(t) vρ j t x v since q(ρ) vρ j and τ(ρ) x v. Let us notice that the ter wρ j t 2ρ ρ j x w +, γ ρ ρ + j x 1 x v δ vρ j t x v +. +, γ ( 1)ρ j ρ + x ( 1) x w δ wρ j t 2ρ x ρ j w + + is ore iportant than the ter β(t) in the service couple given in Corollary 1. Indeed the curve β(t) gives siply the average service of the road, since q(ρ) is the average car-flow and τ ρ is the average travel tie on the road. +.

16 N. Farhi, H. Haj-Sale and J.-P. Lebacque 16 Corollary 2. If α t = σ + rt is an arrival curve for the inflow U to the road, then the following bounds are guaranteed. A axiu bound for the travel tie of cars through the road is τ ax = ax τ + σ q, 2ρ x ρ j w + σ. wρ j A axiu bound for the nuber of cars waiting at the entry of the road (not confuse with the nuber of cars queuing at the exit of the road) is b ax = ax σ + rτ, σ + r 2ρ ρ j An arrival curve for the departure flow fro the road is α t = b ax + rt. x w. Proof. It is well known (Le Boudec & Thiran, 2001) that if a flow with an arrival curve α t = σ + rt is served in a server with a iniu service curve β t = R(t T) +, then the axiu delay is T + σ R, the axiu backlog is σ + rt, and the curve σ + rt + rt is an arrival curve for the departure flow. The result is then obtained by adapting these bounds to the case of couple service instead of iniu service curve (see Appendix B), and by using the couple service given in Corollary 1. In the road traffic, it is probably not interesting to assue arrival curves with no null σ. One ay siply estiate the arrival flow rate (linear arrival curve) and the car-density on the road, at a given tie, and want to deterine the axiu three bounds at the considered tie instant. In this case, we siply have fro Corollary 2: τ ax (ρ) = ax τ ρ, 2ρ x ρ j w = ax 1 v, ρ ρ j ρ and b ax = r τ ax, and α t = r(t + τ ax ). 1 w, 2ρ 1 ρ j w x, (10) The forula (10) tells that the axiu travel tie through the road is greater than the average travel tie only in the car-density interval [ ρ j 2 w v, ρ j 2] when w < v. That is to say that ρ j w 2 v < ρ < ρ j 2 τ ax ρ > τ ρ. (11) In Figure 5, we show the average and the axiu travel ties in function of the car-density in the case where w < v.

17 N. Farhi, H. Haj-Sale and J.-P. Lebacque 17 Figure 5. The average and the axiu travel ties through the road in the case where w = v/2. Let us notice that the choice of the traffic flow odel (cell transission odel) and that of the fundaental traffic diagra (triangular one) are very iportant. Indeed these two choices peritted to write the car-dynaics on the road linearly in the in-plus algebra. This is necessary because the service couple is derived as an ipulse response of a in-plus linear syste. 4 Road network calculus We give in this section the first ideas on how to use the results obtained on a single-lane road to extend the approach to intersections and whole transportation networks. First we notice that for ulti-lane roads, one ay just use one fundaental diagra for all lanes and apply the sae approach as done for a single-lane road. We assue here that the sae odel and results are used for ulti-lane roads. However, other odels siilar to the one given above can also be developed for ulti-lane traffic. Before presenting the procedure of extending our approach to intersections and networks, we need to recall and extend an iportant result of the deterinistic network calculus on the series coposition of servers (it shows in particular the power of the algebraic approach against other approaches). The result tells that a iniu service curve of the series coposition of two servers guaranteeing β 1 and β 2 as iniu service curve for each of the, is siply the curve β 1 β 2. The result can easily be proved by using the associative property of the in-plus convolution; see (Chang, 2000) (Le Boudec & Thiran, 2001) for ore details. Moreover, since the in-plus convolution is also coutative, then the order of the coposition of the two servers is not iportant. For our odel, we need to have a siilar result for a coposition of servers offering iniu service couples (rather than iniu service curves). It is easy to see that the coposition of two servers offering two service couples β 1, λ 1 and β 2, λ 2 guarantees a service couple β 1 β 2, β 1 λ 2 λ 1. Indeed fro Y = β 1 Z λ 1 and Z = β 2 U λ 2, we get Y = (β 1 β 2 ) U (β 1 λ 2 λ 1 ).

18 N. Farhi, H. Haj-Sale and J.-P. Lebacque 18 In counication and coputer networks, one deterines first the (residual) guaranteed service for all (input, output) couples through every switching router. Then, it is sufficient to copose (with a in-plus convolution) all the guaranteed services through all the arcs of a given path, to deterine the service guaranteed through the whole path. Finally, one deterines the axiu end-to-end delays on a counication network by calculating the axiu delay on each (origin-destination) path on the network, by siply using its guaranteed service curve, and considering the whole path as an eleentary 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; see (Chang, 2000) and (Le Boudec & Thiran, 2001) for ore details. We think that we can proceed siilarly for transportation networks. Indeed, the ain difference between data traffic in counication networks and car-traffic in transportation networks is the interaction between particles (drivers observe reaction ties contrary to data packets), expressed in the fundaental diagra of the road. Since this difference is already taken into account in the one road odel (presented above), then to extend the approach to coplicated transportation networks, it reains the adaptation of the residual guaranteed service calculus on input-output couples of routers to apply in intersections, for transportation control policies. Then one only needs to copose eleentary road services and residual guaranteed services, to obtain guaranteed services on whole paths. Maxiu travel ties can then be derived siilarly. We notice here that one of the ost difficult issues to solve, in order to extend the approach presented in this article to big networks, is the presence of cyclic dependencies of inflows arriving to one intersection, even though soe eleentary results exist to deal with that issue (Chang, 2000). Let us explain the calculus of axiu bounds of travel ties in a tree-like network. Let us consider the transportation network of Figure 6. Figure 6. Tree-like network. The car-traffic goes fro the left side to the right side. In Figure 6, the notation A + indicates soe point downstrea of intersection A. The notation C A indicates soe point upstrea of intersection C in the direction of intersection A. All other notations are interpreted siilarly. We assue that we have the fundaental diagras of the roads (A +, C A ), (B +, C B ), (C +, E C ), (D +, E D ) and (E +, F E ). The approach for calculating a axiu bound for the travel tie fro A to F in Figure 6 is the following.

19 N. Farhi, H. Haj-Sale and J.-P. Lebacque Deterine 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 intersections. We siply use the fundaental diagras of the roads, and the odel presented above. 2. Calculate the residual services on the erge C: a. The service guaranteed on C for the aggregate inflows coing fro C A and C B is assued to be siply the guaranteed service on the road (C +, E C ). b. The control policy on C is assued to be known. c. Fro these two inforation (a and b), calculate the residual guaranteed services for the flows C A, C + and (C B, C + ). This shall be done by adapting well known results of network calculus theory on the calculus of residual guaranteed service. (This is not yet done). 3. Calculate the residual services on the erge E. By the sae ethod as in 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 siply (A, F), is given by the series coposition 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 siilarly. The axiu travel ties fro A (resp. B and D) to F are then derived fro the guaranteed services A, F, B, F and D, F respectively, as done on a single road. Conclusion We presented in this article a network calculus traffic odel that perits the derivation of a axiu bound for the travel tie of cars passing through a single-lane road. An iportant advantage of this odel is its algebraic forulation which is very powerful coparing to other forulations (e.g. the series coposition). Even though the odel is basic and eleentary, its developents and extensions ay be proising. Our future work shall be on the realization of the odel extension process presented in section 5. We shall also deonstrate the effectiveness of our approach by perforing nuerical investigations on effective data sets. Appendix A (Details on the signals γ p and δ T ) We give here soe particular signals in F as well as soe properties used in section 3 (traffic odel). The gain signal γ p : The shift signal δ T : γ p t = δ T t = p if t = 0 + for t > 0 0 if t T + oterwise

20 N. Farhi, H. Haj-Sale and J.-P. Lebacque 20 It is then easy to obtain the following signals: The signal γ p δ T : The signal γ p δ T : γ p δ T t = γ p δ T t = p if t [0, T] + if t > T 0 if t 0 kp for kt t < k + 1 > T, k N Moreover, we can easily check (see Figure 7) that γ p δ T t p T t, t. We explain here the convolution of the signals γ p and δ T with a signal f. γ p f t (γ p f) t = f t + p, t. δ T f t δ T f t = f t T, t. γ p δ T f t γ p δ T f t = f t T + p, t. Figure 7.The signals γ 2, δ 3, γ 2 δ 3 and γ 2 δ 3 respectively fro left side to right side. We recall the following additional properties; see (Baccelli, Cohen, Olsder, & Quadrat, 1992) and/or (Le Boudec & Thiran, 2001). (P1): f F, f f f f. (P2): f F, f 0 = 0 f f = f. (P3): f, g F, (f g) = f g. (P4): f, g F, f 0, g 0 [ f g γ a ] + = f [g γ a ] +. (P5): f F, e f f = f. Appendix B (iniu service couple) We clarify here the three bounds of network calculus in the case where we have a iniu service couple β, λ instead of a iniu service curve β. That is to say that we have Y β U λ instead of Y β U.

21 N. Farhi, H. Haj-Sale and J.-P. Lebacque 21 The axiu backlog: B t = U t Y t U t in in 0 s t ax ax 0 s t ax ax z 0 ax ax s 0 ax s 0 U t s + β s, λ t α s β s, α t λ t ax 0 s z α s β s, ax z 0 α s β s, ax s 0 α s (β λ) s. α z λ z α s λ s The axiu delay: Let t 0. Let τ 0 such that τ < d t inf{ 0, U(t) Y(t + )}. Then we have U(t) > Y(t + τ). We have Y t + τ in in U t + τ s + β s, λ t + τ. 0 s t+τ That is s 0, 0 s 0 t + τ, Y t + τ in U t + τ s 0 + β s 0, λ t + τ. Then s 0, 0 s 0 t + τ, U(t) in U t + τ s 0 + β s 0, λ t + τ. Therefore If U(t) U t + τ s 0 + β s 0 then s 0 > τ and thus α s 0 τ U t U t + τ s 0 > β(s 0 ) Hence τ < sup inf 0, α t β t + sup inf 0, α t (β λ) t +. t 0 t 0 If U(t) λ(t + τ) then since U 0 = 0, we have α(t) U(t) λ(t + τ). Hence τ < sup inf 0, α t λ t + sup inf 0, α t (β λ) t +. t 0 t 0 The departure flow: Y t Y s U t ax in 0 z s U s z + β z, λ s ax ax 0 z s ax ax 0 z s U t U s z β z, U t λ s α t s + z β z, α t λ s ax ax α t s + z β z, ax α t s + z λ z 0 z s 0 z s ax α t s + z in β, λ z (α (β λ))(t s). 0 z s References Ain, S., & al. (2008). Mobile century-using GPS obile phones as traffic sensors: a field experient. New York: In 15th World congress on ITS. Baccelli, F., Cohen, G., Olsder, G. J., & Quadrat, J.-P. (1992). Synchronization and Linearity. Wiley. Chang, C.-S. (2000). Perforance guarantees in counication networks. Springer. Claudel, C. G., & Bayen, A. M. (2008). Guaranteed bounds for traffic flow paraeters estiation using ixed Lagrangian-Eulerian sensing. Allerton: 46th Annual Allerton Conference on Counication, Control, and Coputing.

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