Fabien Leurent
email: fabien.leurent@enpc.fr

Vincent Benezech

François Combes

Relstat12 Leurent

Benezech Combes

A RELATIVITY THEORY OF TRAFFIC ALONG A TRANSIT LINE

Keywords: Vehicle load, headway rank, passenger exposure, wait time, run time, journey time, assignment model

Along a transit line, vehicle traffic and passenger traffic are jointly subject to variability in travel time and vehicle load hence crowding. The paper provides a stochastic model of passenger physical time and generalized time, including waiting on platform and in-vehicle run time from access to egress station. Five sources of variability are addressed: (i) vehicle headway which can vary between the stations provided that each service run maintains its rank throughout the local distributions of headways; (ii) vehicle order in the schedule of operations; (iii) vehicle capacity; (iv) passenger arrival time; (v) passenger sensitivity to quality of service. The perspective of the operator, which pertains to vehicle runs, is distinguished from the user's one at the disaggregate level of the individual trip. After recalling the basic properties from a previous paper [0], this paper provides additional properties and explores some consequences for models of traffic assignment to a transit network.

Introduction 1.Background

The operations of a transit line, and even more of a network of lines, are submitted to variability in a number of ways. On the operator side, vehicle type may not be homogeneous, the passenger load depends on the service schedule and varies along the route, traffic disruptions arise due to causes either internal (such as human error, material incident, passenger incident or accident…) or external (such as adverse weather, malevolent intrusion, conflict with another flow…). On the demand side, the passenger experiences travel conditions along his trip, from service waiting and platform occupancy at the access station up to station egress passing by vehicle occupancy and its journey time, which vary according to the occurrence of the trip in a series of reiterations and also between passengers on a given occurrence. A major issue pertains to service reliability: any disruption causing a large delay induces a significant loss in quality of service, and the frequent reiteration of such events will make the passenger reconsider his travel decision of network route and even of transportation mode. Stated Preferences surveys have shown that frequent significant delays amount to additional travel time in a more than proportional way: for instance, the factor of proportionality was estimated to 1.5 for delays of more than 10 minutes occurring three out of 20 times in Paris suburban railways [START_REF] Kroes | On the value of punctuality on suburban trains to and from Paris[END_REF]. Such behavioral patterns must be taken into account in network planning, both within network traffic assignment models and the cost-benefit analysis of transportation projects.

Objective

The paper's objective is to provide a stochastic model of traffic variability and passenger exposure along a transit service. On the side of transit services that are supplied to passengers, the model assumptions involve the statistical distribution of, first, the local vehicle headways at station nodes, second, the local run times along inter-station links, third, vehicle capacity in terms of seated and standing passengers. On the side of passenger demand, a spatial pattern is assumed for the access-egress matrix of passenger flows, together with a statistical distribution (temporal pattern say on a day-to-day basis) of a volume index. The model yields the following outcomes: (i) the distribution of vehicle journey times by pair of accessegress stations, together with the distribution of passenger loading; (ii) the distribution of passenger physical time by access-egress pair; (iii) the distribution of passenger generalized time by access-egress pair, assuming that crowding density adds discomfort cost to travel times. Thus the interplay of operations variability with the spatial pattern and temporal distribution of passenger flows is captured in an explicit and consistent framework. The framework amounts to a line model that can be used in a larger model of traffic assignment to a transit network, which also deals with route choice.

Approach

The model pertains to the physics of traffic operations and passenger exposure to travel conditions both of service operations and vehicle load. The main variables of vehicle traffic, passenger traffic and passenger travel are cast into a probabilistic framework in the form of random variables. Variability sources are identified, among which the major one is the heterogeneity of vehicle headways. Analytical properties are established between the main model variables, in the form of functional relationships linking the CDF, PDF, mean and variance of them. This is achieved by postulating the conservation of headway rank by service run and by deriving a series of consequences on the basis of probabilistic calculus. Overall, the model blends up probabilistic analysis taken mostly from the theory of renewal and survival, with traffic analysis at the two levels of transit vehicles and passengers, respectively. In the basic exposition of the model [START_REF] Leurent | A stochastic model of passenger generalized time along a transit line[END_REF], three steams of related previous work have been identified, namely (i) analysis of vehicle traffic only [START_REF] Carey | Stochastic approximation to the effects of headways on knock-on delays of trains[END_REF], [START_REF] Meester | Stochastic delay propagation in railway networks and phase-type distributions[END_REF], (ii) focus on passenger waiting on platform [START_REF] Bowman | Service frequency, schedule reliability and passenger wait times at transit stops[END_REF], (iii) empirical or simulation-based analysis of travel conditions [START_REF] Islam | Impact of Occupancy Profile for Waiting Time of Bus Transit Users[END_REF], [START_REF] Xue | Empirical Characteristics of Transit Travel Time Distribution for Commuting Routes[END_REF]. To the best of our knowledge, our analytical approach to distinguish the operator and user perspectives is original in the context of transit traffic: it may be called a relativity theory of traffic along a transit service.

Structure

The rest of the paper is organized in six sections. Vehicle traffic is considered first, by focusing on headways and deriving some consequences on journey times by pair of entry-exit stations (Section 2). Then, passenger load by vehicle is characterized with respect to headway rank and the index of demand volume (Section 3). Next, we turn our attention to passenger exposure to in-vehicle crowding, wait time and travel time (Section 4). The consequences of service irregularity and other variations affect not only the physical times but also the "generalized time" which takes into account the discomfort of specific travel states (Section 5). From this stem important consequences for models of traffic assignment to a transit network that are applied to cases that involve both irregularity and congestion (Section 6). Lastly, the conclusion points to the model scope, limitations and potential developments (Section 7).

On vehicle headways and journey times

A transit line operated along a single service route in a single direction is considered. The stations are indexed by M m  and the sections or links between adjacent stations by A a  . Each vehicle run is characterized by a trajectory in space and time. The journey time is made up of the run times on the sections plus the dwell times at the stations. The objective of this section is to model the statistical distribution of vehicle run times between station pairs along the line. The statistical population of interest is the set of runs during a reference period, for instance the morning peak hour of working days. First, we shall model the distribution of vehicle headways ( § 2.1). Second, their propagation between stations is addressed in § 2.2. Then, a postulate is made about the "conservation of headway rank" ( § 2.3), which entails specific properties for the distribution of vehicle headways ( § 2.4) and that of journey times ( § 2.5).

On vehicle headways

] [ m   or the relative dispersion ] /E[ ] [ ] [ m m m      
. Assuming that the incoming passengers at station m arrive independently from one another and from service schedule, their arrivals can be modeled as a Poisson process and, if the process intensity is medium or high, then it can be safely assumed that the number of passengers waiting for a given vehicle is proportional to the headway (neglecting any capacity constraint). Furthermore, the distribution of passenger waiting times at m stems from that of vehicle headway in a specific way (see Section 4).

Spatial propagation

The instant of departure of vehicle i from station m , ) (i h m , is separated from that of the next station, )

( 1 i h m
, by the run time along section )

) ( ) ( ) ( 1 i t i h i h a m m    .
(2.1)

Note that we also have:

) ( ) ( ) ( 1 i h i h i m m m     . So that from vehicle 1  i
to vehicle i , the headways at service stations satisfy:

) ( ) ( ) ( 1 i i i a m m       , (2.2)
wherein ) 1 ( ) ( ) (     i t i t i a a a
is the difference in travel time along a and m .

Service operations and exogenous influences may affect the distribution of a  and, in turn, that of m  . The influences on the mean and variance are of crucial interest. By the linearity of expectation:

] E[ ] E[ ] E[ 1 a m m       , (2.3)
whereas, by the bi-linearity of covariance,

) , cov( 2 ] V[ ] V[ ] V[ 1 1 a m a m m           .
(2.4) Formula (2.2) and its consequences (2.3-4) state the propagation of vehicle headways from station to station.

On the conservation of headway rank

Of course, the conservation of schedule order is assumed along the line, under a First In -First Out discipline. Let us focus on the rank of each run in the "local" distribution of headway, characterized by the fractile ) ( H

m m m   
. In this study, the postulate of conservation of headway rank is made:

) ( ) ( ) ( , , i i i n m i n m         . ( 2

.5)

This states that if a vehicle run is associated to a relatively low (resp. large) headway at a given station, it is associated to relatively low (resp. large) headways at all the stations of the line. However, local magnitudes may differ, only the rank remains stable. The postulate is realistic enough in various instances:  when the operations are regular along the line, the headway at the initial station is maintained from station to station.  If most of traffic disruptions occur on a given section a , then the main source of variation pertains to a  and the rank in its distribution may be assumed to apply on the rest of the line as well. The most noteworthy consequence is the functional dependency between the headways along the line:

) ( H ) ( H 1 1 1             m m m m m m , hence (2.6
)

) ( H H 1 1 ) 1 (       m m m m  . (2.7) Thus 1       m m a
also is a function of

( ] [ / ]) E[ ( 1 1 1               m m m a a a
. This holds notably for perfectly correlated normal variables: in this case a valuable complementary property is that m  is normal, too, yielding normal variables for headway and section time variation along the line.

Vehicle journey time with respect to schedule order

Let us turn to the journey time of each vehicle run with respect to its order in the schedule of operations, denoted by i . Let r denote a reference station and r m  a subsequent station in the selected direction of traffic, ) (i t rm be the journey time of vehicle run i between the instants of departure from r and m , )

(i h m and ) (i h r respectively. It holds that ) ( ) ( ) ( i h i h i t r m rm   , so ) ( ) ( ) 1 ( ) ( i i i t i t r m rm rm       and          i j r m rm rm j j t i t 1 ) ( ) ( ) 0 ( ) ( , (2.9)
wherein vehicle run #0 is an ideal vehicle run of nominal performance which immediately precedes the reference period. By the linearity of expectation, it then follows that

]) E[ ] (E[ ) 0 ( )] ( E[ r m rm rm i T i T      . ( 2

.10)

Under the assumption that the ) (i  are i.i.d., the runs are mutually independent, which implies that:

] V[ . )] ( V[ r m rm i i T     . ( 2

.11)

Under the conservation of headway rank and the assumption of normality,

       ] , [ m r a a r m satisfies that           ] , [ ] [ ] [ ] [ m r a a r m , which entails that ] [ ] [ ] [ ] [ ] , [ r m m r a a r m                .
(2.12)

Combining (2.12) and (2.11), we get that

] [ )] ( [ r m rm i i T       . ( 2

.13)

Of course the assumptions of headway rank conservation and of run independence are likely to interfere in practice. However, eqns (2.10) and (2.13) give some insight into the progressive deterioration of the vehicle journey time with respect to the order of the run in the schedule of operations, when submitted to irregularity and random disruptions.

Vehicle loading

So far, two sources of variability have been made explicit: headway rank, denoted as  , and the order in the schedule, denoted as i . In this section, two other sources are introduced, namely the level of passenger transport demand, denoted as  , and the vehicle capacity, denoted as  . Sources  and  jointly influence the vehicle load in passengers. Sources  ,  and  jointly influence the ratio of load to capacity by vehicle run. This section establishes some analytical properties of the passenger load and load ratio along a transit line, by taking into account the demand (passenger flow) between stations of entry and exit.

Assumptions about passenger demand

A reference period of given duration is considered for line operations. In fact it refers in some average way to a population of periods, for instance the morning peak hour throughout a series of working days.

To depict the variability of periods, let us associate to each period its level  of passenger demand, with CDF B in the population of periods. Within a given period, passenger flow is modeled as a stationary random process, with macroscopic properties as follows: between any pair s r  of stations along the line, the passenger flow arriving at r and destined to s

during time interval ] , [ h h  amounts to ) ( h h q rs   
. Thus the set of trip rates ] : [ s r q rs  describes the spatial structure of passenger demand per unit of time. Across the population of periods, we could define  so as to satisfy that 1 ] E[   ; however we shall keep ] E[ in the formulae for the sake of traceability.

Vehicle loading conditional on 

Assuming that passenger demand is not restrained by vehicle capacity, at each station r of entry a given vehicle run will attract incoming passengers in proportion to its local headway, r  . On section a , the vehicle load denoted by a y consists in those passengers having entered at station a r  and destined to station a s  (with obvious notation for  and  for position along the line):

       a s a r r rs a q y , , . (3.1)
Then, on average:

       a s a r r rs a q y , , ] E[ ] E[ . (3.2)
Keeping to the postulate of conservation of headway rank, the vehicle run is characterized by its fractile

 so that ) ( H ) 1 (     r r . Then          a s a r r rs a q y , ) 1 ( , ) ( H ) ( . (3.3)

Denote by

 , Y a the CDF of a y conditional on  . Then:

        a s a r r rs a q , ) 1 ( 1 , H Y . ( 3.4)
Furthermore, as in the previous section the sum of totally dependent random variables sharing a Gaussian pattern satisfies that

         a s a r r rs a q y ] [ ] [ . (3.5)

Vehicle loading, overall distribution

Let us now aggregate the analysis with respect to  . Denoting

      a s a r
r rs a q , the random variable of reference link flow and by a X its CDF, it holds generally that:

       ) dB( ) / ( X } { Pr ) ( Y z z z a a a . ( 3.6)
In reality, demand level  may influence vehicle operations -for instance because the number of boarding and alighting passengers may determine the dwelling time. However, for simplicity, independence is assumed in this model, yielding that:

      a s a r r rs a q y , ] E[ . ] E[ ] E[ . (3.7) ] V[ . ] E[ ] E[ . ] V[ ] V[ 2 2 a a a y       due to ] V[ XY ] V[ ]² E[ ] V[ ²] E[ X Y Y X   , hence     2 , 2 2 , ] [ . ] E[ ] E[ . ] V[ ] V[              a s a r r rs a s a r r rs a q q y . ( 3
) exp( ] E[ 2 2 1 a a a s m    2 2 2 ] E[ . ) 1 ) (exp( ] [ a a a s      Assuming lastly that ) , LN(     s m , then the link load ) , LN( 2 2 a a a s s m m       .

Vehicle loading ratio

Vehicle capacity, denoted as  , pertains to the number of seats plus a reference number of positions for passenger standing with sufficient comfort (e.g. 4 persons per square meter). Heterogeneous vehicles may be used to operate the transit line, leading to the variability of capacity hence of the ratio of passenger load to capacity. Let us denote that ratio as

      / / a a a y z . (3.9)
While it is quite natural to assume the independence of  and  , it would be a wise policy of line operations to assign vehicle types according to the planned headways, by associating larger capacity to larger headways so as to balance the load ratio across the runs. Under such a balancing policy, the load ratio could be analyzed in the same way as vehicle load by replacing

) (  r with     / ) ( r .
On the contrary, a negligent policy may be modeled by assuming independence between  and  as well as  , yielding straightforward consequences on the mean and variance of load ratio (eqns. (3.10-11) in [START_REF] Leurent | A stochastic model of passenger generalized time along a transit line[END_REF]).

Passenger exposition to traffic conditions

Let us come to the perspective of the user at the level of the individual trip, as opposed to the operator's one at the level of the vehicle run.

User's exposure

Let us recall some basic properties of renewal theory (e.g. [START_REF] Kleinrock | Queuing systems[END_REF]

) ( H ) ( H o u     r r   . ( 4
] E[ / ] ) E[( ] ) E[( o 1 o u r k r k r      . ( 4
) , ( f ) , ( f o u        , (4.3)
wherein o f is the PDF of passenger group sizes from the perspective of the operator. Assuming independence between  and  , then

) ( H ). ( B ) , ( f o o o      r   : thus independence is maintained in the user's perspective, since ) ( H ). ( B ) ( H ). ( B ) , ( f u u o o u           r r     . (4.4) in which ] /E[ ) ( B ) ( B o o u        and ] /E[ ) ( H ) ( H o o u r r r        . As    . u r n , its CDF is            ) ( dB ) / ( H ) ( dB ) ( N } { Pr ) ( N u u u u , u x x x x r r r . ( 4

.5)

The independence property enables us to establish the mean and variance of group size as follows: . Then, conditionally to  :

] E[ ] ) E[( ] E[ ] ) E[( ] E[ ] E[ ] E[ o 2 o o 2 o u u u r r r r n         . (4.6)                                         2 o 2 2 o o 3 o o 2 o 2 o 2 2 o o 3 o o 3 o ] E[ ] ) E[( ] E[ ] ) E[( 2 ] E[ ] ) E[( ] E[ ] ) E[( ] E[ ] ) E[( ] E[ ] ) E[( 2 u u u 2 u u ] E[ ] V[ ] V[ ] E[ ] V[
         a s a r e r rs e a q y , o ) 1 o( u , , , ) ( H H  . ( 4.9)
Eqn (4.9) must be distinguished clearly from (3.3), in which  is the order of fractile and can be seen as a random variable distributed uniformly on [START_REF] Leurent | A stochastic model of passenger generalized time along a transit line[END_REF][START_REF] Kroes | On the value of punctuality on suburban trains to and from Paris[END_REF]. From the equation above stems the unconditional variable u ,e a y . Its CDF is given by:

           d d ) , ( f } { Pr } { Pr ) ( Y u u , , , u , u , z y
                            z z z z q z y a e
           ) ( B d ) / ( X H H d ) ( f } { Pr ) ( Y u o ) 1 o( u u u , ,

Specific properties

In [START_REF] Leurent | A stochastic model of passenger generalized time along a transit line[END_REF] the distribution of vehicle load has been explored by assuming log-normal headways and lognormal vehicle loads in the operator perspective, yielding log-normal vehicle load in the user perspective.

In the general case, by inverting (4.11a), we get that

) ( H H X ) ( Y ) 1 ( u o ) 1 ( o ) 1 ( u , ,         e e a e a   . ( 4

.12)

Then, conditional on  , the expectation of any function f of y is

       d )) ( f(Y )] E[f( 1 0 ) 1 ( u , ,



. By construct, Assuming that o a  and  are statistically independent, then the unconditional moments of the vehicle load in the user perspective are derived from their conditional counterparts in the following way:

1 d d d ) ( H H d ) ( H H ). ( H since / ) ( H / ) ( H ) ( H since ) ( H H ) ( H ). ( H ) ( H H ). ( H d ) ( F d ) 1 ( o o ) 1 ( o o ) 1 ( o o ) 1 ( o o o u ) 1 ( o o o ) 1 ( o ) 1 ( o ) 1 ( o u ) 1 ( o      e e e e x x             Then, as ) ( Y ) ( Y ) 1 ( o , , ) 1 ( u , ,        e a e a if ) ( F    e , o 1 0 ) 1 ( o ) 1 ( o , , u , , / d ) ( H )) ( f(Y )] E[f( e e e a e a y                 . ( 4
                                                    e a e
) 1 ( ] [ ] E[                  e a e o , , o o o o , , o , , o o , , u , , / ] [ wherein
      ) 2 ] E[ ( ) ( ] V[ ] [ 2 ] E[ ] V[ ). ( E )]. ( ) ( ) )( [( E / ) ,
q q q q q y                                                                So: ) 2 ] E[ ( ) ( ] ) E[( ] ) E[( o 3 2 o o 2 o , , 2
) 1 ( ] E[ ] E[ o o o , , u u , e a e a e a y y       , (4.18a) ] ] E[ 1 ].[ V[ . ] E[ )] 1 .( .[ ] V[ ] V[ 2 o 3 o o , 2 u 2 o o o , u u , , e e e a e a e a e a y                . ( 4

.18b)

Formula (4.18a) was obtained as an approximation in [START_REF] Leurent | A stochastic model of passenger generalized time along a transit line[END_REF] under the heuristic assumptions of first normal then log-normal distribution of headways: it turns out that it is an exact property in the general case.

Run time

In section 2.3 some statistical properties of run time have been established for vehicles: schedule order i determines the mean and variance of run time ) (i T rs . Any user that arrives at station r at a given instant

i  . Let also ] E[ ] E[ E r s      . The average run time is E ) 0 ( ]) E[ ] (E[ ) 0 ( )] ( E[ ] E[ 2 1 1 1 1 1 u               I rm I i I r s rm I i rs I rs t i t i t t . ( 4

.19)

By the law of total variance, the variance of the run time is made of an interclass part plus an intra-class part in the following way, in which ] [

r s        :   2 12 1 2 2 1 2 1 2 24 1 2 1 2 1 1 2 2 1 2 1 1 2 1 1 2 u 1 u ) ( E ) ( E ) ( ) ( E )] ( [ ]) E[ )] ( (E[ ] V[ 2                                  I I I I I i I I i I I I i rs I I i rs rs I rs i i i t t i t t (4.20)

Wait time

The user wait time on the station platform, e w , amounts to the residual span (or lifetime) of the on-going headway interval. From survival theory, its PDF is

  ] /E[ ) ( H 1 ) ( W o o e e e x x     (4.21)
This leads to the following relationships between the moments of the two variables: There may be some correlation between the two components. However independence may be assumed as a crude approximation, yielding: , times the time elapsed since the departure of the last vehicle. From survival theory [START_REF] Kleinrock | Queuing systems[END_REF], the latter is the random variable (4.28)

) 1 /( ] ) E[( ] E[ ) 1 ( ] ) E[( ] E[ u o 1 o         k k w k e e k e
] V[ ] V[ ] Ṽ[ u u rs r rs t w t   . ( 4
Independence of  and r  implies that r r w   u is independent of  , yielding ) ( ] E[ ] E[ ] E[ u 2 1      r s rs r r q S (4.29a)   2 u u 2 2 ] E[ ]. V[ ] V[ ] E[ ) ( ] V[ r r r s rs r q S         . ( 4.29b)
The perspective of either the operator or the user is specified by setting the adequate distribution of  .

On passenger generalized time

To a trip-maker, the "generalized time" of travel is a comprehensive disutility to capture both the physical travel time and the quality of service during the trip. Each physical state (e.g. sitting in-vehicle) or transition (e.g. vehicle egress) within the trip sequence, is associated with a specific penalty factor: from 1 for sitting in-vehicle to 2 for standing in-vehicle under dense crowding or more for waiting in crowd with no traffic information. The physical time spent in a given state is multiplied by its penalty factor to yield the generalized time of that state. This is aggregated along the trip sequence to yield the generalized time of the trip. It is used in discrete choice models of network route or transportation mode. It is also the basis to evaluate the benefits and costs of a transport plan to the community.

The formation of generalized time

The notion of generalized time involves penalty factors that vary across the individual trip-makers. Small persons resent standing in a crowd more than tall ones do. In general, old persons move and walk more slowly than younger ones. People are more or less sensitive to fatigue. Let  denote the particular sensitivity of a given individual. Wait time r w and link time a t are transformed into generalized times, denoted as   r and   a , respectively. The generalized travel time amounts to

          [ , ] ,
s r a a r rs .

(5.1)

To model the dependency of  and  on the crowding density, assume that

) ( r r r r S w      , (5.2) ) , ( u u
      a a a a y t . (5.3)
Formulae (5.1-3) provide a basis to analyze the influence of passenger flow on travel disutility. Taking wait time and link time as random variables, then so are   r ,   a and   , rs conditionally to  . From the previous section, r w and r S are correlated. Link loads u a y along successive links are correlated, too. Furthermore, platform variables and link loads are correlated due to headway rank. As all the correlations are positive, the generalized travel time conditionally to  is subject to large relative dispersion.

In-vehicle discomfort

Let us focus on in-vehicle time and the influence of crowding density on its specific penalty factor. A well-known model is the so-called BPR function [START_REF] Spiess | Optimal strategies: a new assignment model for transit networks[END_REF]:

a b a a a a y c y ) / ( . 1 ) , ( u 1      , (5.4)
in which exponent a b takes positive values such as 1 or 4, whereas factor a c takes positive values between 0 and 3 typically. Formulae (5.4) and (5.3) . Then   1.13 and the ratio of sur-costs is varied from 1.13 to 1.65 as b is changed from 1 to 4. Fig. 1

     0 u 0 ) ) / ] (E[ 1 ST).( ( T y c T b a 19.6' if 2  b and 1  c
, whereas a naive evaluation by the operator

would yield     0 o 0 ) ) / ] (E[ 1 ( T y c T b a 10.3' only.
The discrepancy between the two evaluations would be much larger for larger values of exponent b . This demonstrates the need for accurate estimations of penalty functions and a consistent, user-oriented evaluation of vehicle crowding in the cost-benefit assessment of transport plans.

Consequences for traffic assignment models

A model of traffic assignment to a transit network is purported to yield flows of passenger trips by network link -either a service link or a pedestrian link of service access, egress or transfer -on the basis of the performance of transit services and of a matrix of passenger trip flows by origin-destination pair of travel demand zones. Local travel conditions are perceived by a network user at the level of the network path by serial composition of the links along the path. The travel conditions aggregated along the path determine its quality of service, hence its attractiveness and value to the user. By assumption, every user chooses a path of minimum generalized time to him. Thus, the relationship between trip flow and quality of service is a key component in the assignment model. The most widely used model is a "static" model with Markovian transit services, such that path choice amounts to the split of trip flow at any network node between competitive services that are available and attractive there in proportion to their respective frequency [START_REF] Spiess | Optimal strategies: a new assignment model for transit networks[END_REF]. In most applications, trip flows are considered only at the service level, rather than by vehicle. Out of the variety of capacity constraints and associated congestion effects [START_REF] Leurent | Transport capacity constraints on the mass transit system: a systemic analysis[END_REF], crowding discomfort requires to characterize not only the passenger load by vehicle but also the associated in-vehicle capacity. It has been addressed in [START_REF] Spiess | Optimal strategies: a new assignment model for transit networks[END_REF] by assuming a constant vehicle load -which is not consistent with the assumption of Markovian services, which entails an exponential distribution of vehicle headways. This section is purported to provide the properties of exponential headways ( § 6.1) and to derive their consequences in the evaluation of crowding discomfort ( § 6.2). Then, it states the requirements for stochastic modeling in the framework of dynamic transit assignment models ( § 6.3).

Exponential headways and their properties

An exponential distribution of headways at station r , denoted ) / EXP( H f r r   , has a parameter, H f r / , which is the local frequency of service by time period of length H . For simplicity, let us assume here not only the conservation of headway rank between stations, but also the homogeneity of local frequency, yielding

    s a r rs a q Q . As ) / . exp( 1 ) ( H H x f x e    , it holds that f H r / ) 1 ln( ) ( H ) 1 (       . As a lemma, let us evaluate ] ) E[( o , , n e a y  for 1  n : ) 1 ln( variables of change the by d ) exp( d )) ( H ( d )) ( (Y ] ) E[( 0 1 0 ) 1 ( o 1 0 ) 1 ( o , , o , ,                             x x x x Q y n n f H Q n r a n e a n e a a
Integrating by parts, it comes out that

                   0 1 0 1 0 0 d ) exp( d ) exp( )) exp( ( d ) exp( x x x n x x x n x x x x x n n n n So, by recursion, ! d ) exp( 0 n x x x n     if 1  n , yielding that n a n e a n y    ! ] ) E[( o , , , wherein f H Q a a /    . ( 6.1)
Then, from eqn (4.14

), ] ) E[( ) 1 ( ! ) 1 ( / ! ) 1 ( d )) ( H ( ) ( d ) ( H )) ( H ( ) ( / ] ) E[( ] ) E[( o , , 1 1 0 1 ) 1 ( o 1 1 0 ) 1 ( o ) 1 ( o 1 o o o o , , u , ,
Q Q H f Q y y                                    (6.2)

This entails the following consequences

: ] V[ 2 2 ] V[ ] ) E[( 3 6 ] ) E[( ] E[ 2 2 ] E[ o , , 2 u , , 2 o , , 2 2 u , , o , , u , ,               

Interaction with congestion law

In [START_REF] Spiess | Optimal strategies: a new assignment model for transit networks[END_REF], crowding discomfort is modeled by multiplying the in-vehicle time by a penalty factor as follows: The naïve evaluation is consistent neither with the operator nor with the user perspective under irregular headways. Under exponential headways and a fixed  , the exact consequences of the congestion law in the population of passenger trips are , which varies from 2 to 120 when b is varied from 1 to 4 -the instance value taken by analogy to motorway traffic. So it is most important to deal with crowding discomfort in consistency with headway variability. This principle should apply not only to the simulation of network flows but also in any survey of the costs of discomfort across a population of transit passengers.

)! 1 ( ) 1 ( ) / ( )! 1 ( ] ) / ( E[ u         

On stochastic features and dynamic assignment

A dynamic model of traffic assignment to a transit network is able to capture the temporal variation of both demand flows and service times, hence of inter-run headways. A precise determination of headway  will lead to a precise determination of passenger loads. However, even in a dynamic framework it is still necessary to deal with the following stochastic features:  Randomness in any headway around its scheduled value: what is the degree of reliability of the nominal timetable?  Randomness in vehicle capacity -factor  : are there several vehicle types operated on a given service and, if so, is the vehicle type selected ex-ante for each planned service run?  Randomness in passenger flow rate -factor  . In the authors' opinion, the probabilistic model presented here can be used to capture the stochastic features in an enhanced static setting; it could also be adapted to the dynamic setting by considering any headway from the nominal schedule as a random variable influenced by the operating policy.

Conclusion

A model of traffic along a transit line has been provided at both levels of traffic unit, the vehicle versus the passenger. The perspectives of the operator and the user have been identified. Based on a powerful postulate, the conservation of headway rank, it has been shown that service irregularity and demand variations, as well as other factors such as vehicle order in schedule, vehicle size and passenger sensitivity to quality of service, affect the passenger conditions of travel significantly. Crowding density above a ratio of say 80% exerts major influence on generalized travel time. The operator perspective is plagued with bias that must be corrected to represent passenger conditions objectively. The model captures a set of variability sources. Analytical formulae have been established to assess their respective effects. The main postulate is the conservation of headway rank. Gaussian or log-normal approximations have been made to yield convenient approximations; in the authors' opinion their effect is innocuous. The established properties will be useful in models of traffic assignment to a transit network, as they pertain to travel conditions hence to the leg quality of service, which determines the passenger travel choice of a network route. Further work is required to analyze transit lines serviced by a set of routes: vehicle type and load will depend on the route and the joint operations. On the passenger side, between some station pairs a subset of routes will be used, yielding reduced waiting time but more diverse in-vehicle conditions. Another research topic pertains to the feedback of vehicle load on the operating conditions, as in the assignment model of [START_REF] Leurent | User and service equilibrium in a structural model of traffic assignment to a transit network[END_REF].

. 1 )

 1 By integration, the factor of proportionality amounts to ] at the next order:

4 . 2 ..

 42 Vehicle load by link as experienced by the user Depending on his entry station e , the user travelling along link e a  experiences there a vehicle passenger load as follows, wherein u ,e r  depends on the entry station: The conservation postulate in the operator's perspective is maintained in the user's perspective, yielding that )

.27) 4 . 7 .

 47 Platform crowdingA related issue pertains to the number of passengers waiting on platform at a given station r . At any instant, this number is proportional to the level of the incoming flow,    r s rs q .

Fig. 1 .

 1 Fig. 1. Generalized time versus Occupancy ratio, according to variability parameter

  This is consistent with formula (4.16), since 1

3 )

 3 This entails a "naïve" evaluation of the average passenger sur-time on link a as follows:

  Thus the ratio of exact to naïve sur-time amounts to !

  .8)

	To gain further insight into the structure of influences, let us add to the assumption of Gaussian headways
	the approximation of the resulting flow, a  , by a log-normal variable with same mean and standard
	deviation,	]  and E[ a	[ a  	]	. Denote by a m and a s , respectively, the mean and standard deviation of
	a ln . From classical properties of log-normal distributions, these are related to the moments of a   by:

  pp. 169 sq). Denote by o H

r the CDF of headway duration r  and by o H r  its PDF, with superscript o to mark the operator's perspective. A user willing to board at r arrives on platform at a random instant, which will belong to a headway interval of duration  with a probability proportional to  : in the user's perspective, marked by superscript u ,

. 2 )

 2 Consider now the size of the passenger group that includes the individual user, to board in a vehicle run at station r , u

		r n . Its probability density stems from the density	) f u  , ( 	of pair	) (  ,  , which is related to
	the PDF	) f o  , ( 	in the following way:

  .13)

	To sum up,
		E[f(	y	u , a	e	, 	)]		E[f(	y	o , a	e	, 	). 	o e	]	/	E[		o e	]	.	(4.14)
	Furthermore, as	E[	X	. Y	]		E[	X	].	E[ Y	]		cov(	X	,	Y	)	, (4.14) is equivalent to
		E[f(	y	u , a	e	, 	)]		E[f(	y	o , a	e	, 	)]		cov(f(	y	o , a	e	, 	),		o e	)	/	o e 	.	(4.15)
	This general property enables us to derive the conditional moments of u ,  , e a y	. At the first order:
	cov(	o ,	,	,	o	)	/		o														cov(	o	,	o	)	1 o	by	the	y bilinearit	of	covariance
																								cov(	o	o	o	o	,	o	o	o	o	)	o 1
																								o			o	V[	]	since	o	o	o	o	o	o	due	to	total	n correlatio
															o			[		, o		,		]		because	V[	]	1	by	construct

  will board a vehicle of order i which is random due to irregularity, so he will get a random run time.

	9Leurent, Combes, Benezech									RelStat12				9
	The precise definition of	i	(h	)	as a random variable is difficult except for Markovian vehicle runs which
	would yield a Poisson distribution but at the price of assuming a large amount of variability. For
	simplicity, let us assume here that	i	(h	)	has a uniform discrete distribution derived from
	i		1		int[(	h		0 h	)	/E[	r 	]]		on the reference period	[	, 0 h 1 h	]	. Let	I 	i	( 1 h	)	and	1	/	I	be the elemental
	probability of		,.. 1 { I	}

h

  The travel time of a user between stations r and s is composed by the wait time at r , r w , plus the run time between the two stations, u

			e	k																								(4.22)
	So it holds that
	E[	e w		]		E[(	o e 	)	2	]	/		E[ 2	o e 	]		2 1	E[	u e 	]	,	(4.23)
	V[	e w	]		3 1	E[(	u e 	)	2	]		4 1	E[	u e 	] 2		4 1	V[	u e 	]		12 1	E[(  u e	)	2	]	.	(4.24)
	Furthermore, u e  is correlated to e w and so are the headway rank and all derived variables such as o/u ,e a y .
	For instance,	E[	e w	.  e u	]		E[(	u e 	)	2	]	/	2	so	cov[	e w	,	u e 	]		V[  e u	]	/	2	.
	4.6. Travel time
																													rs t :
	u ~rs r u rs t w t  	.																(4.25)
	By the linearity of expectation,
		Ẽ[ t rs u		]		E[	r w		]			E[ t	u rs	]	.	(4.26)

, ,

RelStat12

Acknowledgements

This research was supported by the Research and Education Chair on "The socio-economics and modeling of urban public transport", operated by Ecole des Ponts ParisTech in partnership with the Transport Organizing Authority in the Paris Region (STIF), to which we are grateful.


Charles André
email: candre@sophia.inria.fr

Frédéric Mallet
email: fmallet@sophia.inria.fr

Robert De Simone

Robert De

Simone Time

Time Modeling in MARTE

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. Introduction

Modeling of Time should certainly be a central concern in Model-Driven Engineering dedicated to Real-Time Embedded Systems (RTES). Timed extensions should allow to support a rich design flow, that can encompass both established and emerging new techniques for model-based optimization, transformation and analysis of systems. Indeed, embedded system models very often consists of a predefined set of application functions, and of execution platforms on which to allocate these functions. Application elements are increasingly componentized, may coexist and possibly cooperate concurrently. Execution platforms increasingly comprise parallel resources for both communications and computations.

The design challenge in embedded system modeling is then to provide model-level compilation techniques that provide support for both spatial distribution and temporal scheduling of applications onto platforms (collectively called allocation). This approach is therefore akin to system level design techniques such as advocated in SysML [START_REF]OMG, Systems Modeling Language (SysML) Specification[END_REF]. But SysML, just like UML, hardly formalizes its real-time aspects.

This global issue triggered over the years a number of proposals for specific representation formalisms, and their associated particular design techniques. These models then can be, and often have been, represented inside the scope of the UML [START_REF] Selic | Real-Time Object-Oriented Modeling[END_REF], [START_REF] Douglass | Real-Time UML: developing efficient objects for embedded systems, ser. Object technology series[END_REF], [START_REF] Gérard | Using the model paradigm for real-time systems development: Accord/uml[END_REF], [START_REF] Graf | A real-time profile for UML[END_REF], [START_REF] Apvrille | TURTLE-P: a uml profile for the formal validation of critical and distributed systems[END_REF]. But this is typically done, a) mostly at the metamodel level, and b) without any clear interpretation of any kind of the time annotations in the framework of the UML. This raises the risk of mismatch between the private interpretation inherited from the formalism and the existing UML semantics [START_REF] Simone | Towards a "Synchronous Reactive" UML profile?[END_REF].

The primary objective of the Time subprofile in MARTE [START_REF] Omg | UML profile for Modeling and Analysis of Real-Time and Embedded systems (MARTE)[END_REF] was to provide basic and advanced time modeling concepts, with interpretation inside the UML modeling level, not outside. These time-related concepts could then be used to build various Models of Computation and Communication (MoCC). Importantly, the profile should subsume the former SPT [START_REF]UML Profile for Schedulability, Performance, and Time Specification[END_REF] and the UML2 Simple Time models [START_REF]UML 2.1 Superstructure Specification[END_REF], while extending them towards the possibility of modeling much richer MoCCbased design approaches [START_REF] Lee | A framework for comparing models of computation[END_REF], [START_REF] Buck | Ptolemy: A framework for simulating and prototyping heterogeneous systems[END_REF], [START_REF] Jantsch | Modeling Embedded Systems and SoCs -Concurrency and Time in Models of Computation[END_REF], [START_REF] Benveniste | Composing heterogeneous reactive systems[END_REF].

Time as considered in design can be of physical or logical nature. Physical time is continuous, but can usually be discretized into chronometric clocks under appropriate assumptions. Logical time is less often recognized in itselft as an explicit modeling concept. Processings and execution steps performed at the rate of a processor cycle (which may vary according to power consumption management), or triggered by successive occurrences of an external event (such as completion of an engine revolution) are simple example of that. Often the allocation process may be perceived as this : asynchronous concurrent application components are each considered as being governed by their own (local) logical clock, connected to appropriate events; then the allocation itself consists in fitting these various clock threads onto a single (or at least more correlated) synchronous clock, subject to constraints of various sources abstracted from physical time properties and requirements. The transformation and analysis steps involved in the proper mapping are (at least implicitly) dealing with scheduling objects that are relations between logical and physical clocks attached to the various processings. MARTE Time profile is meant exactly to represent that.

In the sequel, we shall describe the profile in greater details and its position with respect to other parts of MARTE. We start with the domain view and give an overview of the UML representation. Two examples illustrate the usage of the profile.

II. Time domain view

This section describes the MARTE Time domain view, i.e., the main concepts related to time and their relationships. In an MDE approach, this is done through metamodeling. Before reviewing the MARTE Time models, we will have a look at the former UML profile for Time: SPT

A. SPT

The UML Profile for Schedulability, Performance, and Time (SPT) aimed at filling the lacks of UML 1.4 in some key areas that are of particular concern to realtime system designers and developers. SPT introduces a quantifiable notion of time and resources. It annotates model elements with quantitative information related to time, information used for timeliness, performance, and schedulability analyses.

SPT only considers (chrono)metric time, which makes implicit reference to physical time. It provides timerelated concepts: concepts of instant and duration, concepts for modeling events in time and time-related stimuli. SPT also addresses modeling of timing mechanisms (clocks, timers), and timing services. SPT, which relies on UML 1.4, had to be aligned with UML 2.1. This is one of the objectives of the MARTE profile, presented below.

UML 2 has included a new package called Simple-Time, part of the CommonBehavior package. The main new meta-classes are TimeEvent and Observation (Time and Duration). The model is kept very simple and is intended to be extended in specialized profiles. This is the specific agenda of MARTE Time model.

B. MARTE time model

As a successor of SPT, MARTE has to support a metric time with implicit reference to physical time. However, MARTE goes beyond this quantitative model of time and adopts more general time models suitable for system design. In MARTE, Time can be physical, and considered as dense or discretized, but it can also be logical, and related to user-defined clocks. Time may even be multiform, allowing different times to progress in a non-uniform fashion, and possibly independently to any (direct) reference to physical time.

1) Concept of time structure: The building element in a time structure is the TimeBase (Fig. 1). A time base is a totally ordered set of instants. A set of instants can be Time bases are a priori independent. They become dependent when instants from different time bases are linked by relationships (coincidence or precedence). The abstract class TimeInstantRelation in Fig. 1 has Coinci-denceRelation and PrecedenceRelation as concrete subclasses. Instead of imposing local dependencies between instants, dependencies can be directly imposed between time bases. A TimeBaseRelation (or more precisely one of its concrete subclasses) specifies many (possibly an infinity of) individual time instant relations. This will be illustrated later on some time base relations. Time-BaseRelation and TimeInstantRelation have a common generalization: the abstract class TimeStructureRelation. As a result of adding time structure relations to multiple time bases, time bases are no longer independent and the instants are partially ordered. This partial ordering of instants characterizes the time structure of the application.

This model of time is sufficient to check the logical correctness of the application. Quantitative information, attached to the instants, can be added to this structure when quantitative analyses become necessary (see below).

2) Access to time: In real world technical systems, special devices, called clocks, are used to measure the progress of physical time. In MARTE, we adopt a more general point of view: a clock is a model element giving access to the time structure. Time may be logical or physical or both.

A Clock makes reference to a TimeBase (Fig. 2), and thus indirectly to the instants of this timebase. The nature attribute indicates whether the accessed time is dense or discrete. A Clock accepts units (acceptedUnits property). Unit is a concept introduced in the MARTE NFP (Nonfunctional property) domain view. One of these accepted units is the defaultUnit. A clock may own an event

Fig. 3. Logical and chronometric clock

Clock is an abstract concept. There exist two concrete specializations of Clock: LogicalClock and Chronometric-Clock. A chronometric clock is a clock making reference to physical time. A special attention is then put on the quantitative information attached to this model. Non functional properties like stability, skew, etc can be defined for these clocks with respect to some refer-enceClock. On the other hand, a logical clock can be defined by any event (definingEvent property); in this case, the clock ticks at each occurrence of the defining event. Logical time is usually counted in the number of ticks. So, tick is a predefined unit often used as the default unit for a logical clock.

3) Time value specifications: An application may use time in two ways: either as a reference to a time instant or as a time span. So, MARTE introduces two distinct concepts: InstantValue and DurationValue, specializations of the abstract concept of TimeValue. Since the access to time is done through clocks, a TimeValue refers to a Clock (the onClock property). A time value also have an associated unit. When optional property unit is given, it must be used instead of default unit of the associated clock. The attribute nature specifies whether the time values associated with the clock take their values in a dense or discrete domain. Events and behaviors can be directly bound to time. The occurrences of a (timed) event refer to points of time (instants). The executions of a (timed) behavior refer to points of time (start and finish instants) or to segments of time (duration of the execution).

Clock

TimedEvent (TimedProcessing, resp.) is a concept representing an event (a processing, resp.) explicitly bound to time through a clock. In this way, time is not a mere annotation: it changes the semantics of the timed model elements.

Other timed elements-not detailled in this presentation-are also defined in the MARTE Time domain: timed observations, and timed constraints. As timed elements they explicitly bind observations or constraints to clocks.

III. UML view of Time in MARTE A. The Time sub-profile

The time structure presented above constitutes the semantic domain of our time model. The UML view is defined in the "MARTE Time profile". This profile introduces a limited number of powerful stereotypes. We have striven to avoid the multiplication of too specialized

1) ClockType and Clock:

The main sterotypes are presented in figure 8. ClockType is a stereotype of the UML Class. Its properties specifies the kind (chronometric or logical) of clock, the nature (dense or discrete) of the represented time, a set of clock properties (e.g., resolution, maximal value. . . ), and a set of accepted time units. Clock is a sterotype of InstanceSpecification. An OCL rule imposes to apply the Clock stereotype only to instance specifications of a class stereotyped by Clock-Type. The unit of the clock is given when the stereotype is applied. Unit is defined in the Non Fonctional Property modeling (NFPs) subprofile of MARTE; it extends EnumerationLiteral. It is very convenient because a unit can be used like any user-defined enumeration literal, and conversion factors between units can be specified (e.g., 1ms = 10 -3 s). TimedElement is an abstract stereotype with no defined metaclass. It stands for model elements which reference clocks. All other timed stereotypes specialize TimedElement.

2) Clock constraints: ClockConstraint is a stereotype of the UML Constraint. The clock constraints are used to specify the time structure relations of a time domain.

The context of the constraint must be a TimedDomain. The constrained elements are clocks of this timed domain and possibly other objects. The specification of a clock constraint is a set of declarative statements. This raises the question of choosing a language for expressing the clock constraints. A natural language is not sufficiently precise to be a good candidate. UML encourages the use of OCL. However, our clocks usually deal with infinite sets of instants, the relations may use many mathematical quantifiers, which are not supported by OCL. Additionnally, OCL [START_REF] Omg | Object Constraint Language, version 2.0[END_REF] is made to be evaluatable, while our constraints often have to be processed altogether to get a set of possible solutions. So, we have chosen to define a simple constraint expression language endowed with a mathematical semantics. The specification of a clock constraint is a UML::OpaqueExpression that makes use of pre-defined (clock) relations, the meaning of which is given in mathematical terms, outside the UML. Our Clock Constraint Specification Language is not normative. Other languages can be used, so long as the semantics of clocks and clock constraints is respected.

3) TimedEvent and TimedProcessing: In UML, an

Event describes a set of possible occurrences; an occurrence may trigger effects in the system. A UML2 TimeEvent is an Event that defines a point in time (instant) when the event occurs. The MARTE stereotype TimedEvent extends TimeEvent. Its instant specification explicitly refers to a clock. If the event is recurrent, a repetition period-duration between two successive occurrences of the event-and the number of repetitions may be specified.

In UML, a Behavior describes a set of possible executions; an execution is the performance of an algorithm according to a set of rules. MARTE associates a duration, an instant of start, an instant of termination with an execution, these times being read on a clock. The stereotype TimedProcessing extends the metaclasses Behavior, Action, and also Message. The latter extension assimilates a message tranfer to a communication action.

Note that, StateMachine, Activity, Interaction being Behavior, they can be stereotyped by TimedProcessing, and thus, can be bound to clocks.

B. The Time model library

The TimeLibrary (Fig 9) is a user's model libary that provides enumerations related to time and facilities for using the ideal chronometric time (i.e., the time referenced in physical laws). TimeUnitKind contains the main chronometric time units. s (second) is an SI unit. Other units are derived units. All the enumeration literals are stereotyped by Unit. LogicalTimeUnitKind is a special enumeration which contains one enumeration literal only. This literal is tick. The IdealClock and its instance idealClk model the abstract and ideal time which is used in physical laws. It is a dense time. idealClk should be imported in models that refer to chronometric time. TimedValueType is a templated data type. The template parameter is an enumeration which contains time units. Examples of such clock constraints are described in a technical report [START_REF] André | Modeling time(s) in UML[END_REF]. For lack of room we cannot describe them here. Some constraint relations state that variations in rate or jitter between two clocks are somehow bounded, that clocks are related up to some drift, or that a clock has to be a subclock of some other (with the subclocking mechanism following possibly some pattern). Some relations are more imperative in that they denote the single solution to some clock transformations (as for instance in b isPeriodicOn a ofPeriod n).

IV. Examples

The first step for the designer is to construct its own, not always perfect, clocks. The first subsection shows how to create chronometric clocks and the second subsection illustrates the use of logical clocks.

A. Chronometric clock specifications

The MARTE TimeLibrary provides a model for the ideal time used in physical laws: idealClk, which is an instance of the class IdealClock, stereotyped by ClockType (Fig. 10, upper part). idealClk is a dense time clock, its unit is the SI time unit s. After importing the library, new user-defined chronometric clocks can be defined. For instance, Fig. 10 defines the class Chronometric with an attribute resolution of type Real and an operation currentTime that allows for reading the current time. The stereotype ClockType is applied and the tagged values characterize the nature of clocks represented by this class. Here, the clock is a discrete clock and accepts the units defined by the predefined Enumeration TimeUnitKind (see Fig. 9). By default, the clock types are chronometric, not logical.

Actual clocks belong to timed domains, i.e., a package stereotyped by TimedDomain (Fig. 10, lower part). Here, a single time domain is considered. It owns three clocks. idealClk is imported from the library. Two instances of the class Chronometric, cc1 and cc2, are defined. They both use s (second) as a time unit and their resolution is 0.01 s. The three clocks are a priori independent. A clock constraint specifies relationships among them. According to the given constraints, cc1 and cc2 are two 100 Hz clocks, the stability of which is 10 -4 , and with an offset less than 5ms.

B. Logical clock specifications

Fig. 11 illustrates the definition of logical clocks. The discrete, logical, clock type AngleClock is defined. It has three attributes, resolution, offset and maximalValue. The label function angle associates a real value, the clock reading, with each instant of the clock. Each instant is uniquely identified by a natural number, its index, inferred from the linear order defined on the clock instants.

The enumeration AngleUnitKind defines the units, enumeration literals stereotyped by NFP::Unit, that can be used by the clocks.

Two clocks, instances of AngleClock, are then created. crkClk represents the crankshaft revolutions, its unit is

• CRK (degree crank), its resolution is 1 • CRK and its maximal value is 720 • CRK, i.e., two revolutions of the crank shaft. camClk represents the camshaft resolutions, its unit is • CAM (degree cam), its resolution is 1 • CAM and its maximal value is 360 • CAM , one rotation of the camshaft. One revolution of the camshaft (mechanically) implies two revolutions of the crankshaft, hence This example is developed in a previous paper [START_REF] André | A multiform time approach to real-time system modeling: Application to an automotive system[END_REF] dedicated to the use of multiform time, and the modeling of the "knock control" problem with MARTE.

1 • CAM = 2 • CRK.

V. Conclusion

The MARTE Time subprofile provides a limited number of time concepts from which to build Models of Computations with timed interpretations (as examplified by our modeling of AADL aspects in the same proceedings [START_REF] André | Modeling of immediate vs. delayed data communications: from AADL to UML MARTE[END_REF]). These concepts rely on the existence of various time threads (or logical clocks) that may drive application elements. Constraint relations may exist between clocks, from the loose asynchronous compositions to stricter simultaneous coincidence. More constraints are raised as result of scheduling decisions, or by abstraction of timely requirements demanded by the view or imposed by the execution platform. Solving constraints and committing progressively to particular schedule results from the intended flow of design promoted by MARTE in model-based engineering of embedded systems.

Fig. 1 .

 1 Fig. 1. Time structure (Domain view).

1 Fig. 2 .

 12 Fig. 2. Clock

1 Fig. 4 .Fig. 5 .

 145 Fig. 4. Time values

Fig. 7 . 1 Fig. 8 .

 718 Fig. 6. Timed event

Fig. 9 .

 9 Fig. 9. TimeLibrary: a user library for time

Fig. 10 .

 10 Fig. 10. Chronometric clocks.

Fig. 11 .

 11 Fig. 11. Logical Clocks.

Fig. 12 Fig. 12 .

 1212 Fig.12uses the clock camClk to represent a fourstroke engine cycle. This state machine is stereotyped by TimedProcessing. The on attribute identifies the clock used and therefore the unit ( • CAM ).