of the system, i.e. the money spent on the infrastructures will not be well “invested”.
Since these approaches are not acceptable (either for the passengers or from the rail-way system), to prevent the propagation of delays it is necessary to identify the depen-dencies between primary and secondary delays. In such a way we may be able to find an optimal compromise among all the conditions that the system has to satisfy, so that all the passengers will be able to reach their final destination with the minimal delay.
This can imply some missing connections to limit the propagation of the perturbation on the timetable.
Delay propagation analysis can be a useful tool to verify whether the capacity of the system is sufficient for a given timetable at a predefined level of punctuality (if not, the infrastructure management might decide to adjust or extend the existing infras-tructure). In addition it provides new criteria for the timetable designers in order to obtain insights in how to optimize a timetable concept by adding buffer times to the
“most sensitive" train successions.
In this thesis we will focus our attention on the analysis of delay propagation to find appropriate wait and depart decisions in delay management.
1.4 Panorama of possible approaches
The Train Timetable Problem is nowadays a well known problem and much progress has already been achieved in particular concerning the delay management. In a contest of an overall view, we will give in the next pages a short description of a few models that have been applied to delay propagation and/or prediction, drawing attention on some of their drawbacks. In particular we will introduce the following techniques:
Process Algebra (see [5] or [9]), Markov Chain (see [80]), Linear Regression (see [13]), Wakob’s Approach (see [23]). Detailed explanations on the application of the procedure and on the results can be found in the quoted articles.
Instead of the method that will be tested in this work, a stochastic technique originally applied in the genetic field, will be separately presented in Chapter 3.
Process Algebra
Process Algebrais an active area of research in computer science and corresponds to an algebraic approach to the study of concurrent processes to ensure that they are correctly designed.
The wordAlgebradenotes that we take an algebraic or axiomatic approach in describ-ing the behavior. So a Process Algebra is then any mathematical structure satisfydescrib-ing the axioms given for the basic operators. Within this structure calculations with pro-cesses that are the elements of the Process Algebra can be performed (by the axioms).
The basic operators and the axioms must be defined according to the concurrent sys-tem under examinations (see [5]). Hence a deep knowledge of the process that has to be simulated is required.
Concurrent systems consist of a possibly huge number of components that not only work independently but also communicate with each other from time to time (e.g the railway signaling system). If the total amount of states is quite large, the number of possible actions can become too big to be considered.
In the deterministic case, the focus of Process Algebra is on verifying that the ex-ectution of specific actions is guaranteed by a fixed deadline after some event has happened, e.g that if a train is approaching a railroad crossing, then bars must be guaranteed to be lowered on due time.
In the stochastic case, instead, are considered systems, whose behavior cannot be deterministic predicted as it fluctuates according to some probability distribution.
Due to economical reason, these systems are referred to as shared resources systems because there is a varying number of demands competing the same resources. The consequences are mutual interference, delays due to contention, and varying service quality. In this case, the focus of process algebra is on evaluating the performance of the systems: e.g. in the railway system we may be interested in minimizing the average train delay or studying the characteristics of the flow of passengers.
Since these process can describe just certain aspects of behavior, disregarding oth-ers, they are always considered an abstraction of the “real” behavior of the system.
Moreover a huge amount of details has to be taken into account (e.g. interconnec-tion and synchronizainterconnec-tion structure, allocainterconnec-tion and management of resources, real time constraints,. . . ) so that it is necessary to have a close collaboration of many people with different skills in the project.
As applicative example we introduce the case of train-gate-controller presented in [40]. The problem is composed by three components: a train, a gate and a controller.
A train approaches a gate from a great distance with a speed between48and52m/s.
As soon as it passes the detector signal placed (1000m) backward from the gate, an approaching signal is sent to the controller. The train may slow down (speed between 40and52m/s) and pass the gate. As soon as it passes the detector placed (100m) forward from the gate, an exit signal is sent to the controller. A new train may come after the current one has passed the second detector, but only at a security distance (1500m). The gate is able to receive lower and raise signals from the controller at any time. As soon as the gate receives a lower signal, it lowers from90◦ to0◦. As soon as it receives a raise signal, it raises from0◦to90◦. The controller is able to receive approaching and exit signals from the train detectors at any time. When the controller receives an approaching signal, it sends a lower signal to the gate. When it receives an exit signal, a raise signal is sent to the gate. Because of security procedure, an approaching signal should always cause the gate to go down, and exit signals should be ignored while the gate is going down. The train gate controller specifications have following environment variables:
• xfor the distance of train from gate;
• rfor the angle of gate with the ground (90◦up,0◦down);
• dfor possible delay controller;
• yfor speed of the train.
It is assumed that initially there is no train at a distance smaller than1400m from the gate, the gate is open and the controller is idle. In transition labels,x,r,ddenote values of variables before the transition. The primed variables x0,r0,d0 represent values of variables in the new location after the transition. The problem can be represented with three graphs as in Figure 1.4. The complete system of commands to describe the system requires more than three pages and it can be found in [40]. In any case it should be evident that the need of different graphs (one for the trains, one for the signals and one for the infrastructure) make the procedure neither easy to define nor easy to understand.
In the last years many Process Algebras have been formulated, extended with data, time, mobility, probability and stochastic (e.g. [9]). Unluckily they have not always been satisfactory, because the presence of concurrency, communication, synchroniza-tion and nondeterminism makes the study of the correctness of concurrent systems particularly difficult, expecially when the structure is not regular.
Markov Chain
Markov Chains are a special case of random processes which can be used to model
1.4 Panorama of possible approaches 9
Figure 1.4: Train gate controller automat (from [40]) various processes in queuing theory and statistics.
A random process is a collection of random variablesXindexed by some setTtaking values in some setsI.
• T is the index set, usually time (in the delay managementT =Zbut it can also be chosen asRorR+);
• Iis the state spaces (in the delay managementI =Z+but it can also beRor {1 . . .n}or{a,b,c}).
We classify random processes according to both the index set (discrete or continuous) and the state space (finite, countable or uncountable, continuous) A random process is called Markov Chain if conditioned on the current state of the process, its future is independent of its past. Mathematically we can write this property as
P(X(t+1) =it+1|X(t) =it, . . . ,X(1) =i1,X(0) =i0) =P(X(t+1) =it+1|X(t) =it) whereP(A|B)is the conditional probability defined by
P(A|B) = P(A∩B) P(B)
The definition states that only the present state gives information on the future be-havior of the process. Knowledge of the history of the process does not add any new
information.
The controlling factor in a Markov Chain is thetransition probability, i.e. a condi-tional probability for the system to go to a particular new state given the current state of the system. This means that we get fairly efficient estimates if we can determine the proper transition probabilities.
A Markov Chain can be applied to predict the next state of a system, given the infor-mation of the system in the previous states. Under the assumption that theMarkov propertyis valid, it is possible to neglect all the information coming from the history of the process except the most recent one. So the transition probability is defined as
p(i,j) =P(X(t+1) = j|X(t) =i)
and it does not depend on the timet. Intuitively this value gives the “rules of the game"
since it is the basic information needed to describe a Markov Chain and, due to the definition of probability, we must have
p(i,j)≤1∀i,jand X
j
p(i,j) =1∀i
A state jis said to be accessible from statei(writteni → j) if, given that we are in statei, there is a non-zero probability that at some time in the future, we will be in state j.
In the railway system, the variablesX1, . . . ,Xn ∈ Zcan represent the delays of the considered trains in the system. Then the Markov property can be explained as fol-low: the value of the delays of the trains at timet,(x1,t, . . . ,xn,t), depends only on the previous measurement of their value, i.e.(x1,t−1, . . . ,xn,t−1).
A recent application of Markov chains for the railway-timetable problem is [80], in which different distributions (and hence probability functions) are considered to eval-uate the level of punctuality (e.g number of punctual trains) of the systems in case of delay.
Regression Model
Another approach often used in the railway delay management is the linear regression model (see [13] or [49]). Let’s consider a response variable, also called dependent variable,Y ∈ R(in our case the delay of an event), and some explanatory variables (independent variables)X1, . . . ,Xp−1∈R(delays of ”previous” events).
The regression model tries to explainYthrough a systematic component based on the Xiand an errorto cover possible discrepancy:
Y =r(X1, . . . ,Xp−1) +
As first assumption we suppose that the functionr(·)is linear, so that we can rewrite the model as
1.4 Panorama of possible approaches 11 containing the values of the p explanatory variables, we can rewrite the model as
Y =βX+.
As second order hypothesis we consider (homoscedasticity):
E() = 0
Var() = σ2In ∃σ2> 0unknown rank(X) = p
so that
E(Y) =µ=βXand Var(Y) =σ2In
If all that hold, we have defined a linear model that is particularly suited for the mathematical description of problems arising from controlled experiments, where experimenters can control the values taken by relevant factors to examine the corre-sponding values of the response variables. In this setting,Xcontains the value of the experimental factors, which are non-stochastic since they are chosen by the experi-menters. The error term is due to measuring errors (which explains its name) and, if the instruments are not biased, it follows that E() = 0. Finally, if the various experiments are conducted in such a way not to influence each other, then the stochas-tic independence assumption is satisfied, implying uncorrelated errors. To be able to define the probability distribution of Y, we introduce an alternative criterion to the Likelihood principle that would require additional hypothesis on the distribution of. We chooseβsuch that it minimizes||Y−µ||= p
(Y−βX)T(Y−βX) =Q(β)(Least Squares Error). This method will be applied in Chapter 5 to evaluate the coefficient of the ”virtual activities”, i.e. the capacity constraints related to the outcome of the Tri-graph method (see Sections 2.11, 3.5.3 and 5.6).
In case the matrix Xis regarded as non-stochastic, we can use the model to predict the values of Y. Consideration of non-stochastic X is supported by the following argument. In most common cases, the distribution of the explanatory variables does not contain any information on the relationship with the response variables, since we are interested in making interferences on this relationship not on the distribution of the explanatory variables. Therefore, we examine the variables conditionally on the values taken byX. In other words, we can operate within the conditional principle, which stipulates that interferences should be based not on the distribution itself but on the conditional distribution.
The main problem we found out in the application of this procedure is the choice of the variableXthat should be considered in the definition of the model to evaluate the propagation of the delays. In fact this choice is strictly dependent on the knowledge of the set of dependencies of the system, that is the aim of this thesis. Therefore there is a really good interaction between Tri-graph method and linear regression: the first procedure identifies the dependencies, whereas the second one gives the specific
“degree” of dependencies, i.e. the slope (see Chapter 5).
Wakob’s Approach
Wakob has proposed an analytical framework for capacity assessment of railway sta-tions which is based on queuing theory. More precisely, he applies queuing theory to predict the waiting time incurred by the simultaneous arrival and random processing of two trains as isolated part of the infrastructure.
Wakob’s approach does not provide a queuing model for an entire railway station, but it proposes an analytical framework for capacity planning. It partitions the station into specific parts of the infrastructure (blocks, switches, platforms,. . . ) and it describes the performance of them as single element (single server) instead of the station as a whole.
Let a setCconsists of basic infra-elements1,s2, . . . ,spand assume thatCcarries the single server identity. If an arbitrary infra-element si,i ∈{1, . . . ,p}ofCis occupied
by one train, then the server identity implies that all other infra-elements insideCare blocked. ConsequentlyCcan not be used by any other train during the same time slot.
SoCcan be seen as a "Teilfahrstraßenknoten" (TFK), i.e. as common parts of several routes. Wakob’s Approach assumes that all the TFKs have infinite queuing space, to prevent a train from getting locked/blocked if a queue has reached its capacity, and that allows the approach to consider the TFKs separately. Since the size of a TFK is small compared to the length of a train, several TFK will be occupied simultaneously.
Other assumptions of the method are Erlang distribution for the interarrivals processed at a TFK and for the service times, and random arrival order of the trains. The method evaluates the average time of a queue by the Pollackzek-Khintchine formula. In order to calculate the total amount of waiting time it evaluates the mean queue length by Little’s formula, so that the total waiting time is computed by multiplicating the ex-pected queue length by the predefined observation period.
The approximation is very accurate but the waiting times are generally larger than those obtained via simulations. Moreover, it is a “timetable”-free approach, hence it cannot be compared with daily observations since a specific timetable can not be used to verify or falsify it. Therefore this method should only be adopted as a first approxi-mation for the capacity assessment of railway stations. The studied cases indicate that the approach is indeed able to locate the bottlenecks section in a station. However, it seems to be rather uncomfortable for the practical use by railway staffdue to the substantial efforts that are required to implement and to mantain the algorithms.