NOTE ON THE COORDINATION OF PERIODIC PROCESSES IN TRANSPORTATION SYSTEMS NOTE ON THE COORDINATION OF PERIODIC PROCESSES IN TRANSPORTATION SYSTEMS

The article deals with a coordination of periodic processes in transport systems. It introduces some basic mathematical models useful in process optimization. It also shows some examples of everyday transport practices where this optimization can be used.


Introduction
Users of transport systems often declare their interest in velocity. However, it does not mean technical velocity of transport means. They concern the "velocity of displacement", i.e. the distance between the origin and destination divided by the transport time. And it is often true that this value is strongly influenced by coordination between transportation processes.
Let us consider several examples: A. A passenger travels from the village V to the town W using first a local train from V to a station S and afterwards an express train from S to W. It is obvious that the travel time from V to W is strongly influenced by the waiting time for the train changing in S and, consequently, by the coordination of the transport processes of local and express train operation. B. A passenger walks from his house H to an urban bus stop S 1 , then he/she travels by bus from the stop S 1 to the stop S 2 , using any from the two routes r 1 , r 2 operating between S 1 and S 2 . It is obvious that the total duration of the trip from H to S 2 is strongly influenced by the waiting time for a bus at the stop S 1 and, consequently, by the coordination of the transport processes on the two routes. C. A wagon is loaded at a station A and its destination is the station B. However, it has first to use a train to a marshalling yard Y and after some manipulations there to use another train from Y to B. It is obvious that the total duration of the trip from A to B is strongly influenced by the manipulation time in Y and, consequently, by the coordination of the processes in Y. D. A person drives his car from his home H to his office O passing through a signalized road intersection I. It is obvious that the total duration of the trip from H to O is strongly influenced by the waiting time in front of the junction I and, consequently, by the coordination of the processes at I.
If we consider the cases A., B. and D. we can imagine that the displacement activities may be more complicated: A. There may exist many local trains coordinated with several express trains in several stations.
B. There may exist many routes coordinated at many common legs. D. There may exist many processes at several junctions to be coordinated.
The complex problems of optimal coordination can be a bit simplified introducing a periodicity: A. To introduce a periodic timetable (a "Takt Fahrplan" in German), e.g. to repeat the departures after each hour. B. To introduce a periodic time table, e.g. to repeat the departures after each 12 minutes. C. To introduce a periodic schedule, e.g. to repeat the processes each day (it was already done in the major part of marshalling yards). D. To introduce a "cyclic" mode of operation, i.e. to repeat the green signals e.g. each 80 seconds (a big part of junctions work in a periodic regime now).
In the sequel we shall try to formulate the coordination problems in a general mathematical way.

NOTE ON THE COORDINATION OF PERIODIC PROCESSES IN TRANSPORTATION SYSTEMS Jan Černý -Vlastislav Mojžíš *
The article deals with a coordination of periodic processes in transport systems. It introduces some basic mathematical models useful in process optimization. It also shows some examples of everyday transport practices where this optimization can be used.

Coordination
Having two processes at the same place it is quite natural to require some type of coordination between them. In general, we can meet different requirements:

Single and simple linking.
We speak about single linking if we have two processes p 1 , p 2 with only one linking between them, e.g. p 2 linked to p 1 , symbolically p 1 → p 2 . We call this linking simple if both processes have the same number n of time-points in one period p and the process p 2 ϭ t 21 , t 22 , …, t 2n (ϩ kp) is linked to the process p 1 ϭ t 11 , t 12 , …, t 1n (ϩ kp). The quality of this linking can be expressed by the differences d 1 ϭ t 21 Ϫ t 11 , d 2 ϭ ϭ t 22 Ϫ t 12 , …, d n ϭ t 2n Ϫ t 1n put into an objective function f(d 1 , d 2 , …, d n ), e.g.
Among the abovementioned examples only the A can serve as the illustration of single linking, but, unfortunately, it is not simple (we have 2 arrivals within 60 min. but only one departure).

Double and multiple linking.
We speak about double linking if: a) We have both p 1 → p 2 and p 2 → p 1 . b) There exists another pair p 3 → p 4 .
In the case a) we speak about double mutual linking of the pair p 1 , p 2 .
We speak about multiple linking if there exist more than one linked pair (i.e. double linking is a particular case of multiple linking). We suppose we are given a set of time-point periodic processes P ϭ {p 1 , p 2 , …, p m } with periods p 1 , p 2 , …, p m and the common period p ϭ (p 1 , p 2 , …, p m ) ϭ the least common multiple of the periods p 1 , p 2 , …, p m . The mutual linking could be expressed by means of a linking digraph (ϭ oriented graph) G ϭ (V, A), where the vertex set V ϭ {1, 2, …, m} represents the processes p 1 , p 2 , …, p m , the arc set A represents linking, i.e. the arc a ϭ (i, j) ʦ A represents linking p j to p i i.e. p i → p j .
Writing the time-point one has to take into account the common period p, i.e. if a time-point is represented by a number t ik then t ik ʦ {0, 1, …, p} must hold; if not, then it must be reduced mod p.

General single linking.
We speak about general single linking p 1 → p 2 if it is not simple. In the case of general single linking the processes p 1 , p 2 may have different numbers n 1 , n 2 of time-points in one period p and the differences d i may be calculated from some selected n-tuples t 11 , t 12 , …, t 1n , t 21 , t 22 , ..., t 2n , n Յ n 1 , n Յ n 2 .
Especially, if the linking p 1 → p 2 intends to express a train changing, we denote T 1 the set of all time-points (= arrivals to the change station plus some time for walking from one train to another) of the process p 1 in one period p and T 2 is the set of all time-points (departures from the change station) p 2 , but moreover both the sets T 1 , T 2 are extended by adding the first time-point from the next period (i.e. we add the time-point t ϩ p where t is the first time-point from the period p). Then the number n is the maximum natural number allowing the n-tuples t 11 , t 12 , …, t 1n , t 21 , t 22 , …, t 2n to have the following properties of "closeness": t 1i ϭ ϭ max{tʦT 1 : t Յ t 2i }, t 2i ϭ min{tʦT 2 : t Ն t 1i }. If in such a manner t 1n ϭ t 11 ϩ p and t 2n ϭ t 21 ϩ p then we omit t 1n , t 2n and we put n Ϫ 1 instead of n.
In our example A we have T 1 ϭ {28, 58, 88}, T 2 ϭ {42, 102} corresponding to the local train arrivals 6:28, 6:58, 7:28 and the express train departures 6:42, 7:42. Obviously, the first value of n ϭ 2, t 11 , t 12 ϭ 28, 88, t 21 , t 22 ϭ 42, 102 . However, p ϭ 60 and both 88 ϭ 28 ϩ p, 102 ϭ 42 ϩ p. Thus, we omit 88, 102 and put n ϭ 1.  In practice it can happen that these processes are originated in different locations and their influence represented by a relation p i → p j "works" in a third location which needs some equalization of time-points. Usually, we use a value o(i, j) said the offset which has to be added to the time-points of the process p i before their comparison with the time-points of p j .

Coordination for changes
Let us turn to our examples. B) concerns the well known problem of coordination of public transport on common legs. C) deals with freight train formations and the methods of solution are similar to the previous ones. D) is the well known problem of signalised intersections. All three are described by many authors in many books and papers, see. e.g. the monograph [1]. One can say that the available methods satisfy the practical needs.
On the other hand the problem A) cannot be considered satisfactorily solved. In [1] and [2] one can find a heuristics and a linear programming model for time shift optimization, having given the set of trains operating on a general network. Other authors, e.g. [3], study the same problem, but limited to the "herring-bone type of network" using congruency calculations.
However, the authors have not yet met any paper dealing with the coordination problems, optimizing: a) the size of trains for the given number of trains during the common period together with their time shifts b) the number of trains during the common period together with their size and time positions.
The authors hope they have an idea of solution of the problem a) using mathematical programming. The problem b) seems to be open.