Delay in a Railway Network 135

7.3. Delay in a Railway Network 135

orders-of-magnitude reduction. Referring back to the process definition in Table 7.6,

we see that ′ P 3−03 and P 1−53 are two of the most complex and involved processes: they both contain two parallel composition (or maximum) operations. Without the use of

the reduction algorithm, indeed these processes have the largest state spaces for all different sizes of the basic running-time distributions. The impact of the reduction algorithm on these processes is impressive but somewhat irregular in size. The reason for this irregularity is, in several cases, the hyper-Erlang representations produced by G-F IT are far from minimal, and, therefore, often can be reduced by the reduction algorithm even before they are used in further compositions. Furthermore, the hyper- Erlang representations have a nice structure with many states having similar total out- going rates. As a result, some processes may have smaller final state spaces when the size of the representations of the basic running-time distributions is increased. Even though the produced hyper-Erlang representations can be reduced, G-F IT guarantees that they are the best in the given number of states, as the tool searches for the best fitting from all possible configurations or hyper-Erlang structures.

The computation times of the reduction procedure are dominated by the transfor- mations (S PA ), although the difference between the computation times of the transfor- mation and reduction is not as large as that of the previous two case studies. In several cases, the computation times are shorter when the original state space is larger, for in- stance process P ′ 3−03 in size 30 is obtained quicker than it is in size 10, 15, or 20. This is due to the above-mentioned irregularity of the size of the reduction’s results.

The resulting delay distributions of the processes at time point 3 (in minutes) fol- low no apparent patterns with the varying sizes of the APH representations used to approximate the basic running-time distributions. This means that fitting traces to APH representations of larger size does not necessarily produce better approximations. This is because the stopping criterion for the iterative procedure in almost every fitting tools including G-F IT is based on the difference between the values of some variables

(such as the approximated parameters or the likelihood) in subsequent iterations, in- stead of the absolute value of the optimized measure of closeness. The absence of patterns also forbids us to predict the location of the “precise” probabilities.

In the Erlang experiments (cf. Table 7.9) we also obtain significant reductions in the size of the state spaces of the processes. Furthermore, there are clear patterns in how the reduced state spaces grow with the increasing size of the basic running-time distributions. This can be explained by the regularity of the APH representations of Erlang distributions used as the basic running-time distributions. Similar to the first

experiments, ′ P 3−03 and P 1−53 experience the largest reductions. In the experiments, the size of the original state spaces and the number of states that can be removed from them by the reduction algorithm mostly determine the com- putation times needed. Indeed, we see that the reduction of process P ′ 3−03 takes the

most time. In several cases, however, the reductions take more time than the transfor- mations do. Investigating the cases more carefully, we found that in these cases, the states that can be removed are concentrated in the end of the chains of the ordered bidiagonal representations. Hence, the need to solve the systems of linear equations

(when removing the states) arises in the end, and, as a result, the systems are big and take much more time to solve.

Contrary to the Weibull experiments, the resulting delay distributions of the pro- cesses at time point 3 (in minutes) follow a particular pattern, namely the larger the size of the used APH representations of the basic running-time distributions, the

136 Chapter 7. Case Studies

Table 7.9: Summary of the Result of the Erlang Experiments

Process State Spaces Comp. Time (sec.) Pr(D ≤ 3) Size

APH

No Red. Red.

S PA

Red.

5 P ′ 1−03

<1 0.91717075 P ′ 1−53

<1 0.90995698 P ′ 2−04

<1 0.94416608 P ′ 2−54

<1 0.94609606 P ′