DEVS Modelling and Simulation

## General information

• The due date is December 20th, before 23:55.
• Submissions must be done via BlackBoard. Beware that BlackBoard's clock may differ slightly from yours. All results must be uploaded to BlackBoard and accessible from links in the main (index.html) file.
• The assignment must be made in groups of maximum 2 people. It is understood that all partners will understand the complete assignment (and will be able to answer questions about it). Clearly identify who did what.
• Grading will be done based on correctness and completeness of the solution. Do not forget to document your requirements, assumptions, design, implementation and modelling and simulation results in detail!
• To get feedback about the assignment workload, provide the number of hours you spent on this assignment.
• Contact: Simon Van Mierlo.

## Goals

This assignment will make you familiar with modelling, simulation, and performance analysis in DEVS.

## The problem

The purpose of this assignment is to model the behaviour of train traffic on a straight stretch of train track. In this assignment, the tracks are uni-directional: trains travelling in opposite directions will use different tracks. The stretch is made up of a sequence of smaller railway segments. All segments have the same length. For safety reasons, a single railway segment (also known as "block" in railway signalling terminology) can hold at most a single train, even if the segment is much longer than the train. Each segment is guarded by a traffic light (also known as "signal") in the beginning, which will be red if there is a train on that segment, or green otherwise.

You will use your created model to determine the optimal length of a railway segment. The optimal railway segment length will find a balance between capacity as smaller segments decrease the space between trains and cost as smaller segments means more traffic lights which increases the infrastructure cost. Two different configurations are depicted in Figure 1. You may assume a simple cost function: transit time in seconds to traverse the entire track + 10 * number of lights . This is a weighted sum of the two performance criteria. You will have to determine, for this cost function, and for a few different operating conditions, what is the optimal length of a single railway segment.

Figure 1: different railway segment lengths lead to different performance.

The model will consist of a Coupled DEVS model named TrainNetwork. This model is made up of a concatenation of one Generator Atomic DEVS model, followed by a series of RailwaySegment models (either Coupled or Atomic DEVS models), and terminated by a Collector Atomic DEVS model, as depicted in Figure 2.

Note how the modelling view we will take is called "active resource" (as opposed to the "active entity" or "agent-based" view). In the active resource view, it is the resources for which there is competition (and as a result, queueing), in our case, the railway segments, which are modelled as dynamic entities. In contrast, the trains moving around will be modelled as passive entities, passed around as (structured/parametrized) events between the active resources. This is an antropomorphic view, as railway segments are obviously not active, and do not make any decisions. It is however a convenient abstraction/view which is commonly used when modelling queueing systems.

Figure 2: overview of the TrainNetwork structure.

The following is a detailed description of the three different DEVS models, as well as of the Train, Query and QueryAck entities sent (as events) between the DEVS building blocks.

### Train

Trains are instances of the Train class, and are generated by the Generator model. They pass through a sequence of RailwaySegment models, and finally end up at the Collector model. The Generator models the input, from the environment, to the railway track system and the Collector models the environment receiving the trains leaving the railway track system to the environment. Together they are used to collect performance information: the "transit time", the time it takes a single train to traverse the entire track.

The Train class is a container for all relevant information pertaining to a train, at our chosen abstraction level. It has the following attributes set at instantiation time:

• ID: a unique ID for the train (to be able to compute transit time).
• a_max: the maximal acceleration of the train in its current configuration (a positive number). It is different for each train, and is therefore sampled from a random distribution by the Generator.
• departure_time: the time at which the train leaves the Generator. It is the Generator's responsibility to assign this value.

We give the Train class the following additional attributes, which are modified during simulation:

• v: the current speed at which the train is moving.
• remaining_x: the remaining distance on this railway segment.

### Query

The moment a Train approaches (i.e., gets close enough to) the next segment, a Query message will be sent to the next railway segment. This Query is used to model, at a discrete-event level of abstraction, the train driver's observation of the next railway segment for the current state of the traffic light. The railway segment will reply to the Query with a QueryAck, described next.

### QueryAck

The QueryAck is sent as the reply to the Query message. It contains information on the state of the traffic light: whether it is green or red.

### Generator

A Generator generates Train instances on its output port. The Inter Arrival Time (IAT) of trains is sampled from a uniform distribution over the interval [IAT_min, IAT_max[. Every time a Train instance is generated, it is passed a value for a_max by the Generator. This value is sampled from a uniform distribution over the interval [a_min, a_max[. a_max is a strictly positive number. For sampling from a uniform distribution, you should use Python's random module which implements pseudo-random number generators for various distributions. The initial velocity is always 0. For the same seed of the pseudo-random number generator, exactly the same stream of pseudo-random numbers is generated. This guarantees reproducibility of the simulation experiments.

The basic behaviour of a Generator is: keep output-ing trains, with IAT and acceleration sampled from their respective distributions. It is however likely that at the moment of departure from the generator, there is still a train in the first railway segment.

So, the Generator should more accurately reflect what happens in the real world. You might have "scheduled" to leave at a certain time, but the moment you want to leave, you first look ahead. If there is nothing in your way, you will leave, otherwise you will wait until there are no more obstacles and you are able to leave. This is what we need to model.

Even if the previous train cannot leave yet, the next train might already be scheduled to leave. The Generator will therefore need to have a queue of trains that are ready to leave (in First-In First-Out order). It will poll every 1 second to check if the traffic light of the segment became green. If so, it will output the first train of the queue to the first railway segment.

### Collector

The Collector will receive Trains and will act as if it is a railway segment with a traffic light that is always green. It will therefore always respond with a QueryAck which allows the train to enter.

When the Collector receives a train, it will calculate performance metrics. In our case, it will compute the transit time of each train: how long it took between the generation of the train, and its arrival (of that specific train) in the Collector. This transit time includes the time spent in the Generator's queue.

Ideally, we would like as much insight as possible in the distribution of the above performance metric. For simplicity, you should however not collect the distribution in a table, but only the average.

### Railway Segment

A RailwaySegment has the following parameters:

• L: the length of the railway segment
The RailwaySegment also keeps track of the train that is currently present on that segment.

When a Query arrives in the railway segment, it will reply immediately (i.e., with no delay) with a QueryAck. This will grant permission (green light) if there is no train present in the RailwaySegment at the moment, or will refuse entry (red light) if there is a train on the RailwaySegment. Upon receiving a QueryAck, the resulting action depends on whether the RailwaySegment is clear or not.

If the RailwaySegment is clear (green light), the train will accelerate as fast as possible. It will keep going full throttle for the remainder of the railway segment, of course capped by its maximum speed. Some math is required to determine how long it takes before the end of the railway segment is reached, and at what velocity the train will leave the railway segment. This has been solved for you: the file formulas.py includes 2 functions which define both the acceleration and braking behaviour. The first function (acceleration_formula), computes the acceleration behaviour, and takes the parameters:

• v_0: the current velocity, expressed in meters per second.
• v_max: the maximum allowed speed, expressed in meters per second.
• x_remaining: the length of the railway segment that needs to be traversed, expressed in meters.
• a: the maximum acceleration of the train, expressed in meters per second squared.
It will return a tuple containing the new velocity (in meter per second) at the end of the provided distance, and the time it will take to reach that distance (in seconds). The second function (brake_formula), computes the braking behaviour, and takes the parameters:
• v_0: the current velocity, expressed in meters per second.
• t_poll: the time between two polls.
• x_remaining: the total distance that remains to be travelled before the red light.
It will return a tuple containing the new velocity (in meter per second) after t_poll seconds have passed, and the distance travelled in that period (in meters).

After the time, returned by the acceleration_formula function, has passed, the train needs to leave the current railway segment, and enters the next railway segment. The train will keep accelerating here too, but only until the traffic light of the next railway segment is within sight. We assume that a traffic light can be seen from a distance of 1000 meters. This means that your first call to the acceleration_function has to pass the distance of the first part of the segment (the total distance minus 1000). Therefore, the train will keep accelerating until it is 1000 meters before the traffic light, and will then look at the traffic light. During this observation, the train will just maintain its previous acceleration. From there on, it continues as specified above. Note that, if the length of the railway track is less than 1000, the driver will sometimes see multiple traffic lights. The driver will then only be interested in the traffic light of the nearest (next) track, and not look at the further ones (yet).

If the railway segment is occupied (red light), the train will brake gradually. While it is possible to keep accelerating, and finally brake as hard as possible at the last second, this is not very comfortable for the passengers, nor is it particularly safe. Therefore, the train will decrease its velocity linearly, by using a negative (but constant) acceleration. This is automatically computed by the provided formulas, and will make it such that the train would stop right in front of the traffic light. We assume that the tracks are sufficiently long for a train driving at the maximum speed, to come to a halt before reaching the traffic light. As such, there is no need to model cases where the train can not stop in time.

The train driver will poll (look at) the traffic light every second, by sending a new Query. If entry is still prohibited (red light), this behaviour will be repeated. If entry is allowed (green light), the train will again accelerate for the remaining distance of the track. Note that we assume that it is impossible for a train driver to see a green light, which later on turns red.

Figure 3: The evolution of the speed in different phases.

An example is shown in Figure 3. At point 1, the segment is entered, and the train accelerates until it reaches v_max. At point 2, v_max is reached, and the speed is capped. At point 3, the first Query message is sent to the next railway segment. During this phase, the train does not change its velocity yet. At point 4, a QueryAck is received from the next railway segment, which says that it is red. Therefore, the train starts to brake. At point 5, the second Query is sent. The train keeps braking until a reply is received. At point 6, the QueryAck is received, which still results in a red light, thus further braking the train. At point 7, the same happens as on point 5. At point 8, finally, a permissive QueryAck (green light) is received. The train starts to accelerate again, until it leaves the railway segment.

With the provided formulas, you would need to call the acceleration_formula at point 1, which will return you the time of point 3. Point 2 is completely taken care of by the functions. At point 3, a Query is sent, but the train keeps accelerating, so the acceleration_formula is again needed. At point 4, the train knows that it has to brake, and it will call the brake_formula, and will poll again after the polling time. At point 5, a new Query is sent, but because we keep braking, the brake_formula is called, thus computing the distance travelled during the wait for poll. At point 6, exactly the same happens as in point 4. Point 7 is again identical to point 5. Then, at point 8, the train gets a green light, and therefore starts accelerating again for the remaining distance. Here again, the acceleration_formula is called, which will return the time at which the railway segment has to be exitted.

Note that in our abstraction, a driver who saw a green light will never respond to a red light anymore. This is because it is a single (no merging tracks) straight stretch of track, and the light would have no reason to switch to red again after it was green.

## Simulation

Make a plot showing both the performance and the cost of the railway network for varying length of segments. Do this for a fixed set of parameters. Plot the value of the cost function for a fixed set of parameters, thus showing which length is ideal for these parameters.

Finally, run these simulations for a varying IAT, plotting the ideal railway segment size for each possible configuration of the IAT. Plot your results. You can do the same while varying other parameters.

Perform a reasonable number of simulation experiments and be sure to sample with enough trains. Make sure that the total length of the railway track is equal across all experiments: 10 segments of 1 kilometer should be compared to 5 segments of 2 kilometers! Always choose the simulation duration sufficiently long enough to get statistically relevant measurements. Discuss your results!

## Practical issues

You will use the PythonPDEVS simulator. You are strongly advised to first study the Classic DEVS TrafficLight example in the examples directory before starting on this assignment. Installing PythonPDEVS can be done by executing python setup.py install --user. Note that you need Python 2.7 to be able to run PythonPDEVS. Finally, you can run the simulation by executing python experiment.py.

Note that in PythonPDEVS, when an external transition is triggered, this means that some external input has arrived on one or more of the ports. The inputs will be passed to the method in the form of a dictionary, which is the only argument of the extTransition method. The key values of this dictionary are the ports. If a port is not present in the dictionary, there was no input on that port. The elapsed time can be accessed using the elapsed attribute, but note that this is only correct in the extTransition method. It is therefore not allowed to access the elapsed attribute in an intTransition (and its value will be undefined).

Apart from the model, you will also need to create an experiment file. In that file, you instantiate the model and the simulator, and run the simulation with a specific configuration. Make sure that you run the simulation as Classic DEVS, with the configuration option setClassicDEVS(True). After the simulation, it is possible to access the model and its updated state.

As you will be comparing different approaches, you would normally have to create multiple simulators, one for each model being compared. The statistics can be printed after the simulate call and printed to the console, to be plotted at the end (e.g., using gnuplot).

In Python, you can use the "infinity" value using the float("inf") construct.