Euler got to Version 19.5.  There is now a script mode with the parameter -script in the command line. No window will appear, unless there is an error in the script. The scripts are ordinary Euler files. To pass parameters, either read a file, or use a command line parameter. There is a new function eulerarg(i) to read the i-th parameter.

Another feature are hidden comment lines. These lines start with „%% „.

I have update the notebook about the average waiting time. The general case is a graph with transition probabilities.  The probabilities do not add to 1 in each node, and the rest is the probability to exit the graph from this node. It is interesting that the average waiting time to exit from the graph can be computed very easily with the formula

\((I-P) w = 1\)

where I is the identity matrix, and 1 is the vector with ones everywhere. The vector w contains the average waiting times w[i] for the exit, if we start at node i.

This trick can be applied to events of all sorts. E.g., the waiting time for all dices to occur can be modeled.