Is it possible to generate this kind of random curves?
I've tried IMagick bezier curves (see http://www.php.net/manual/en/function.imagickdraw-bezier.php), but even with 20-30 points they do not look like this. Here is my sample http://mechanicalzilla.com/sandbox/imagick/curve.php
Thank you.
I bet you could write an algorithm which would basically take x number of random twists before going straight to the exit coordinates. This also assumes that algorithm is smart enough to check the angle of the turn. (assuming you don't want to endup in knot-web)
However, assuming that this isn't your graduation task or that you are paid on per-hour basis to work on this, this would be a waste of time and success is highly doubtful.
Even if you'd manage to generate single line algorithm, doing it so that the lines wouldn't come too close to each other is close to impossible. You will end up with something like this: 
Looks like:
x = 0; y = 0; angel = 0;
while (true) {
angel = angel + 0.5 - random(1);
x1 = x + 0.1 * cos(angel);
y1 = y + 0.1 * sin(angel);
if (abs(x1 - x) + abs(y1 - y) < 10)
drawline(x,y,x1,y1);
x = x1; y = y1;
if (x < 0) x = width;
if (y < 0) y = height;
if (x > width) x = 0;
if (y > height) y = 0;
}
This is far from a complete answer, but in my mind's eye seems like it could help you:
Instead of drawing curves from the start to the end point of the entire line, consider subdividing your board into a evenly spaced grid. Each square of one column of the grid is entitled to have one point of one curve in it, and you'd steadily advance from left to right (at first? for simplicity's sake.).
The randomness would come into play by picking a square for a curve - to prevent it from getting too chaotic, you could give this randomness bounds, say, "you're not allowed to pick a square that (if a distance from square to square is considered 1) violates abs(current vertical position - new vertical position) <= 5 unless none such is free anymore at this point" or some other arbitrary restraint. ("unless none such is free anymore at this point" is important, otherwise it's possible to lock yourself into an unsolvable state.)
(Sorry, drawing curves with my mouse -> worst/no interpolation ever. Catmull-Rom interpolation will probably be your friend here, though, I imagine.)
The display should be loose enough given that your curve points cannot arbitrarily scatter together given a grid, but it's probably very difficult to get the curve to connect to the end point 'fluidly' - might be a good solution if you don't mind arbitrary end points, though, read as, the algorithm can decide for itself where it wants the line to end.
Think this idea might help you with your curves?
One way to approach this would be to first generate a set of random curves and then use a physics solver to apply repulsion forces between them to avoid clumping.
Here's a quick proof of concept:
I created this using a very niche tool (for anyone interested: Kangaroo Physics solver, a plugin for Grasshopper, visual scripting language for Rhinoceros3d) but you can probably recreate the same concept in any mainstream programming language, eg. Python.
