int rate = 30;

float a = 2 * 255.0 * rate / DISPLAY_TIME;

void setup() {
// ...
  frameRate(rate);
}

Uncomment your bound test.

hi phi.lho,
thank you for the answer.
i changed the code, but it is not exactly what i am trying to achieve.
the first string displayed fades out --- the second fades in and so forth
at the moment it is to fast and it looses sync.
finally the strings (johnCage quotes) are diplayed 18000ms

regards
rettnoise

PFont font;
int counter;
final int DISPLAY_TIME = 3000; // 2000 ms = 2 seconds
int lastTime; // When the current image was first displayed

float rx = 120;
float ry = 160;
float alphaVal = 255;
int rate = 30;
float a = 2*255.0*rate/DISPLAY_TIME;

void setup() {
  size(640, 480);
  background(255);
  font = loadFont("ArialUnicodeMS-48.vlw");
  textFont(font);
  textSize(random(22, 35));
  fill(66, 190, 255);
  lastTime = millis();
  frameRate(rate);

}

void draw()
{
  background(255);
  if (millis() - lastTime >= DISPLAY_TIME) // Time to display next image
  {
    counter = int(random(quotes.length));
    textSize(random(20, 30));
    rx = random(10, 220);
    ry = random(10, 120);
    // alphaVal = 255 
    lastTime = millis();
  }
  fill(66, 140, 255, alphaVal);
  text(quotes[counter], rx, ry, 400, 300);
  alphaVal -= a;
 
  if (alphaVal < 0 || alphaVal > 255) {
   a *= -1;
 }
}


String[] quotes = {
  "string_1",
  "string_1",  
  "string_1",
  "string_1",  
 
};

On my computer, with the above code, I see string_n appearing, fading out then in, then another string_n (I changed the number of verify they are distinct) appears elsewhere, fades out then in, and so on.
That's what you asked (fade out, then in).

If you want the reverse effect, just set (and reset) alphaVal to 0 instead of 255, and use alphaVal += a; (instead of -=).