Spawn rates and variable time steps

Question

How would I work a spawning algorithm into a sim engine with variable time steps?

If each engine step is, say, 1/30 of a second and I have some event occur with a 0.001 probability each step, how do I translate this to an engine with a variable time step? At the end of 1 second I could have nothing spawned or (in a very unlikely event) I could have as many as 30 spawned.

Now if the engine hasn't done an update for a full second, how can my spawner simulate the same behaviour and probabilites? What if my steps are much smaller, say 1/100 or 1/1000 of a second?

What algorithms can I use?

Answer 1

I'd go with something like this (pseudo-code):

spawnProbability     = 0.001 // Your original probability at fixed timestep
spawnAtThisFrequency = 1/30  // The original fixed timestep dt
dtAcc                = 0     // An accumulator that will accumulate the variable dt

update( dt )
{
  dtAcc += dt; // Accumulate

  while( dtAcc >= spawnAtThisFrequency ) // If enough time has accumulated...
  {
    dtAcc -= spawnAtThisFrequency // remove from the accumulator
    tryToSpawnWithProbability( spawnProbability ) // and try to spawn
  }
}

With this, you take into account that the dt can be smaller than what it was with a fixed timestep, and you can also handle larger dt.

Answer 2

The number of spawns will be distributed as a binomial, the sum of the Bernoulli RVs of your spawn windows.

E.g., say for a window w of some number of ticks per spawn opportunity, and an elapsed ticks e since last check, with Bernoulli probability of p per spawn window, the number of spawns will be distributed as

B(e/w,p)

So, at each spawn check (end of period e), take one sample from the resulting distribution, and you have the total number to spawn.

Obviously, e/w is not always an integer, so round/floor/ceiling as desired.

Answer 3

I just used Random.Range for my spawn times. Could you use the same?