Initial bias and its elimination

Initial bias (initialization bias, start-up problem) is the error in the estimated steady-state measures caused by the artificial starting conditions of a simulation run. A run usually starts "empty and idle", which is unlike the long-run behaviour of the system, so the early observations are not representative of steady state and pull the average away from the true value.

For example, a bank simulation starting with no customers shows short waiting times during the first hour. Including these data underestimates the true mean waiting time.

Methods to reduce or remove initial bias

1. Intelligent initialization: start the model in a state that is typical of steady state, such as with the average number of customers already in the queue. The state can be taken from the real system or from a pilot run.
2. Deletion of initial data (warm-up period): run the simulation for a warm-up time T0, discard all observations collected before T0, and compute the statistics only from the remaining data. For a run of length TE, the estimate is the average over (T0, TE].
3. Long runs: make the run so long that the initial transient has little effect on the overall average. This is simple but costly.
4. Replication with deletion: make R independent replications, delete the first d observations of each, and average the rest.
5. Batch means: divide one long run into equal batches after deleting the warm-up, and use batch averages as nearly independent observations.

Choosing the deletion point: Welch's method

1. Make R (at least 5) replications of length m, and compute the average Y_j over replications for each observation index j.
2. Smooth the averaged series with a moving average of window w to remove noise.
3. Plot the smoothed series and find the point where it flattens.
4. Take that index as the number of observations to delete.

 value
   |  .                  steady state
   |    .  ___ ___ ___ ___ ___ ___
   |     ./
   |  transient | 
   +------------+------------------> time
              T0 (delete before)

A warm-up that is too short leaves bias; one that is too long wastes data and raises the variance. In practice it is common to choose it generously.