Output Analysis for Terminating Simulations

• Use independent replications, i.e. the simulation is repeated a total of R times, each run using a different random number stream and independently chosen initial conditions.

• Let $Y_{ri}$ be the ith observations within replication r, for $i = 1, 2, ... n_r$ and r = 1, 2, ..., R.

• For a fixed r, $Y_{r1}, Y_{r2}, ...$ is an autocorrelated sequence. For different replications $r \ne s$ $Y_{ri}$ and $Y_{sj}$ are statistically independent.

• Define a sample mean
  
  $$\hat{\theta_r} = \frac{1}{n_r} \sum_{i=1}^{n_r} Y_{ri}, ~~ r = 1,2,...,R$$
  
  

• There are R samples, so R sample means, the overall point estimate is
  
  $$\hat{\theta} = \frac{1}{R} \sum_{r=1}^R \hat{\theta_r}$$