Types of output analysis; infinite population

The question has two parts: the types of output analysis, and the method suited to an infinite-population (non-terminating) model. This answer treats "infinite population" as a system that runs continuously with no natural end, so its long-run (steady-state) behaviour is required.

Types of simulation output analysis

Simulations are classified by how they end, and the analysis depends on this.

1. Terminating (finite-horizon) simulation: the run ends at a natural event E, and the initial conditions are part of the system. Examples: a bank that opens at 9:00 and closes at 17:00, or a shop that closes when the last customer leaves. Measures are of the transient period.
   • Analysis: independent replications, each of the same fixed length or until E occurs. Use the mean of replication outputs and the t-based confidence interval.
2. Non-terminating (steady-state) simulation: the system runs for a very long time with no specific end event, such as a telephone exchange, a network router or a 24-hour hospital. Measures are long-run averages that do not depend on initial conditions.
   • Analysis: steady-state methods: replication with deletion of the warm-up, batch means, regenerative method, or autoregressive/spectral methods.

Both can also be classed as transient (depends on the start) and steady state (independent of start).

Method for infinite population

With an infinite (unlimited) calling source, arrivals never stop, and the system settles into a stationary distribution if traffic intensity ρ < 1. So the non-terminating, steady-state analysis applies.

Why:

• There is no natural stopping event, so replications of a terminating type have no definition of "end".
• The aim is the long-run mean, so the effect of the empty initial state (initial bias) must be removed.
• The usual procedure is to run long, delete the initial transient (Welch's method), divide the remaining data into k batches, and form the confidence interval from batch means: Ȳ ± t(α/2, k-1)·S/√k.
• Batching makes the autocorrelated observations nearly independent, which the confidence-interval formula needs.

Detail of the steady-state methods

• Replication with deletion: run R replications, each long, delete the first d observations, average the rest in each replication, then apply the t-interval to the R averages. Simple, but the warm-up is paid R times.
• Batch means: one very long run is cut into k batches of equal size (k is commonly 10 to 30). Batch means are almost independent when the batch is long, so one warm-up is enough and the cost is lower.
• Regenerative method: the output is split at times when the system returns to the same state (for example, the system becomes empty). Cycles are independent, so no warm-up is needed, but such returns may be rare in busy systems.

Example

A router queue with Poisson packet arrivals (unlimited source) is run for 110000 packets. The first 10000 are deleted as warm-up. The remaining 100000 are divided into 20 batches of 5000, and batch average delays are computed. If their mean is 12.4 ms and S = 1.1 ms, then with t(0.025, 19) = 2.093 the 95% interval is 12.4 ± 2.093 × 1.1/√20 = 12.4 ± 0.51 ms.

Summary