Simulation run statistics

Simulation output is random, so a run produces a set of observations from which run statistics are computed to estimate the system's performance measures. The statistics are calculated over the whole run (or over a chosen time period T), and a single run of length T gives one estimate for each measure.

Types of run statistics

1. Observation-based (tally) statistics: computed from a list of discrete observations, such as the delay of each customer. Mean = (sum of x_i) / n and variance S² = sum(x_i - mean)² / (n - 1).
2. Time-persistent statistics: computed for variables that hold a value for a duration, such as queue length or server status. The time-weighted average is the area under the curve divided by run length T: Q̄ = (1/T) · ∫ Q(t) dt = Σ (Q_i · duration_i) / T.
3. Counts and rates: number served, throughput = number completed / T, number lost or balked.
4. Extremes: maximum queue length, maximum waiting time.

Procedure

1. Decide the measures and the run length (or terminating event).
2. Collect data during the run, usually updating running sums at each event.
3. At the end, compute the statistics, and, from several runs, form confidence intervals.

Example: single-server queue, T = 10 min

Queue length: 0 for 0-2 min, 1 for 2-5 min, 2 for 5-6 min, 1 for 6-10 min. The server is idle for 0-2 min.

• Average queue length = (0·2 + 1·3 + 2·1 + 1·4) / 10 = 9/10 = 0.9.
• Utilization = busy time / T = 8/10 = 0.8.

Five customers wait 0, 2, 3, 1, 4 min.

• Mean delay = 10/5 = 2.0 min.
• S² = (4 + 0 + 1 + 1 + 4)/4 = 2.5, so S = 1.58 min.
• Maximum delay = 4 min.

These are estimates from one run; another random stream gives other values, so the statistics from independent runs are used for confidence intervals.