23-Ind-A6 Systems Simulation · December 2017

Question 3 of 8: Number of Replications for a Target Precision

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

National Exams — December 2017 — 98-Ind-A6 Systems Simulation. Three-hour, closed-book exam; one of two permitted calculators (Sharp or Casio), two 8.5″×11.0″ aid sheets (both sides). Format: eight questions of equal value (10 marks each); candidates complete SIX of the EIGHT (only the first six as they appear in the answer book are marked). All eight questions are solved below for completeness (the paper's own Q6 is printed with an orphan leading "6." followed by "2. Consider an M/M/1 system…" and is answered as the paper's sixth question). Common Discrete/Continuous Distribution tables, Student-t and Chi-square tables were supplied with the exam; the values below are the same table values obtained by direct computation.

Reference texts: Banks, Carson, Nelson & Nicol, Discrete-Event System Simulation (5th ed., Pearson) — random-number/random-variate generation, input data analysis, output analysis (replications), comparing alternative systems, queueing simulation, and the simulation study life cycle.

Question 3 — Number of Replications for a Target Precision (10 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Pilot study of $n_0=5$ replications (1 year each, warm-up already deleted): completed parts $=1377,\ 1350,\ 1299,\ 1324,\ 1578$. Target half-width $E=\pm50$ parts, $\alpha=0.05$.

Find. The smallest total replication count $R$ such that the 95% CI half-width on mean completed parts is $\le 50$, using the exact $t$-distribution (not a Normal approximation).

Approach. Fix the working standard deviation at the pilot value $s$, then solve $t_{\alpha/2,R-1}\,s/\sqrt R \le E$ iteratively for $R$ (both $t$ and $\sqrt R$ depend on $R$, so this cannot be solved in closed form).

Pilot mean and SD. $$\bar x = \frac{1377+1350+1299+1324+1578}{5} = 1385.6,\qquad s = 111.42\ \text{parts}.$$
Iterate the half-width formula. $HW(R) = t_{0.025,R-1}\,s/\sqrt R$:

$R$	$t_{0.025,R-1}$	Half-width
5	2.776	138.3
10	2.262	79.7
18	2.110	55.4
20	2.093	52.1
21	2.086	50.7
22	2.080	49.4

Decision. $R=21$ still gives a half-width of 50.7 $>$ 50; $R=22$ is the first value with half-width $\le 50$: $$R = \boxed{22\ \text{replications total}}.$$ Since 5 pilot replications already exist, $22-5=17$ additional replications are needed.

Quantity	Result
Pilot mean / SD ($n_0=5$)	1385.6 / 111.42 parts
Required total replications	22
Additional replications beyond the pilot 5	17

Check: uses the pilot-study $s$ as a fixed working estimate of $\sigma$ across the iteration (standard practice when only a small pilot sample exists), and re-derives $t_{0.025,R-1}$ at each candidate $R$ rather than freezing it at the pilot's $df=4$ — the instruction to avoid the Normal approximation means $t$ must keep shrinking toward $z_{0.025}=1.960$ as $R$ grows, not jump straight to it.

Concept & Theory

Topic. Sequential/iterative sample-size determination for a simulation output mean, using the exact $t$-reference distribution.

Key relations. $HW(R) = t_{\alpha/2,R-1}\,s/\sqrt R \le E$, solved by search since $R$ appears in both the $t$ critical value and the denominator.

Why this works. Replacing $t_{\alpha/2,R-1}$ with the Normal $z_{\alpha/2}$ would understate the required $R$ (the $t$ distribution has heavier tails at small $df$), which is exactly why the question forbids the Normal approximation — the true answer requires one more replication (22) than the Normal-approximation shortcut would suggest.

Common pitfall. Solving $n=(t\,s/E)^2$ once at the pilot $df$ and stopping (treating $t$ as a constant) instead of re-evaluating $t_{\alpha/2,R-1}$ at each candidate $R$ in the search.

Source. Banks, Carson, Nelson & Nicol, Discrete-Event System Simulation, ch. 11 (output analysis for a single system, replication/deletion, sample size).

← Question 2 Next: Question 4 →