23-Ind-A6 Systems Simulation · December 2013
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
National Exams — December 2013 — 98-Ind-A6 Systems Simulation. Three-hour, closed-book exam; one of two permitted calculators (Sharp or Casio), one 8.5″×11.0″ aid sheet (both sides). Format: 14 sub-questions across four parts — Part A (do 1 of 2, 25 marks), Part B (do 3 of 5, 15 marks), Part C (do 1 of 3, 10 marks), Part D (do 2 of 4, 20 marks); 7 questions, 70 marks constitute a complete paper. All fourteen sub-questions are solved below for completeness (the source restarts its own numbering at 1 within each Part). Statistical tables (Normal, t, chi-square, F) 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) — simulation study design, random-number generation, input/output data analysis, variance reduction, verification & validation, queueing simulation; Montgomery, Design and Analysis of Experiments (9th ed., Wiley) — factorial designs and ANOVA.
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.
Common random numbers (CRN). When comparing two or more system configurations (e.g. ship-heavy vs. truck-heavy transport policy), CRN drives each configuration's simulation with the SAME underlying random-number streams, synchronized so that, say, the third simulated transit-time draw uses the same $U(0,1)$ value in every configuration. If the configurations' outputs respond similarly to the same random inputs, this induces positive correlation between the two configurations' output estimates; since $\mathrm{Var}(\hat\theta_1-\hat\theta_2)=\mathrm{Var}(\hat\theta_1)+\mathrm{Var}(\hat\theta_2)-2\,\mathrm{Cov}(\hat\theta_1,\hat\theta_2)$, positive covariance directly shrinks the variance of the DIFFERENCE estimator that a policy comparison actually cares about.
Antithetic variates. Within a SINGLE configuration, run each replication as a pair: one driven by random numbers $U_i$, its partner driven by $1-U_i$ (which is also $U(0,1)$). Because the inverse-transform outputs from $U$ and $1-U$ tend to move in opposite directions (a large $U$ giving a large transit time in one run pairs with a small transit time in its partner), the PAIR's average has lower variance than two independent runs would, provided the output is a monotonic function of the underlying random numbers.
Drawbacks and detecting failure. Both techniques can backfire: CRN can INCREASE variance if the synchronization accidentally induces negative rather than positive correlation (e.g. mismatched event sequencing across configurations, so the "same" random number is consumed by different events in each run); antithetic pairing backfires if the output is NOT monotonic in the random numbers (the negative correlation the method relies on simply does not materialize, or reverses). Failure is identified empirically, not assumed: run the variance-reduced design AND a plain independent-replications design at the same computational budget, and compare the resulting variance of the estimator directly — if the "reduced" design's variance is not smaller, the technique has failed for this model and should be dropped.
Conditions for guaranteed success, and their realism here. CRN is guaranteed to help only when the response is monotonic in each underlying random number and synchronization across configurations is maintained event-for-event; antithetic variates are guaranteed to help only when the output is a monotonic function of the random-number stream. For a system this size — seasonal mode choice, multiple parallel subcontractors, UNIF sub-assembly sizes all interacting through one shared, capacity-limited yard — strict monotonicity is unlikely to hold everywhere (a slightly longer transit time can, through the yard's capacity constraint, sometimes IMPROVE a downstream metric by smoothing an arrival spike), so these conditions are not fully realistic here; the techniques should be tried and empirically validated per output measure, not assumed to work.
Control variate. A control variate exploits a second output, $C$, whose expected value $E[C]$ is known ANALYTICALLY and which is correlated with the output of interest $Y$; the adjusted estimator $\hat Y_c = \hat Y - c\,(\hat C - E[C])$ (for an optimally chosen coefficient $c$) has lower variance than $\hat Y$ alone whenever $C$ and $Y$ are correlated. Here, a natural control variate is the yard's occupancy under the simplified M/M/2-type queueing approximation from Question 12 (D2), whose steady-state behaviour has known closed-form moments and is correlated with the full model's simulated yard occupancy.
