23-Ind-A6 Systems Simulation · Undated paper

Question 8 of 10: Variance Reduction Techniques — Common Random Numbers and Antithetic Variates

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

Question 8 (Part B.3): Variance Reduction Techniques — Common Random Numbers and Antithetic Variates (15 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.

This is an essay question on output-analysis methodology; no numeric data is supplied, so the answer below is flowing prose organized by sub-part.

(a) How variance reduction cuts computation time. A performance estimate's precision (its confidence-interval half-width) shrinks with $1/\sqrt n$ for a fixed per-observation variance, so reaching a target precision by brute force means running enough independent replications to drive $n$ up. A variance-reduction technique (VRT) instead lowers the variance per replication (or, more precisely, the variance of the estimator built from a fixed computational budget) by exploiting known structure in how the random inputs are used — most commonly by deliberately correlating the random-number streams across runs being compared, rather than letting them be independent. Because the required number of replications for a target precision scales with the variance, a technique that meaningfully reduces variance lets the same precision be reached with far fewer (and therefore computationally cheaper) simulation runs, directly cutting total run time.

(b) Common random numbers (CRN). When comparing two or more alternative system configurations (e.g. two staffing policies), CRN drives each configuration's replications with the same underlying random-number streams (same seeds, with each stream dedicated to the same source of randomness — e.g. the same stream for arrival times in every configuration). Because both configurations then see the identical sequence of “random” events, differences between their outputs are due to the change in configuration itself rather than to different luck in the random draws. If the two configurations' outputs respond similarly (positively correlated) to the same random inputs — the usual case when comparing similar systems — this induced positive correlation cancels out of the variance of the difference between the two outputs, $\operatorname{Var}(\bar X_1-\bar X_2)=\operatorname{Var}(\bar X_1)+\operatorname{Var}(\bar X_2)-2\operatorname{Cov}(\bar X_1,\bar X_2)$, which is smaller than the independent-streams case whenever $\operatorname{Cov}>0$.

(c) Antithetic random variates. Within a single configuration, antithetic variates pair each replication driven by uniform stream $U$ with a companion replication driven by $1-U$ (using the same stream, complemented). If the simulation's output response is a monotonic function of the underlying uniforms, $Y(U)$ and $Y(1-U)$ tend to move in opposite directions from run to run, inducing a negative correlation between the pair. Averaging the pair, $\bar Y=(Y(U)+Y(1-U))/2$, then has variance $\tfrac14\big[\operatorname{Var}(Y(U))+\operatorname{Var}(Y(1-U))+2\operatorname{Cov}(Y(U),Y(1-U))\big]$, which is reduced below the two-independent-runs case precisely because the covariance term is negative.

(d) Drawbacks and difficulties. CRN requires careful, disciplined stream management — the same logical source of randomness (e.g. “patient arrival times”) must be assigned its own dedicated stream in every configuration being compared, and any accidental desynchronization (e.g. one configuration consuming a different number of random draws per event than another, shifting the streams out of alignment) can destroy the intended correlation or even induce the wrong sign, making the variance worse rather than better. Antithetic variates add implementation complexity (generating and correctly pairing $U$ with $1-U$ throughout the whole model) and, if the response is not actually monotonic in the uniforms, can fail to produce the needed negative correlation. Both techniques also add analyst effort and code complexity relative to plain independent replications, and neither is guaranteed to help — a technique applied blindly can, in the worst case, increase variance instead of reducing it.

(e) When are these techniques assured to work, and is that realistic? CRN is assured to reduce variance of a comparison when the compared configurations' outputs are positively correlated under synchronized streams — which holds when the systems are structurally similar and the streams are properly synchronized event-for-event. Antithetic variates are assured to reduce variance when the simulation's output is a monotonic function of the driving uniform random numbers, so that $U$ and $1-U$ reliably push the output in opposite directions. $$\boxed{\text{Neither condition is universally realistic}}$$ Complex simulations with many interacting random inputs, conditional branching, or non-monotonic responses (e.g. a queueing system where a random input can push a metric up in one region of its range and down in another) can violate monotonicity or induce weak/negative correlation between compared systems, in which case CRN or antithetic variates can fail to help, or even hurt. In practice, an analyst should verify the expected sign of the correlation (pilot-test the technique and inspect the resulting variance) rather than assume a VRT will work simply because it was applied.