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David D. Yao coauthored the 1992 work on common random numbers with Paul Glasserman. Their analysis gives conditions under which coordinated random inputs improve comparisons of simulated systems, emphasizing monotonicity and continuity of event times.^[ref] The cited primary byline retains his middle initial without expanding it.

An original queueing theory calculation illustrates the event-time viewpoint. Start an initially empty, first-come-first-served single-server queue with arrivals $A=(0,1,3)$. For service times $S_n$, its single-server queue departure-time recursion isThis is the paper's service-completion recursion in renamed notation.^[ref] With service times $(2,2,2)$, the departures are $(2,4,6)$. Reducing all services to $(1,1,1)$ gives departures $(1,2,4)$ using the same arrival input.

At time $2.5$, the first system has completed one job and the second has completed two. The states differ even though corresponding job labels and arrival times are coordinated. In a stochastic comparison, random inputs should likewise be assigned consistently to the intended jobs or events; a raw position in a mixed event list need not identify the same physical event in both simulations.

The deterministic numbers above demonstrate the recursion, not a measured variance reduction. A useful random coupling must preserve each system's marginal input law and have the response properties required for the chosen performance comparison. Shared random seeds alone do not establish those properties.

Paul Glasserman studies simulation and statistical methods. His 1992 joint paper with David D. Yao investigates when common random numbers beneficially couples comparisons of stochastic systems, including parameter-sensitivity comparisons.^[ref]

An original finite-difference example shows how the noise scale can depend on coupling. For $\theta >0$, let $S_{\theta }=\theta E$, where $E$ has an exponential distribution of mean and variance one. Since $\mathbb{E}[S_{\theta }]=\theta$, its expectation derivative is $1$. Using the same $E$ for both parameter values giveswhose variance is $1$ regardless of positive $h$.

If the two simulations instead use independent random variables $E_1,E_2$ with that same exponential law, the estimator remains unbiased but has varianceFor chosen $\theta =1$ and $h=0.01$, this is $20201$, compared with $1$ under the shared-input coupling. Averaging independent replicate pairs divides either variance by their number. These are exact properties of the stipulated linear simulation model, not benchmark measurements.

The useful cancellation occurs because the two outputs respond smoothly to the same random input. Other models can give a less favourable covariance or discontinuous response. The paper's conditions, rather than the act of reusing a seed alone, justify its general variance-reduction conclusions. Common random numbers explains the underlying covariance identity and a counterexample where sharing inputs makes a comparison noisier.

Primary paper, Management Science 38(6), 884–908 (1992), by Paul Glasserman and David D. Yao. Page884 gives the covariance identity for a difference of simulated outputs; the abstract and later results state conditions for effective common random numbers.

Additional locators: §3, equations 8–9, printed p. 894, arrival and service-completion recursions; §4.2, Example 4.6, printed p. 897, initially empty single-server queue and event-time properties.