Question 2 of 10: Linear Congruential Generator — Output, Period, Repeatability
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Exams May 2019 — 17-Ind-A6, Systems Simulation, 3 hours duration. Section A: do 3 of 5 (30 marks); Section B: do 1 of 2 per the front-page summary table — the Part B section instructions on the page itself read “do 2 of 3” instead, a genuine inconsistency in the source paper's own front matter (flagged below); Section C: do 1 of 2 (20 marks). This study resource answers all ten questions across the three parts.
Check — front-page/section instructions disagree. The cover page's summary table states “Section B: Do 1 of 2 Questions, Total marks: 20,” but Part B's own section banner (page 4) reads “Complete two of the following three sets of questions,” and Part B in fact contains three questions worth 15 marks each. The cover page's own overall total (70 marks across “5 Questions”) is likewise only consistent with the “2 of 3” reading (30+30+... does not match 70 either way exactly, since 30(A)+2×15(B)+20(C)=80, one section choice short of 70); this is an internal inconsistency in the exam's own printed materials. All three Part B questions are answered in full below regardless.
Reference texts. Banks, Carson, Nelson & Nicol, Discrete-Event System Simulation (5th ed.) — primary text for random-variate generation, input/output analysis, variance reduction, verification & validation, and design of simulation experiments.
Find. (a) $R_1,R_2$; (b) the generator's maximum possible period and the condition that grants it; (c) the meaning and simulation importance of repeatability.
Approach. Iterate the recurrence twice for (a); check the Hull–Dobell full-period conditions for (b); answer (c) conceptually.
(a) Iterate the LCG.
$$X_1=(13\times12+13)\bmod16=169\bmod16=9,\qquad R_1=X_1/m=9/16=\boxed{0.5625}$$
$$X_2=(13\times9+13)\bmod16=130\bmod16=2,\qquad R_2=X_2/m=2/16=\boxed{0.1250}$$
(b) Maximum possible period. For a mixed congruential generator, the Hull–Dobell theorem gives full period $m$ iff: (i) $\gcd(c,m)=1$; (ii) $a-1$ is divisible by every prime factor of $m$; (iii) if $m$ is a multiple of $4$, $a-1$ must also be a multiple of $4$. Here $m=16=2^4$: $\gcd(13,16)=1$ ✓; the only prime factor of $m$ is $2$, and $a-1=12$ is divisible by $2$ ✓; $m$ is a multiple of $4$ and $a-1=12$ is also a multiple of $4$ ✓. All three conditions hold, so
$$\boxed{\text{maximum possible period}=m=16}$$
(every one of the $16$ residues $0,\dots,15$ is visited exactly once before the sequence repeats).
(c) Repeatability. A random number generator is repeatable when re-seeding it with the same $X_0$ (and the same $a,c,m$) reproduces the identical stream of pseudo-random numbers on demand — the sequence is fully determined by the seed, not by external system noise. Repeatability matters for a simulation because it lets an analyst re-run a scenario under the exact same underlying randomness while changing only one design variable at a time (a controlled experiment), reproduce and debug an unusual observed result, and apply variance-reduction techniques such as common random numbers, all of which require driving two or more runs from an identical (or a precisely related) stream. In practice, repeatability is assured by recording the seed(s) used for each stream and by assigning logically distinct streams (or widely separated sub-sequences of one very-long-period generator) to different sources of randomness in the model, so each source can be repeated or resynchronized independently.
