Solution. \(7, 6, 1, 8, 11, 10, 5, 12, 15, 14, 9, 0, 3, 2, 13, 4\), and then \(7\) again: period \(16\), the full modulus. This agrees with the Hull–Dobell theorem: \(c = 3\) is coprime to \(16\), and \(a - 1 = 4\) is divisible by \(2\), the only prime factor of \(16\), and by \(4\), since \(16\) is a multiple of \(4\).

Solution. \(F(x) = x^2/2\) for \(0 \leq x \leq 1\) and \(1 - (2 - x)^2/2\) for \(1 \leq x \leq 2\). Solving \(F(x) = u\): \(x = \sqrt {2u}\) for \(u \leq \frac 12\) and \(x = 2 - \sqrt {2(1 - u)}\) for \(u > \frac 12\).

Solution. \(1.96 \times 2/\sqrt R \leq 0.05\) requires \(R \geq (1.96 \times 2/0.05)^2 = 6147\) replications; halving the half-width would need four times as many.

Solution. \(\operatorname {Var}e^U = \frac {e^2 - 1}2 - (e - 1)^2 = 0.2420\) and \(\operatorname {Cov}(e^U, e^{1 - U}) = e - (e - 1)^2 = -0.2342\). Two independent draws have mean with variance \(0.1210\); an antithetic pair has \(\frac 12(0.2420 - 0.2342) = 0.0039\). The ratio is \(31\), matching the \(30\) of Software 16.5.3.

Solution. Check that each part behaves correctly by tracing events by hand on a short run; run special cases with known answers (one doctor, exponential times: an M/M/1 queue); check identities that hold whatever the details, such as Little’s law between the simulated mean number present, arrival rate and mean time in the department; and compare the simulated waits with the department’s records for the present staffing before using the model to predict the effect of changes. The first steps are verification (is the program right?), the last is validation (is the model right?).

Solution. (a) Linear programming, with sensitivity analysis for the limits. (b) Integer programming (a facility-location model with binary site variables). (c) Queueing, an M/M/\(c\) model with a cost or service-level criterion, or simulation if arrivals vary through the day. (d) The newsvendor model. (e) A replacement model, or a Markov decision process if the buses’ condition is observed and deteriorates at random.