Errors in the book

Errata for 'Mathematical Foundations of Computer Networking'

My apologies for the following errors in the book.

Chapter 1: Probability

• Example 1.15, Page 12: The fact “P(52) = 0.54” is not known at this point in the book. It is derived in Example 1.16.
• Section 1.3.4, Eq. 1.19, Page 20: E[g(C)] should read E[g(X_C)]
• Section 1.3.5, The equation for variance should read V[X] = (Σ x_i²)/n - (Σ (x_i/n))². For practical use of this formula, please refer to this site.
• Section 1.4.1 Note that skewness is defined as M³_μ/(M²_μ)^1/2
• Section 1.5.1, Page 25, last sentence: Variance should be a(1-a)
• Section 1.6.3 Example 1.33, the calculation should read 1-F(6) = 1- 1- e^-6/3 = e^-2 = 13.5%
• Section 1.7.3 Example 1.38 Page 39, the last equation should read 1-(p(X=0) + p(X=1)... rather than (1-p(X=0) + p(X=1) ...
• Section 1.8.1, Page 47, the LHS of the last equation on the page should read p(L|R) not p(PLR)
• Exercise 13 should read "Prove ...N(0,1) without using MGFs.

Chapter 2: Statistics

• Section 2.4.5 Example 2.10 has an error in computation. s² should be 0.243, s should be 0.493, so the t variate is 0.016 (not 0.0048).
• Section 2.4.5 Equation 2.23 has an error: in the denominator, the left hand term's denominator should read n(n-1) and the right hand term's denominator should read m(m-1)
• Section 2.4.6 In equation 2.24, the LHS should read P(o₁^n₁o₂^n₂...o_k^n_k)
• Section 2.4.6, Example 2.12: "...any value greater than 3.84 occurs with probability less than 95%" - this should read 5%
• Section 2.5 Example 2.17: The mean throughput should read 60.4kbps and mean delay 766.66 ms. The value of r should be -0.82.
• Exercise 5: this exercise is incorrect, since the Chi-squared test should be done using counts, not percentages.

Chapter 5: Signals, Systems, and Transforms

• Section 5.6, Page 198, just above equation 5.33, replace "... each with a real magnitude c_k" with "... each with a magnitude c_k" (because this magnitude may be complex).
• Page 207, Table 5.2, the transform for sin \omega_0t is missing a factor of -1.
• Page 212, Example 5.21. Note that the zero is outside the ROC.
• Page 213, Table 5.3: there is an error in the ROC with frequency shifting. The ROC should be \alpha + Re(a) < Re(s) < \beta + Re(a).
• Page 213, Table 5.3: Note that the differentiation result in the table is for the bilateral Laplace transform. The unilateral transform of dx(t)/dt = sX(s) - x(0).

Chapter 6: Queueing theory

• Section 6.1.2, Page 238-239: The terms “waiting time” can mean different things. In the theorem as well as this proof, it refers to the time waiting in the queue, but not for service. However, the theorem applies to the latter case as well.
• Section 6.2 On pages 240/241, the subscript for X should be k rather than i, to avoid confusion later, where i is used to refer to the state, rather than time step
• Section 6.2.1 Example 6.5: The example should read 'Continuous Space Process'.
• Section 6.2.2, Page 244, Example 6.8. P(X=1) should read "P(X_2=1)".
• Example 6.9, Page 245, The last sentence should read ".... can compute the distribution of X_{n+1} given that of X_n for all values of n."
• Section 6.2.7, Page 249, The statement "If all states in a Markov chain are ergodic, other than for a finite set of transient states..." should read "If all states in a Markov chain are ergodic, other than for a finite set of transient states such that transient states always lead to an irreducible recurrent chain, then the chain itself is ergodic".
• Section 6.2.3, Equation 6.2, Page 246: The suffix of the first p in the sum should read “ik” instead of “ij”.
• Section 6.2.8, Theorem 6.11 is actually Theorem 6.1
• Section 6.2.11 ...the probability of staying in the state for m steps is given by p_jj^m-1(1-p_jj) not p_jj^m(1-p_jj)
• Section 6.3.3, Page 254, first sentence after Equation 6.9: Should read “during a small interval” instead of “during any interval”.
• Section 6.3.3, Page 254, first sentence after Equation 6.9: Should read “conditional on its already being at state i at time t” instead of “conditional on its already being at state i”.
• Equation 6.11, Page 254 “(t)” should be removed from the equation
• Section 6.4, Page 255, last sentence before Equation 6.15: Should read “Given this fact, Equation 6.10, and Equation 6.14, we find that”
• Section 6.4.2, Page 256, The first sentence in this section should read "The stationary probability distribution of an ergodic birth-death process is given by Equation 6.13. ".
• Section 6.4.2, Page 256, In the second sentence, delete the extraneous P_j.
• Section 6.4.4, Page 259, Eq. 6.24 the right hand side should read \lambda t e^{-\lambda t}.
• Section 6.4.4, Page 260, P(0 customer arrivals in time (0,t)) = \pi_0(t), not 1-\pi_0(t).
• Section 6.4.5: Page 261, The condition for ergodicity stated is incomplete. The full conditions can be found here.
• Example 6.22, Page 264, first sentence: Reference should go to Example 6.21, not “6.19”. Also, the observation that "we expect three in the queue and one in service" is only approximate. The mean number of packets in service is not 1, but \rho, but when the queue length is long (the case we care about), \rho is close to 1.
• Section 6.5, Page 265, just before Eq. 6.38, the equation is (rho/(1-rho)) not (rho/(1-rho)c
• Example 6.23, Page 266, first sentence: Reference should go to Example 6.21, not “6.20”.
• Page 267: just after Eq. 6.42, both references to customers in the queue should read "customers in the system."
• Example 6.25, Page 269, first sentence: Reference should go to Example 6.21, not “Equation 6.21”.
• Section 6.7.1, Page 270, replace "interdeparture time (a constant)" by "departure rate (a constant) "

Chapter 7: Game theory

• Section 7.2, Page 292, 4th line from bottom should read "If either prisoner informs on the other (defects, or D), that prisoner is set free, ...
• Section 7.2, Page 293, payoff matrix for Prisoner's Dilemma game, the tuple in row 1 col 2 should read (-4, 0)
• Section 7.2.4, Page 294, just before example 7.18, for the maximin strategy, the equation should read

<math>\pi(s^{min}_{-i}(s_i^*), s_i^*) /geq \pi(s^{min}_{-i}(s_i^j), s_i^j) \forall s_i^j \neq s_i^*</math>

• Section 7.3.1, Page 302: an English auction does not reveal the winner's true valuation, since this valuation can be higher than the winning bid
• Section 7.3.2, Page 303: In the discussion on independence of irrelevant alternatives, the text should read "...the other alternatives could be arbitrarily ranked, but as long as a majority of individuals rank a higher than b, so too should the social-welfare function."
• Section 7.3.3, Page 306: The utility of a bid is given by v-p, not v-b
• Section 7.3.7, Page 310, Eq. 7.6: there are extra parentheses that can be removed
• Solution to Exercise 5, Page 445: this solution is incorrect. Instead, there should be 8 rows, corresponding to the pure strategies (Y,Y,Y), (Y,Y,N), (Y,N,Y), ... (N,N,N). For each cell, however, the payoffs are as shown in the table on page 445. For example, for the row (Y,Y,Y), for column L,the payoffs are (1, a-1), for column M, the payoffs are (2, a-2), and for column H, the payoffs are (3, a-3). This reflects the idea that the row player can choose one of their 8 strategies at the start of the game and have it played for them no matter what the column player does.
• Solution to Exercise 12, Page 446: should read "Any mixed strategy of R2 and R3 that plays R2 with a probability ..." instead of plays R3.
• Exercise 18, Page 318 has some extra '*' symbols that should be ignored. The solution on page 447 also has an error. v₂(x^-1) + v₃(x^-1) -(v₂(x^*) +v₃(x^-*)) = 120(1-e^-0.5*2.197) - 40*2.197 -( 120(1-e^-0.5*2.5055) - 40*2.5055) = -17.838. Similarly, the corrected values of p₂ and p₃ are -23.559 and -34.977.

Chapter 8: Control theory

• Section 8.2.2, Page 325. The sentence "We first consider a single-input single-output system, with scalar input u and a scalar output y." should read "We first... and a scalar output y."
• Section 8.2.2, Page 326. The sentence beginning "In this case..." should read "In this case, the input u is replaced by the vector u, and the output y by the vector y.
• Section 8.2.2, Page 327. In Equation 8.4, the products are Hadamard products, where the corresponding rows of each vector are multiplied.
• Section 8.2.2, Page 328. Just above Example 8.4, the equation A t^r/r! e^{alpha t} should be replaced by A t^{r-1}/(r-1)! e^{alpha t}.
• Example 8.4: Page 328, here, since we're working with a causal system, we need to take the unilateral Laplace transform. Note that <math> y(t) = x(t), so \dot{y} = \dot{x} = v </math>. The unilateral Laplace transform of <math> \dot{y} = sY(s) - y(0).</math>
• Section 8.3, Page 329, the last paragraph of this section should refer to Section 8.9, not 8.8.
• Section 8.4.2, Page 332, the third term in the second line of equations should read \frac{K\omega_n^2_n}{s(s^2 + 2\zeta \omega_n s + omega^2_n}
• Section 8.4.2, Page 332, the alternative form of \zeta is used later in this example, in Eq. 8.11, and in Figures 8.5 and 8.6.
• Section 8.4.3, Page 334, replace all \omega by \omega_n
• Example 8.9, Page 342, the equation should read Y = R/1+s\tau + W\tau/1+s\tau
• Section 8.6.2, Page 344: the sentence just about 8.6.3 should read "Note that the larger the loop gain, the faster the oscillation frequency."
• Section 8.8.1, Page 355, The last bulleted item should end with the sentence "Roots on the j axis lead to marginally stable systems, and repeated roots on the j axis or roots in the right half of the complex s plane lead to unstable systems."
• Section 8.8.1, Page 356, Example 8.18. The roots are at <math>\pm j \sqrt{K_i}</math>

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My thanks to the following readers for taking the time to point out these errors:

• Martin Jacobsson, Communication Research (CoRe), Dept. of IT, Uppsala University, Sweden
• Jinwen Lu, Yerbol Aussat, Qiushi Jiang, Anthony Anthony University of Waterloo
• D.A. Boyle and V. Agarwal at the University of Cambridge.