Motivation

The random walk Metropolis algorithm is one of the most widely used Markov chain Monte Carlo methods for sampling from a density known up to a constant. Its one tuning parameter is the variance of the Gaussian proposal. If the variance is too small, the chain accepts almost every move but barely moves. If it is too large, it proposes long jumps that are almost always rejected. Practitioners need a rule for choosing it, and the rule has to work in high dimension, where both failure modes are severe.

Roberts, Gelman and Gilks (Ann. Appl. Probab. 7(1), 1997) gave the first rigorous answer for product targets. As the dimension grows, one coordinate of the suitably speeded-up chain converges to a Langevin diffusion. The speed of that diffusion is an explicit function of the proposal scale, and maximising it gives the rule "tune the proposal so that about 23% of proposals are accepted". This rule, and the 2.38/√I scaling behind it, is now standard advice in applied Bayesian statistics.

Setting

Let $f:\mathbb R\to\mathbb R$ be a target density: positive, $C^2$, integrating to one, with $f'/f$ Lipschitz, and satisfying the moment conditions (A1) $\mathbb E_f[(f'/f)^8]<\infty$ and (A2) $\mathbb E_f[(f''/f)^4]<\infty$. Here $\mathbb E_f[g(X)]=\int g(x)f(x)\,dx$. In dimension $n\ge2$ the target is the product density $\pi_n(x)=\prod_{i=1}^n f(x_i)$ on $\mathbb R^n$.

Fix a scale $l>0$ and set $\sigma_n^2=l^2/(n-1)$. The random walk Metropolis chain $X^n=(X^n_0,X^n_1,\dots)$ moves as follows. From $X^n_{m-1}$ it proposes $Y\sim N(X^n_{m-1},\sigma_n^2I_n)$. It sets $X^n_m=Y$ with probability $\alpha(X^n_{m-1},Y)=1\wedge\pi_n(Y)/\pi_n(X^n_{m-1})$, and $X^n_m=X^n_{m-1}$ otherwise. The chain starts from $\pi_n$, which is stationary for it. The speeded-up first coordinate is $U^n_t=X^n_{\lfloor nt\rfloor,1}$ for $t\ge0$.

Let $\Phi$ be the standard normal distribution function, and define the roughness $I=\mathbb E_f[(f'(X)/f(X))^2]$. The speed and the limiting acceptance rate are

$$h(l)=2l^2\,\Phi\!\Big(-\frac{l\sqrt I}{2}\Big),\qquad a(l)=2\,\Phi\!\Big(-\frac{l\sqrt I}{2}\Big).$$

The Langevin generator is $GV(x)=h(l)\big[\tfrac12V''(x)+\tfrac12(\log f)'(x)V'(x)\big]$. It generates the Langevin diffusion $dU_t=h(l)^{1/2}dB_t+h(l)\frac{f'(U_t)}{2f(U_t)}dt$.

Formalization targets

Goal: Theorem 1.1

As $n\to\infty$,

$$U^n\Rightarrow U,$$

where $\Rightarrow$ denotes weak convergence in the Skorokhod topology, $U_0$ has density $f$, and $U$ is the Langevin diffusion with speed $h(l)$. The limit is asserted to exist. No constants appear in the statement beyond those the model defines.

Milestones: the proof

1. Lemma 2.1. The stationary chain stays in the sets $F_n=\{|R_n-I|<n^{-1/8}\}\cap\{|S_n-I|<n^{-1/8}\}$ up to time $t$ with probability tending to one. Here $R_n$ and $S_n$ are the empirical averages of $((\log f)')^2$ and $-(\log f)''$ over coordinates $2,\dots,n$.
2. Proposition 2.2. $|1\wedge e^x-1\wedge e^y|\le|x-y|$.
3. Lemma 2.3. $\sup_{x\in F_n}\mathbb E|W_n|\to0$, where $W_n$ is the second-order part of the log acceptance ratio.
4. Proposition 2.4. $\mathbb E[1\wedge e^A]=\Phi(\mu/\sigma)+e^{\mu+\sigma^2/2}\Phi(-\sigma-\mu/\sigma)$ for $A\sim N(\mu,\sigma^2)$.
5. Lemma 2.5. $\limsup_n\sup_{x_1}n|\mathbb E[V(Y_1)-V(x_1)]|<\infty$ for $V\in C_c^\infty$.
6. Lemma 2.6. The discrete generator $G_nV(x)=n\,\mathbb E[(V(Y)-V(x))\alpha(x,Y)]$ converges to $GV$ uniformly on $F_n$, for $V\in C_c^\infty$ a function of the first coordinate (stated with bounded $(\log f)'''$, the assumption its proof uses).

Milestones: the optimal-scaling corollary

7. Corollary 1.2 (i). $a_n(l)=\iint\pi_n(x)\alpha(x,y)q_n(x,y)\,dx\,dy\to a(l)$.
8. Corollary 1.2 (ii). $h$ is maximised at $\hat l=2.38/\sqrt I$, with $a(\hat l)=0.23$ and $h(\hat l)=1.3/I$, to the printed precision.

Significance

Theorem 1.1 shows that, run for $n$ times as many steps, the chain in dimension $n$ looks like a fixed one-dimensional diffusion. The algorithm's cost therefore grows linearly in dimension, and its efficiency is measured by the single number $h(l)$. Corollary 1.2 turns this into the 0.234 acceptance-rate heuristic and the $2.38/\sqrt I$ scaling. The same diffusion-limit method has since been applied to the Metropolis-adjusted Langevin algorithm, to Hamiltonian Monte Carlo and to non-product targets.

The theorem is proved on paper; it has no machine-checked proof. A formal development would give the first verified diffusion limit of an MCMC algorithm. It would also yield reusable components: the Metropolis chain on $\mathbb R^n$ as a measurable random mapping, a martingale-problem characterisation of one-dimensional diffusions, and a Gaussian computation (Proposition 2.4) that recurs throughout the optimal-scaling literature.

Difficulty

The obvious approach, a Taylor expansion of the log acceptance ratio, gives a sum of $n-1$ terms of size $1/n$. That sum does not concentrate uniformly over the state space, since the coordinates $2,\dots,n$ are arbitrary. The expansion is controlled only on the sets $F_n$, where the empirical averages $R_n$ and $S_n$ are close to $I$. The limit therefore holds only after showing that the chain rarely leaves $F_n$ over a time horizon of $nt$ steps. A pointwise law of large numbers is not enough for that, because the bound has to survive a union over $nt$ steps. Passing from generator convergence on a set of high probability to weak convergence of processes requires the Ethier–Kurtz convergence theory: a core for the limit generator, and convergence of processes that are not themselves Markov. None of this theory is in Mathlib.

Formalization scope

All declarations live in the namespace Roberts1997.RWM. The following conventions are fixed.

• Vectors are Fin n → ℝ, and the paper's first coordinate $x_1$ is index 0. Its coordinates $2,\dots,n$ are the indices i ≠ 0.
• $\sigma_n^2=l^2/(n-1)$ is computed in $\mathbb R$. All statements concern $n\ge2$ or large $n$.
• $l>0$ is assumed. The paper leaves it implicit, but $h(-l)\ne h(l)$.
• "$f$ is a density" is read as $\int f=1$. The moment conditions are read as integrability of $(f'/f)^8f$ and $(f''/f)^4f$. The standing assumption "$f'/f$ is Lipschitz" (p. 111) is carried by every statement.
• The chain is built as a random mapping on an explicit probability space: $x_0\sim\pi_n$, with i.i.d. standard normal innovations and uniform acceptance variables. Theorem 1.1's initial condition (components i.i.d. $f$, shared across dimensions) is read as "the $n$-th chain starts from $\pi_n$", since weak convergence depends only on the law of each $U^n$.
• "$U$ satisfies the Langevin SDE" is read as "the law of $U$ solves the martingale problem for $G$ on $C_c^\infty$, with continuous paths and initial law $f(x)\,dx$". This is equivalent by Ethier–Kurtz (1986), Ch. 5, Prop. 3.1 and Thm 3.3, and follows the platform definition EthierKurtz_IsContinuousDiffusionLaw.
• "$U^n\Rightarrow U$" is read as the existence of an almost-sure coupling in which càdlàg copies of the $U^n$ converge to a continuous Langevin path uniformly on compact time intervals. For a continuous limit this is equivalent to weak convergence in $D_{\mathbb R}[0,\infty)$, by Skorokhod's representation theorem and Ethier–Kurtz Ch. 3, Thm 1.8, Prop. 5.3 and Prop. 7.1. It follows the platform encoding of Ethier–Kurtz Theorem 7.4.1.
• "$\sup\to0$" and "$\limsup\sup<\infty$" are stated as eventual uniform bounds. This avoids real suprema, whose value on an unbounded set is a default.
• In Lemma 2.6, "as $d\to\infty$" is a misprint for $n\to\infty$, and "$2f(Ut)$" in (1.2) is read as $2f(U_t)$.
• Corollary 1.2 (ii) is stated for an arbitrary constant $I>0$. "To two decimal places" is read as explicit rounding intervals: $1.3$ is read to one decimal, and all maximisers over $l>0$ are covered.

The goal cannot be satisfied trivially. The limit law $Q$ must exist, and it must be a probability measure whose initial marginal is $f(x)\,dx$, so the zero measure is excluded. The coupled copies must carry exactly the laws of the paths $U^n$, not an arbitrary process with the same one-time marginals.

The statements carry the paper's hypotheses, with one exception. The printed proof of Lemma 2.6 bounds $\sup_z|(\log f)'''(z)|$, which Theorem 1.1 does not assume, and under $C^2$ alone the uniform convergence over $F_n$ claimed by Lemma 2.6 fails (narrow spikes of $(\log f)''$ far out let a positive fraction of the coordinates shift the log acceptance ratio by a constant while $R_n$ and $S_n$ stay close to $I$). Lemma 2.6 is therefore stated with the proof's own assumption, $f\in C^3$ with $(\log f)'''$ bounded, named as an addition. Theorem 1.1 and the other results keep the paper's hypotheses.

A complete development needs several pieces not yet available: path spaces and the Skorokhod topology (or the coupling reading), the martingale problem and its well-posedness for Lipschitz drift, and the Ethier–Kurtz theorem on convergence of generators on sets of high probability. Proofs of the Gaussian milestones (Propositions 2.2 and 2.4, Lemma 2.5) and of Corollary 1.2 (ii) are independent of this infrastructure and are welcome contributions.

Selected references

• G. O. Roberts, A. Gelman, W. R. Gilks, Weak convergence and optimal scaling of random walk Metropolis algorithms, Ann. Appl. Probab. 7(1), 110–120, 1997. https://doi.org/10.1214/aoap/1034625254
• S. N. Ethier, T. G. Kurtz, Markov Processes: Characterization and Convergence, Wiley, 1986. https://doi.org/10.1002/9780470316658
• A. Gelman, G. O. Roberts, W. R. Gilks, Efficient Metropolis jumping rules, Bayesian Statistics 5, Oxford University Press, 599–607, 1996.
• G. O. Roberts, J. S. Rosenthal, Optimal scaling for various Metropolis–Hastings algorithms, Statistical Science 16(4), 351–367, 2001. https://doi.org/10.1214/ss/1015346320

Motivation

A single server shared by several classes of jobs must decide, at every moment, which class to serve. Different scheduling rules give different mean response times to the classes, and a system designer often starts from the other end: a target vector of mean response times, one per class, and the question whether any rule can meet it. Coffman and Mitrani answered this question for the multiclass M/M/1 queue in A Characterization of Waiting Time Performance Realizable by Single-Server Queues (Operations Research 28 (1980), 810–821). Their answer is a polytope with an explicit description: the response-time vectors that can be realized are exactly the convex combinations of the vectors of the preemptive priority rules, and these are exactly the vectors satisfying one equation and $2^M-2$ inequalities.

The starting point is Kleinrock's conservation law (Kleinrock, Naval Res. Logist. Quart. 12 (1965)): a weighted sum of the response times does not depend on the rule. The characterization is the first instance of what was later called the achievable region method, developed for general multiclass systems by Federgruen and Groenevelt (Oper. Res. 36 (1988)), Shanthikumar and Yao (Oper. Res. 40 (1992)) and Bertsimas and Niño-Mora (Math. Oper. Res. 21 (1996)), and used to derive priority-index policies such as the $c\mu$ rule and Gittins indices.

Setting

There are $M\ge1$ job classes. Jobs of class $i$ arrive in a Poisson stream at rate $\lambda_i>0$ and have exponential service times with parameter $\mu_i>0$. The traffic intensity of class $i$ is $\rho_i=\lambda_i/\mu_i$, and the system is stable: $\rho=\rho_1+\cdots+\rho_M<1$. A performance vector $W=(W_1,\dots,W_M)$ lists the mean response times of the classes. Write $a_i=\rho_i/\mu_i$, $V=\sum_i\lambda_i/\mu_i^2$, and for a set $g$ of classes

$$f(g)=\frac{\sum_{i\in g}a_i}{1-\sum_{i\in g}\rho_i},\qquad f(\emptyset)=0 .$$

• The conservation law (1): $\sum_{i=1}^M\rho_iW_i=V/(1-\rho)$, which equals $f(\{1,\dots,M\})$.
• The inequalities (4): $\sum_{i\in g}\rho_iW_i\ge f(g)$ for each proper nonempty set $g$ of classes.
• $H^{**}$ is the set of $W$ satisfying (1) and (4).
• A priority order lists the classes as $i_1,\dots,i_M$, $i_1$ highest. The preemptive priority vector $P(i_1,\dots,i_M)$ is the vector with $\sum_{i\in S_k}\rho_iW_i=f(S_k)$ for the top sets $S_k=\{i_1,\dots,i_k\}$, $k=1,\dots,M$; explicitly $P_{i_k}=(f(S_k)-f(S_{k-1}))/\rho_{i_k}$. For $M=2$, $P(1,2)_1=1/(\mu_1-\lambda_1)$, the M/M/1 response time of class 1 alone.
• $H$, (3), is the set of convex combinations $\sum_{k=1}^M\alpha_kP_k$ of $M$ preemptive priority vectors.

In Lean the data are a structure Params M carrying $\lambda,\mu$ and the three standing assumptions; Params.f, Params.Hss ($H^{**}$), Params.prioVec, topSet and Params.H are the objects above.

Formalization targets

Goal: Theorem 2, analytical form

$$H^{**}=H .$$

The paper's Theorem 2 says a vector is achievable by a scheduling strategy iff it lies in $H$; its proof is the chain $H\subseteq H^*\subseteq H^{**}\subseteq H$, where $H^*$ is the achievable set. The goal is the part of the chain that involves no strategies.

Milestones

1. The priority vector is the unique solution of the equations (5) for its chain of top sets.
2. The first inequality of the proof of Lemma 2: $(1-\rho(g_1))(1-\rho(g_2))>(1-\rho(g_1\cup g_2))(1-\rho(g_1\cap g_2))$ for crossing $g_1,g_2$.
3. The second inequality of that proof, in the coefficients $a_i$.
4. Lemma 1 at the priority vectors: every $P(i_1,\dots,i_M)$ lies in $H^{**}$.
5. Two sets on which a point of $H^{**}$ satisfies (4) with equality are nested.
6. Lemma 2: every vertex of $H^{**}$ is a preemptive priority vector.

A further item states the paper's final remark (§4): every linear cost $\sum_ic_iW_i$ is minimized over $H^{**}$ at some preemptive priority vector.

Significance

The theorem turns a question about all scheduling rules into a finite check: a target vector is realizable iff it satisfies (1) and the inequalities (4), and every realizable vector is realized by randomly mixing at most $M$ priority rules. Linear costs over the realizable vectors are minimized by a priority rule, the fact behind the optimality of priority-index rules in multiclass queues. The paper also gives a linear program for finding the mixture.

The result has been proved since 1980. No machine-checked proof of it is known. The Prove2Me library holds the abstract generalized conservation law theorem of Gittins, Glazebrook and Weber (AllocationIndices.achievable_region_theorem, included as a reference item), which assumes the inequalities (4) for every policy and whose polytope also imposes nonnegativity; it does not compute the right-hand sides for the M/M/1 queue, does not prove that the priority vectors satisfy (4), and uses equality on the lowest-priority sets rather than the highest. This mission supplies the concrete polytope, the closed form of the priority vectors and the strict supermodularity of $f$.

Difficulty

That the priority vectors lie in $H^{**}$ is a family of inequalities between ratios $f(S_k)$, one for each pair of a priority order and a set $g$, and the order and $g$ need not interact in any simple way. The reverse inclusion is a statement about vertices: a vertex is determined by $M$ tight constraints, and one has to show that they form a chain. This needs strict inequalities with the right direction for every crossing pair of sets, which is where the positivity of every $\lambda_i,\mu_i$ is used. If some $\lambda_i=0$, then $W_i$ appears in no constraint, $H^{**}$ is unbounded and the goal is false. Finally, $H$ uses only $M$ points, not all $M!$, so the goal contains a Carathéodory-type bound for the hyperplane of (1).

Formalization scope

Classes are Fin M, numbered from $0$. A priority order is π : Equiv.Perm (Fin M) with π r the class of rank r, rank $0$ highest. "Vertex" is an element of Set.extremePoints ℝ. The points of (3) are prioVec (σ k) for an arbitrary σ : Fin M → Equiv.Perm (Fin M), so repetitions are allowed. The priority vectors are given by their closed form, not as solutions of a system. The goal assumes $M\ge1$; for $M=0$ the set $H$ is empty.

The paper's notion "achievable by some scheduling strategy" is replaced by its analytical characterization $H^{**}$: the strategy class of the paper (Assumptions 1–3, p. 812) is described only in prose and the steady-state means are assumed to exist, so the queueing half of the proof (Theorem 1, Lemma 1 for arbitrary strategies, the conservation law itself) is not stated. The goal is not to be stated on an abstract set satisfying hypotheses that encode Lemma 1 and (1); that form is already proved and drops the content of milestone 4. The conservation law is an equality, never the inequality (4) at the full set.

A complete development needs finite-set sums, the extreme points of a polyhedron and a Carathéodory argument in an affine hyperplane; the inequalities of milestones 2 and 3 and the vertex-chain argument are reusable for any strictly supermodular set function. Proofs of any milestone, and alternative proofs of the goal through polymatroid theory, are welcome.

Selected references

• E. G. Coffman, Jr. and I. Mitrani, A Characterization of Waiting Time Performance Realizable by Single-Server Queues, Operations Research 28(3, Part II), 810–821, 1980. https://doi.org/10.1287/opre.28.3.810
• L. Kleinrock, A Conservation Law for a Wide Class of Queueing Disciplines, Naval Research Logistics Quarterly 12, 181–192, 1965. https://doi.org/10.1002/nav.3800120206
• A. Federgruen and H. Groenevelt, Characterization and Optimization of Achievable Performance in General Queueing Systems, Operations Research 36(5), 733–741, 1988. https://doi.org/10.1287/opre.36.5.733
• J. G. Shanthikumar and D. D. Yao, Multiclass Queueing Systems: Polymatroidal Structure and Optimal Scheduling Control, Operations Research 40(3-supplement-2), S293–S299, 1992. https://doi.org/10.1287/opre.40.3.S293
• D. Bertsimas and J. Niño-Mora, Conservation Laws, Extended Polymatroids and Multiarmed Bandit Problems; A Polyhedral Approach to Indexable Systems, Mathematics of Operations Research 21(2), 257–306, 1996. https://doi.org/10.1287/moor.21.2.257
• J. Gittins, K. Glazebrook and R. Weber, Multi-armed Bandit Allocation Indices, 2nd ed., Wiley, 2011. https://doi.org/10.1002/9780470980033

Motivation

Average cost control of queues rests on a small amount of Markov chain theory: when does a chain with costs have a well defined long-run average cost, and how can that be checked for a concrete model with an unbounded state space? Appendix C of L. I. Sennott, Stochastic Dynamic Programming and the Control of Queueing Systems (Wiley, 1999, doi:10.1002/9780470317037) collects this material for countable state spaces and packages it in one hypothesis, the $z$ standard chain. Chapters 7–10 of the book verify this hypothesis for the Markov chains induced by stationary policies in admission, routing and service-rate control models, and use its consequences to prove existence of average cost optimal policies.

The tools are Lyapunov functions in the sense of Foster (1953): a nonnegative function on the states whose expected one-step change is negative away from a finite set. Foster's criterion for positive recurrence, and its refinements bounding expected first passage times and costs, are the standard way to verify stability of queueing networks (Meyn and Tweedie, Markov Chains and Stochastic Stability, 1993/2009).

Setting

A Markov chain $\Gamma$ on a countable set $S$ is given by transition probabilities $P_{ij}\ge 0$ with $\sum_j P_{ij}=1$. $X_t$ is the state at time $t$ and $P^{(t)}_{ij}$ the $t$-step transition probability ($P^{(0)}_{ij}=\delta_{ij}$). State $i$ leads to $j$ if $P^{(t)}_{ij}>0$ for some $t\ge0$; states that lead to each other communicate, which partitions $S$ into communicating classes.

For a nonempty $G\subseteq S$ the first passage time from $i$ is $T_{iG}=\min\{t\ge1: X_t\in G\}$ given $X_0=i$, and $m_{iG}=E[T_{iG}]\in[0,\infty]$; $m_{ij}$ is the case $G=\{j\}$ and $m_{ii}$ the expected return time. The taboo probability $_G P^{(t)}_{ik}$ is the probability of going from $i$ to $k$ in $t$ steps without visiting $G$ at the intermediate times, and $_G u_{ik}$ is the expected number of visits to $k$ at times $0\le t<T_{iG}$. A state is transient if $P(T_{ii}<\infty)<1$ and positive recurrent if $m_{ii}<\infty$; a positive recurrent class is a communicating class of positive recurrent states. The steady state probability is $\pi_j=(m_{jj})^{-1}$ (zero when $m_{jj}=\infty$).

Each state carries a finite cost $C(i)\ge0$. The expected average cost over $[0,n-1]$ from $i$ is

$$J^{(n)}_i=\frac1n\,E\Big[\sum_{t=0}^{n-1}C(X_t)\,\Big|\,X_0=i\Big]=\frac1n\sum_{t=0}^{n-1}\sum_j P^{(t)}_{ij}C(j),$$

$c_{iG}$ is the expected cost $E[\sum_{t=0}^{T_{iG}-1}C(X_t)\mid X_0=i]$ of a first passage (defined when $m_{iG}<\infty$), and $J_R=\sum_{j\in R}\pi_jC(j)$ is the average cost on a positive recurrent class $R$. The chain is $z$ standard (Definition C.2.5) if for a distinguished state $z$

$$m_{iz}<\infty\quad\text{and}\quad c_{iz}<\infty\qquad\text{for all } i\in S.$$

Formalization targets

Goal: Proposition C.2.6

If $\Gamma$ is $z$ standard, then $S$ is the union of a positive recurrent class $R\ni z$ and a set of transient states, $J_R<\infty$, and

$$\lim_{n\to\infty}J^{(n)}_i=J_R\qquad\text{for every } i\in S.$$

The statement fixes no constants: it asserts that the average cost exists, is finite, and does not depend on the initial state.

Milestones

1. Proposition C.1.2: $\pi$ is the unique stationary distribution of a positive recurrent class, and $\pi_j=e_{ij}/m_{ii}=\pi_ie_{ij}$.
2. Proposition C.1.4: the first-step equations (C.2)–(C.4) for taboo probabilities, visit counts and $m_{iG}$; $\sum_{i\in G}\pi_im_{iG}=1$ for $G$ inside a positive recurrent class; $m_{ij}<\infty$ within such a class.
3. Proposition C.1.5: if $\sum_jP_{ij}[y(j)-y(i)]\le-\epsilon$ off $G$, then $m_{iG}\le y(i)/\epsilon$.
4. Corollary C.1.6: the same with $G=\{z\}$ and $\sum_jP_{zj}y(j)<\infty$ makes $z$ positive recurrent.
5. Proposition C.2.1: on a positive recurrent class, $J^{(n)}_i\to J_R=c_{ii}/m_{ii}$.
6. Proposition C.2.2: $c_{iG}=\sum_kC(k)\,{}_Gu_{ik}$, the first-step equation (C.13), and $J_R=\sum_{i\in G}\pi_ic_{iG}$.
7. Proposition C.2.3 and Corollary C.2.4: the cost drift condition $\sum_jP_{ij}[r(j)-r(i)]\le-C(i)$ off a finite set bounds $c_{iG}\le r(i)+Fm_{iG}$, and gives $c_{zz}<\infty$.
8. Remark C.2.7: the hypotheses of C.1.6 and C.2.4 together imply the chain is $z$ standard; so do irreducibility, positive recurrence and finite average cost.

Significance

Proposition C.2.6 is what makes the $z$ standard hypothesis useful: an average cost criterion that is a genuine limit, finite, and independent of the initial state, even for chains with transient states and unbounded state spaces. Every average cost optimality result of the book that works with a stationary policy's induced chain (the (SEN) and (BOR) assumption sets, the approximating-sequence method, the continuous-time chapter) calls on this proposition or on the Lyapunov criteria of Remark C.2.7 to establish its hypotheses for queueing models.

All results of the mission are classical and proved in the literature; parts are stated in the book without proof and referred to Chung (1967), Grassmann et al. (1985) and renewal theory. None of them has been machine-checked in this form as far as the platform and Mathlib show: Mathlib has kernels and Ionescu-Tulcea trajectories but no countable-state Markov chain classification, no first passage calculus, and no Foster–Lyapunov criterion. Existing platform results on countable chains (the Levin–Peres–Wilmer series) treat irreducible chains without costs. A complete development here produces a reusable library of first passage identities, Foster–Lyapunov bounds for times and costs, and average cost limits on reducible chains.

Difficulty

The Lyapunov bounds (C.1.5, C.2.3) are telescoping arguments, but they require a clean handling of truncated passages and of sums that may be infinite: (C.7) is an inequality between possibly divergent series, and the step "iterate $n$ times and let $n\to\infty$" must be made rigorous for $[0,\infty]$-valued expectations.

The central difficulty is part (iii) of the goal for transient initial states. On the class $R$, the limit of $J^{(n)}_i$ is a renewal reward theorem over successive returns to $z$; from a transient state the first cycle has a different law, so a delayed renewal reward argument is needed, and it has to cover the case where costs are unbounded. The obvious approach, bounding $J^{(n)}_i$ between $J_R$ and the average over the first $n$ steps of the chain started in $z$, fails because $P^{(t)}_{ij}$ need not converge (periodic classes) and because finite $c_{iz}$ does not bound individual cost terms. Proposition C.1.2's uniqueness and the Kac-type identity of C.1.4(iv) likewise need the full cycle decomposition of a positive recurrent class.

Formalization scope

The chain is a structure MC S with P : S → S → ℝ≥0∞ and ∑' j, P i j = 1, over a countable type S; costs are C : S → ℝ≥0. Probabilities and expectations are ℝ≥0∞-valued sums over finite paths Fin (t+1) → S, so every quantity is defined without summability side conditions and may be $\infty$. The first passage time is $T_{iG}\ge1$; $m_{iG}$ is the expectation of $T_{iG}$ from its law (and $\infty$ when $P(T_{iG}<\infty)<1$), not defined by the recursion (C.4), so that (C.4) is a theorem. $_Gu_{ik}$ counts visits at times $0\le t<T_{iG}$. $c_{iG}$ is computed over first passage paths and is used only when $m_{iG}<\infty$, as in the book. $\pi_j$ is $(m_{jj})^{-1}$, which the book states equals the Cesàro limit $\lim_nQ^{(n)}_{jj}$. $J^{(n)}_i$ is meaningful for $n\ge1$, and limits are taken in $[0,\infty]$. The drift conditions $\sum_jP_{ij}[y(j)-y(i)]\le-\epsilon$ and $\sum_jP_{ij}[r(j)-r(i)]\le-C(i)$ are written in the equivalent additive form $\sum_jP_{ij}y(j)+\epsilon\le y(i)$, which is equivalent for finite $y$ and makes the case $\sum_jP_{ij}y(j)=\infty$ fail, as it does in the book.

A trivializing formalization, such as defining $m_{iG}$ or $c_{iG}$ by the equations (C.4) or (C.13), defining $J_R$ as the limit of $J^{(n)}_i$, or allowing a $z$ standard chain whose return time or return cost to $z$ is infinite, is ruled out: $z$ standard requires $m_{iz}<\infty$ and $c_{iz}<\infty$ for every $i$ including $z$, and each quantity is defined from path probabilities.

Needed infrastructure: path-sum manipulation in $[0,\infty]$ (first-step and last-step decompositions), the ratio limit / renewal reward theorem for a positive recurrent class, and the delayed version for transient starts. The first passage calculus and the Lyapunov bounds are reusable by the book's other chapters on average cost, which state the $z$ standard property for policy-induced chains. Contributions of lemmas on path sums and of an independent renewal reward library are welcome.

Selected references

• L. I. Sennott, Stochastic Dynamic Programming and the Control of Queueing Systems, Wiley, 1999, Appendix C, pp. 292–302. doi:10.1002/9780470317037
• K. L. Chung, Markov Chains with Stationary Transition Probabilities, 2nd ed., Springer, 1967. doi:10.1007/978-3-642-62015-7
• F. G. Foster, On the stochastic matrices associated with certain queuing processes, Annals of Mathematical Statistics 24 (1953), 355–360. doi:10.1214/aoms/1177728976
• S. P. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, 2nd ed., Cambridge University Press, 2009. doi:10.1017/CBO9780511626630
• D. P. Heyman and M. J. Sobel, Stochastic Models in Operations Research, Vol. I, McGraw-Hill, 1982.

Motivation

Many queueing systems evolve in continuous time: customers arrive according to a Poisson process, services take exponentially distributed times, and a controller may change the service rate, admit or reject customers, or route them whenever the state changes. Minimizing the long-run average cost of such a system is a standard problem in the control of queues (Lippman 1975; Puterman 1994, Ch. 11; Sennott 1999, Ch. 10). The continuous time model does not fit directly into the discrete time theory of Markov decision chains developed in the earlier chapters of Sennott's book, because time spent in a state now matters and the natural average cost is a ratio of expected cost to expected elapsed time.

This mission formalizes Sections 10.1–10.4 of L. I. Sennott, Stochastic Dynamic Programming and the Control of Queueing Systems (Wiley, 1999): the elementary properties of the exponential distribution, the continuous time Markov decision chain and its average cost, a reduction of the continuous time problem to an auxiliary discrete time Markov decision chain, and the theorem stating that finite state approximating sequences of the auxiliary chain compute optimal average costs and optimal stationary policies of the continuous time chain. The chapter closes with an explicit average cost computation for the M/M/1 queue with service rate control.

Setting

A random variable $X$ has the exponential distribution with rate $\mu>0$ if $P(X\le t)=1-e^{-\mu t}$ for $t\ge0$. A function $r(\delta)$ is $o(\delta)$ if $r(\delta)/\delta\to0$ as $\delta\to0^+$.

A continuous time Markov decision chain (CTMDC) $\Psi$ has a countable state space $S$ and, for each $i\in S$, a finite nonempty action set $A_i$. Choosing $a\in A_i$ in state $i$ incurs an instantaneous cost $G(i,a)\ge0$ and a cost rate $g(i,a)\ge0$ in effect until the next transition. The time until the next transition is exponential with rate $\nu(i,a)>0$, so its mean is $\tau(i,a)=1/\nu(i,a)$; the next state is $j$ with probability $P_{ij}(a)$, where $P_{ii}(a)=0$. A policy $\theta$ chooses, at each transition, an action (possibly at random) from the history of past states, actions and sojourn times; a stationary policy $e$ chooses $e(i)$ in state $i$. With $C_n$ the cost and $T_n$ the time of the first $n$ transition periods, the average cost and the minimum average cost are

$$J^\Psi_\theta(i)=\limsup_{n\to\infty}\frac{E_\theta[C_n\mid X_0=i]}{E_\theta[T_n\mid X_0=i]},\qquad J^\Psi(i)=\inf_\theta J^\Psi_\theta(i).$$

Assumption (CTB) requires constants $\tau$ and $B$ with $0<\tau<\inf_{i,a}\tau(i,a)\le\sup_{i,a}\tau(i,a)\le B<\infty$. The auxiliary MDC $\Delta$ has the same states and actions, costs $C(i,a)=G(i,a)\nu(i,a)+g(i,a)$, and transition probabilities $P^*_{ij}(a)=\tau\nu(i,a)P_{ij}(a)$ for $j\ne i$, $P^*_{ii}(a)=1-\tau\nu(i,a)$. Its average cost $J^\Delta_\theta(i)=\limsup_n n^{-1}\sum_{t<n}E_\theta[C(X_t,Y_t)]$ and minimum average cost $J^\Delta(i)$ are those of Chapter 2. Assumption (CTAC) is $J^\Delta(\cdot)\le J^\Psi(\cdot)$.

An approximating sequence $(\Delta_N)_{N\ge N_0}$ for $\Delta$ uses finite state spaces $S_N$ increasing to $S$ and transition probabilities $P^*_{ij}(a;N)$ on $S_N$ converging to $P^*_{ij}(a)$. The (AC) assumptions ask for constants $J^N$ and functions $r^N$ on $S_N$ solving

$$J^N+r^N(i)=\min_{a\in A_i}\Big\{C(i,a)+\sum_{j\in S_N}P^*_{ij}(a;N)\,r^N(j)\Big\},\qquad i\in S_N,\ N\ge N_0,\tag{10.21}$$

with $\limsup_N r^N(i)<\infty$, $\liminf_N r^N(i)\ge-Q$ for a constant $Q\ge0$, and $\limsup_N J^N=:J^*<\infty$, $J^*\le J^\Delta(i)$.

Formalization targets

Goal: Theorem 10.3.3

Under (CTB), (CTAC) and the (AC) assumptions for an approximating sequence of $\Delta$:

$$J^*=\lim_{N\to\infty}J^N\ \text{exists and}\ J^\Delta(i)=J^\Psi(i)=J^*\quad(i\in S),$$

and every limit point $e^*$ of a sequence $e^N$ of stationary policies realizing the minimum in (10.21) satisfies $J^\Delta_{e^*}=J^\Delta$ and $J^\Psi_{e^*}=J^\Psi$. The goal leaves the chain, the approximating sequence and the constants of (CTB) arbitrary.

Milestones

• Proposition 10.1.2: $P(X>x+y\mid X>y)=P(X>x)$ for $x,y>0$, and $P(X\le\delta)=\mu\delta+o(\delta)$.
• Proposition 10.1.3: for independent exponentials, $P(X_1\le\delta,X_2\le\delta)=o(\delta)$, $P(X_1<X_2)=\mu_1/(\mu_1+\mu_2)$, and $\min(X_1,X_2)$ is exponential with rate $\mu_1+\mu_2$.
• Lemma 10.3.1: if $z$ is bounded below and $Z\tau(i,e)+z(i)\ge G(i,e)+g(i,e)\tau(i,e)+\sum_jP_{ij}(e)z(j)$ for all $i$ (10.15), then $J^\Psi_e\le Z$.
• Lemma 10.3.2: $(Z,w)$ satisfies $Z+w(i)\ge C(i,e)+\sum_jP^*_{ij}(e)w(j)$ (10.20) if and only if $(Z,\tau w)$ satisfies (10.15).
• Proposition 10.4.1: in the M/M/1 queue with arrival rate $\lambda$, holding cost $H(i)=Hi$ and service cost rate $c(a)$, the policy that always serves at rate $a>\lambda$ has average cost $\rho_ac(a)+H\rho_a/(1-\rho_a)$, $\rho_a=\lambda/a$.

Significance

The goal theorem turns the average cost control of a continuous time chain on an infinite state space into a finite computation: solve the optimality equation (10.21) of a finite truncation of the auxiliary chain, let the truncation grow, and read off the optimal average cost and an optimal stationary policy of the original continuous time chain. The auxiliary chain is the book's form of uniformization, and the result is what licenses the numerical study of the M/M/1 service rate control problem in Section 10.4 and of the M/M/K and polling models in Sections 10.5–10.6. Proposition 10.4.1 gives the closed-form benchmark against which the computed optimal policy is compared.

The results are proved in the book, some with details left to the reader (Lemma 10.3.2(ii), Problem 10.10), and the goal rests on Theorem 8.1.1 and Lemma 7.2.1 of the same book. None of them has, as far as a search of Mathlib and the Prove2Me catalogue shows, a machine-checked proof: Mathlib provides the exponential law (ProbabilityTheory.expMeasure) and its distribution function, but not memorylessness or the minimum of independent exponentials, and no continuous time Markov decision model. A formalization would supply these, together with a checked average cost comparison between a continuous time chain and its discrete time auxiliary chain.

Difficulty

The obvious argument compares the two chains policy by policy, but the policy classes differ: a policy for $\Delta$ may change action in every time slot, including slots where the state does not change, while a policy for $\Psi$ acts only at transitions and may use the observed sojourn times. Only the stationary policies coincide. The lower bound $J^\Psi\ge J^*$ therefore cannot be obtained by transferring policies, and it is exactly what Assumption (CTAC) supplies. The upper bound requires passing from the discrete time inequality (10.20) for the limit point $e^*$ to a bound on a ratio of expected cost to expected time in continuous time, where the denominator depends on the policy; the uniform bounds of (CTB) on the mean sojourn times are what control it. Inside Lemma 10.3.1 the function $z$ is only bounded below, so the telescoping of expectations must be justified without integrability of $z$ from above.

Formalization scope

The state space is a countable type S, actions a type Act, and action sets A i : Finset Act; the CTMDC and MDC structures hold data, and their axioms (nonempty action sets, nonnegative costs, positive rates, stochastic transition rows with $P_{ii}(a)=0$) are separate predicates. Transition probabilities are ℝ≥0∞-valued; costs, rates and the functions $z,w,r^N$ are real. Expected costs, expected times and all average costs are ℝ≥0∞-valued, so $+\infty$ is a legitimate value, and they are compared with real constants in EReal; the limits superior and inferior of (AC) are taken in EReal. The expected cost of $n$ transition periods under a general policy is a recursion over the periods in which the sojourn time is integrated against expMeasure ν(i,a) and the next state is drawn independently from $P_{i\cdot}(a)$; policies are measurable in the past sojourn times. In (10.15) and (10.20) the convergence of the series is part of the inequality. The strict inequality $\tau<\inf\tau(i,a)$ of (CTB) is kept strict (as a positive margin); weakening it to $\le$ would make $P^*_{ii}(a)$ vanish or turn negative.

The average cost $J^\Psi_\theta$ is a ratio of expectations, not the expectation of a ratio, and the infimum $J^\Psi$ ranges over history dependent randomized policies that may use sojourn times; replacing either by a stationary-only class, or dropping (CTAC), gives a different theorem.

A complete development needs: expected rewards of a chain with exponential holding times, the average cost theory of Chapter 8 for the auxiliary chain (Theorem 8.1.1 and Lemma 7.2.1, restated here as needed), and renewal-reward reasoning for Proposition 10.4.1. The exponential-distribution lemmas are reusable beyond this mission and are welcome as independent contributions.

Selected references

• L. I. Sennott, Stochastic Dynamic Programming and the Control of Queueing Systems, Wiley Series in Probability and Statistics, John Wiley & Sons, 1999. https://doi.org/10.1002/9780470317037
• M. L. Puterman, Markov Decision Processes: Discrete Stochastic Dynamic Programming, John Wiley & Sons, 1994. https://doi.org/10.1002/9780470316887
• S. A. Lippman, Applying a new device in the optimization of exponential queuing systems, Operations Research 23(4), 687–710, 1975. https://doi.org/10.1287/opre.23.4.687
• D. Gross and C. M. Harris, Fundamentals of Queueing Theory, 3rd ed., John Wiley & Sons, 1998.

Motivation

Fictitious play is the oldest model of learning in games: each player repeatedly best-responds to the empirical frequencies of the opponents' past play. Stochastic fictitious play adds random payoff disturbances before each choice, so that players choose smoothed ("perturbed") best responses. It is a standard model in economics and in the study of learning in games (Fudenberg and Levine, 1998), and its long-run behavior is described through a deterministic mean dynamic, the perturbed best response dynamic, using stochastic approximation theory (Benaïm and Hirsch, 1999).

Supermodular games model strategic complementarities: the gain from moving to a higher strategy increases when opponents move to higher strategies. They arise in coordination, oligopoly and macroeconomic models (Milgrom and Roberts, 1990; Vives, 1990). Hofbauer and Sandholm (Econometrica 2002) show that in such games stochastic fictitious play converges almost surely whenever its mean dynamic has a unique rest point. This mission formalizes that result and the chain of lemmas behind it (Section 5 and the Appendix of the paper).

Timeline. Benaïm and Hirsch (1999) observed that supermodular games with exactly two strategies per player yield strongly monotone perturbed best response dynamics. Hofbauer and Sandholm (2002) extended this to any number of strategies by introducing stochastic dominance coordinates, and proved Theorem 6.1(iv). Benaïm (2000) supplied the low-dimensional convergence theorem used for the dimension ≤ 2 clause.

Setting

A $p$ player normal form game gives player $\alpha$ an ordered finite strategy set $S^\alpha = \{0,\dots,n^\alpha - 1\}$ and a utility $u^\alpha$ on pure profiles. $\Sigma = \prod_\alpha \Delta S^\alpha$ is the set of mixed profiles, and player $\alpha$'s payoff vector is $U^\alpha_i(x^{-\alpha}) = \sum_{s:\, s^\alpha = i} u^\alpha(s) \prod_{\beta \ne \alpha} x^\beta_{s^\beta}$.

The game is strictly supermodular if for all distinct players $\alpha \ne \beta$ and all profiles $s, \hat s$ with $s^\alpha > \hat s^\alpha$ and $s^{-\alpha} = \hat s^{-\alpha}$, the difference $u^\alpha(s) - u^\alpha(\hat s)$ is strictly increasing in $s^\beta = \hat s^\beta$.

Each player $\alpha$ has a shock density $f^\alpha$ on $\mathbb R^{n^\alpha}$, strictly positive, whose choice function $C^\alpha_i(\pi) = P(\operatorname{argmax}_j \pi_j + \varepsilon_j = i)$ is continuously differentiable. The perturbed best response is $\tilde B^\alpha(x^{-\alpha}) = C^\alpha(U^\alpha(x^{-\alpha}))$, and the perturbed best response dynamic is

$$(\mathrm P)\qquad \dot x^\alpha = \tilde B^\alpha(x^{-\alpha}) - x^\alpha .$$

$RP(\mathrm P)$ is its set of rest points in $\Sigma$ and $CR(\mathrm P)$ its chain recurrent set.

In standard stochastic fictitious play the shocks $\varepsilon^\alpha_t$ have densities $f^\alpha$ and are independent over time and across players. From an arbitrary initial pure profile $\zeta_1$, each player at time $t+1$ plays the pure strategy maximizing $U^\alpha_k(Z_t^{-\alpha}) + (\varepsilon^\alpha_t)_k$, where $Z_t = \frac1t \sum_{u \le t} \zeta_u$ is the vector of empirical frequencies.

The stochastic dominance coordinates are $(T^\alpha x^\alpha)_i = \sum_{j > i} x^\alpha_j \in \mathbb R^{n^\alpha - 1}$, the mass on strategies above $i$; $T x \le T y$ means each $y^\alpha$ stochastically dominates $x^\alpha$. In these coordinates (P) becomes a dynamic (T) on $T(\Sigma)$.

Formalization targets

Goal: Theorem 6.1(iv), unique rest point clause

For a strictly supermodular game with $p \ge 2$ players and shock densities as above, if $RP(\mathrm P) = \{x^*\}$ then

$$P\Big(\lim_{t\to\infty} Z_t = x^*\Big) = 1,$$

on every probability space, for every independent shock family with the given densities and every initial profile.

Milestones

• Lemma A.2, eq. (14), Lemma A.3 and eqs. (17)–(18): the order-theoretic and differential facts behind monotonicity.
• Theorem 5.1: $T^{-\alpha} y^{-\alpha} \ge T^{-\alpha} x^{-\alpha} \Rightarrow T^\alpha \tilde B^\alpha(y^{-\alpha}) \ge T^\alpha \tilde B^\alpha(x^{-\alpha})$.
• Theorem 5.2: there are rest points $\underline x, \bar x$ with $RP(\mathrm P) \subseteq [\underline x, \bar x]$.
• Proposition 5.3: (P) and (T) are linearly conjugate.
• Theorem 5.4: (T) is cooperative and irreducible.
• Corollary 5.5(i)–(ii): (P) is strongly monotone, and $CR(\mathrm P) \subseteq [\underline x, \bar x]$, so $CR(\mathrm P) = \{x^*\}$ when the rest point is unique.

Significance

The result gives global almost-sure convergence of a stochastic learning process in a broad class of games with many strategies, where earlier results needed two strategies per player or specific payoff structures. It also shows which properties of the choice function matter: only eqs. (17)–(18), not the symmetry of its derivative used for potential and zero-sum games.

The paper's proof is complete but relies on outside results: stochastic approximation (Benaïm–Hirsch 1999, Benaïm 1999), monotone dynamical systems (Smith 1995), and the inclusion of the chain recurrent set in the global attractor (Robinson 1995). None of these, nor the Hofbauer–Sandholm theorem itself, is formalized in any proof assistant as far as known. The mission produces a machine-checked version of the theorem and of the Section 5 monotonicity theory.

Difficulty

The monotonicity lemmas are finite-dimensional calculus and summation by parts. The substantial steps are elsewhere. Strong monotonicity of a cooperative irreducible system (Corollary 5.5(i)) is a theorem of monotone dynamical systems that Mathlib does not have, and it must hold on the closed, non-open state space $T(\Sigma)$. The step from the ODE to the random process needs the stochastic approximation theorem: the empirical frequencies are an asymptotic pseudotrajectory of (P), and their limit set is almost surely internally chain transitive. Knowing that $x^*$ is globally asymptotically stable for (P) does not by itself give almost-sure convergence of $Z_t$. The limit set of the random process must be related to the chain recurrent set of (P), which is where Corollary 5.5(ii) enters.

Formalization scope

Players are Fin p, strategies Fin (n α) (0-based, same order as the paper), with every $n^\alpha \ge 1$. Mixed profiles live in the ambient space $\prod_\alpha \mathbb R^{n^\alpha}$ with the sup norm. The fields of (P) and (T) are defined on the whole ambient space, so partial derivatives are ordinary Fréchet derivatives. Densities are $[0,\infty]$-valued. Solutions are forward solutions staying in the state space. The chain recurrent set quantifies over solutions, which are unique here because the fields are $C^1$. The process $Z_t$ is defined pathwise from the shocks, with ties in the argmax broken by the smallest index (a null event). Densities may differ across players.

A statement about the ODE (P), or about a single noise law such as logit, would be a different and much weaker theorem. The goal is about the random process $Z_t$, on every probability space carrying independent shocks with the given densities. Irreducibility and strong monotonicity additionally assume that two distinct players each have at least two strategies; when one player owns every stochastic dominance coordinate and has at least two of them, supermodularity holds vacuously while irreducibility fails.

A complete development needs: random utility choice functions and their derivatives; monotone and cooperative ODE theory on convex sets; the chain recurrent set and the global attractor; stochastic approximation for processes with step size $1/t$. The last three are reusable well beyond this mission. Contributions to any milestone, and general-purpose lemmas on cooperative systems or stochastic approximation, are welcome.

Selected references

• J. Hofbauer and W. H. Sandholm, On the Global Convergence of Stochastic Fictitious Play, Econometrica 70 (2002), 2265–2294. https://doi.org/10.1111/1468-0262.00376 (formalized from the authors' manuscript of February 21, 2002)
• M. Benaïm and M. W. Hirsch, Mixed Equilibria and Dynamical Systems Arising from Repeated Games, Games and Economic Behavior 29 (1999), 36–72. https://doi.org/10.1006/game.1997.0636
• M. Benaïm, Dynamics of Stochastic Approximation Algorithms, Séminaire de Probabilités XXXIII, Lecture Notes in Mathematics 1709, Springer (1999). https://doi.org/10.1007/BFb0096509
• M. Benaïm, Convergence with Probability One of Stochastic Approximation Algorithms Whose Average is Cooperative, Nonlinearity 13 (2000), 601–616. https://doi.org/10.1088/0951-7715/13/3/305
• H. L. Smith, Monotone Dynamical Systems, AMS Mathematical Surveys and Monographs 41 (1995). https://doi.org/10.1090/surv/041
• P. Milgrom and J. Roberts, Rationalizability, Learning, and Equilibrium in Games with Strategic Complementarities, Econometrica 58 (1990), 1255–1277. https://doi.org/10.2307/2938316
• D. Fudenberg and D. K. Levine, The Theory of Learning in Games, MIT Press (1998).

Motivation

Mission XII (12-packet-networks-model) built the discrete-time, slotted packet-network model from scratch and proved the chapter's own version of the fluid-to-stochastic stability bridge (Theorem 12.10). That result is only useful once paired with an actual control policy whose fluid model can be shown stable. J. G. Dai and J. Michael Harrison's Processing Networks: Fluid Models and Stability (Cambridge University Press, forthcoming; cited here from the authors' pre-publication draft, 2020-4-2, http://spnbook.org) supplies exactly such a policy in Sections 12.4-12.5: back-pressure control (called max-weight when the network is single-hop), a rule that has become the default choice in the switching and wireless-scheduling literature because it requires no advance knowledge of arrival rates and achieves the largest possible stability region. This mission formalizes back-pressure control for the discrete-time packet network model and proves it maximally stable — the chapter's own counterpart to mission VIII's continuous-time Theorem 9.12.

Setting

At the start of each timeslot, having observed the current buffer contents $z$, the max-weight/ back-pressure (MW/BP) policy solves $\max_{s\in S(z)} z\cdot Rs$ (Eq. 12.41), where $S(z) := \{s\in S : Bs\le z\}$ restricts the schedule set $S$ (mission XII) to what is actually available given $z$. Equivalently (Eq. 12.42-12.43), writing $w_j(z) := z_{u(j)} - z_{d(j)}$ (with $z_0 := 0$) for the "weight" of activity $j$, the policy maximizes $\sum_j w_j(z) s_j$ — the schedule that clears the most "backlog pressure" per timeslot. The policy is deterministic (no randomization variable is needed, since (12.41) is a genuine optimization problem, not one that inherently requires randomized tie-breaking).

Formalization targets

Goal: Theorem 12.16 — aperiodicity, irreducibility, and conditional positive recurrence

Consider a packet network satisfying (12.1) and Assumption 12.1, operating under back-pressure control. The DTMC $Z$ is aperiodic and irreducible unconditionally. If the stability condition (12.26) — the existence of $\hat s\in\langle S\rangle$ with $\lambda < R\hat s$ — is additionally satisfied, $Z$ is positive recurrent. This is the mission's headline result, and it packages both a structural fact (irreducibility/aperiodicity, needed regardless of load) and a conditional stability fact (positive recurrence, needed only under (12.26)) in one theorem.

Supporting milestones

Lemma 12.17 shows the optimization problem (12.41) always has a nonzero solution whenever the network is nonempty — the fact that back-pressure never "idles unnecessarily," which Lemma 12.18 uses to show every state can reach the empty state (hence irreducibility) and that state $0$ has period $1$ (hence aperiodicity, via (12.1)'s own assumption that no external arrivals is possible with positive probability). Lemma 12.20 is the back-pressure fluid equation (12.44), the discrete-time analog of mission VIII's Theorem 9.8: at each regular point, the fluid-scaled buffer content dotted with the fluid departure rate equals the maximum of that same dot product over the convex hull of the schedule set. Lemma 12.21 shows this fluid model is stable whenever (12.26) holds, via an explicit quadratic Lyapunov function.

Significance

The result itself. Theorem 12.16 shows that back-pressure control — a rule requiring no knowledge of arrival rates, computed fresh from the current buffer contents every timeslot — is maximally stable: combined with Theorem 12.8 and Proposition 12.9 (mission XII), it stabilizes every packet network arrival-rate vector that any policy could possibly stabilize. This is the discrete-time, slotted analog of mission VIII's Theorem 9.12, and the two proofs share their essential Lyapunov argument (the book's own text calls Lemma 12.21's proof "almost identical to that of Theorem 9.12"), even though the two chapters' back-pressure equations are built from different underlying objects — mission VIII's continuous-time allocation polytope versus this chapter's convex hull of a discrete schedule set.

Formalizing it. A live prior-art check (GET /theorems?q=back-pressure, q=max-weight) finds no relevant hits, matching mission VIII's own finding for the continuous-time version. This mission formalizes the back-pressure optimization problem and policy, the back-pressure fluid equation, and its stability from scratch, restating (per this series' convention for concurrently-drafted chunks sharing a sub-namespace) the minimal apparatus needed from mission XII: the packet network model, Assumption 12.1, the raw processes, fluid limit paths, the fluid equations (12.31)-(12.36), and the ambient-chain machinery, plus a new Aperiodic predicate this chunk needs that mission XII's own results do not.

Difficulty

The obvious approach to Lemma 12.20's back-pressure fluid equation — directly differentiate the discrete system equation — obscures the actual argument, which reduces a maximum over the (potentially non-polytope) discrete schedule set $S$ to a maximum over its convex hull $\langle S\rangle$ (justified because a linear functional on a compact polytope is maximized at an extreme point) and then shows that every schedule with a strictly smaller objective value than the maximizer contributes zero derivative to the usage-counting process $\hat T_s$ — the discrete-time analog of mission VIII's Lemmas 9.10-9.11. A second difficulty is connecting the maximum-over-a- finite-set at the fluid-scaled level to Definition 12.3's discrete, arrival-indexed back-pressure policy: this mission's own Lemma 12.20 item states the fluid-relevant consequence of the discrete policy as a documented hypothesis (mirroring mission XI's own scope decision for its structurally identical WWTA fluid equation) rather than re-deriving it from the raw stochastic recursion, which would require reconstructing infrastructure outside this chapter's own numbered results.

Formalization scope

The packet network model, Assumption 12.1, the raw processes, and the fluid-limit apparatus are restated verbatim from mission XII (12-packet-networks-model), since concurrently-drafted chunks sharing a sub-namespace do not import one another's Lean files. IsBPOptimal is phrased via domination over the feasible-at-$z$ schedule set, not sSup/argmax, per this series' junk-value-avoidance convention; the same convention is applied to the maximum over $\langle S\rangle$ in the back-pressure fluid equation. The formalization does not admit a trivializing reading: irreducibility and aperiodicity are the same non-vacuous renewal-theoretic notions used throughout this series (Aperiodic requires a genuine positive self-transition probability, not a vacuously-true condition), and the goal's positive-recurrence conjunct is genuinely conditional on (12.26), stated with the correct strict inequality rather than weakened to $\le$. Contributions completing the five by sorry proofs are welcome, particularly Lemma 12.17's hop-count induction and Lemma 12.20's extreme-point/zero-derivative argument (mirroring mission VIII's own Lemmas 9.10-9.11, adapted to discrete time).

Selected references

• J. G. Dai and J. Michael Harrison, Processing Networks: Fluid Models and Stability, Cambridge University Press (forthcoming), pre-publication draft 2020-4-2. http://spnbook.org

Motivation

Internet routers, wireless base stations, and data-switch fabrics all face the same recurring decision: in each discrete time slot, which of many possible transfer operations should be executed, given the packets currently queued and the physical constraints (link capacities, interference between simultaneous transmissions) on what can be done at once? J. G. Dai and J. Michael Harrison's Processing Networks: Fluid Models and Stability (Cambridge University Press, forthcoming; cited here from the authors' pre-publication draft, 2020-4-2, http://spnbook.org) devotes Chapter 12 to exactly this question, under the name packet networks. Unlike every prior chapter of the book, which works in continuous time with Poisson-driven Markov chains, this chapter builds its stability theory from scratch for a discrete-time, slotted model — the natural setting for a system that makes one scheduling decision per clock tick. This mission formalizes the chapter's foundational layer: the model itself, its notion of a feasible schedule and control policy, subcriticality, and the discrete-time fluid-limit machinery that Chapters 13 and 14 (back-pressure control and random proportional scheduling, respectively) build their own stability proofs on top of.

Setting

A packet network (Section 12.1) has I packet classes and J activities (service types); activity $j$ transfers a packet from its input class $u(j)$ to its output class $d(j)$ (or removes it from the network if $d(j)=0$), giving an $I\times J$ input-output matrix $R$. A processing plan for class $i$ (Definition 12.2) is a chain of activities from $i$ to exit; Assumption 12.1 requires every class to have one and forbids cycles. In each timeslot the system manager chooses a schedule $s\in\mathbb Z^J_+$ from a feasible set $S$ satisfying the packet-availability constraint $Bs\le z$ (the current buffer contents); $S$ is typically built (Section 12.2) from a link usage matrix $A$ and a set $C$ of feasible link configurations via $S_c := \{s : As\le c\}$, $S := \bigcup_{c\in C} S_c$. A Markovian control policy (Definition 12.3, Eq. 12.8) chooses $s(\tau) = f(Z(\tau-1), U(\tau))$ from the current buffer contents and an independent randomization variable; it is admissible if it never overdraws a buffer, and stable if the resulting discrete-time Markov chain $Z$ is irreducible and positive recurrent.

Formalization targets

Goal: Theorem 12.10 — fluid limit stability implies positive recurrence

If the DTMC $Z$ under a Markovian policy $f$ is irreducible and its fluid limit is stable (Definitions 12.14-12.15), then $Z$ is positive recurrent. This is the chapter's own version of Theorem 6.2 (mission III) — the technical fulcrum that lets a deterministic fluid-model stability argument certify the stability of the original discrete, stochastic system — restated for a genuinely different probabilistic model, since Theorem 6.2 was built under continuous-time, Poisson-arrival hypotheses this chapter does not share.

Supporting milestones

Propositions 12.6-12.7 characterize the convex hulls $\langle S_c\rangle$ and $\langle S\rangle$ as explicit polytopes cut out by the link usage matrix — the combinatorial core that Theorem 12.8 (stability implies subcriticality, this chapter's analog of Theorem 5.2) and Proposition 12.9 (a necessary condition for subcriticality) both build on. Lemma 12.11 records that the schedule-usage counting process is Lipschitz; Lemma 12.12 is the functional strong law of large numbers the external arrival process satisfies; and Theorem 12.13 combines them to establish existence of discrete-time fluid limits satisfying the chapter's own fluid equations (12.31)-(12.36) — the chapter's analog of Theorem 6.5.

Significance

The result itself. Theorem 12.10 is what makes the rest of Chapter 12 (and Chapters 13-14) tractable: rather than analyzing an infinite-state discrete-time Markov chain's positive recurrence directly — a notoriously hard problem in general — it suffices to exhibit a deterministic fluid model and show every solution of that fluid model empties in finite time. This is the same strategy Chapter 6 established for the book's continuous-time model, but Chapter 12 cannot simply invoke that earlier theorem: the packet network model uses discrete time slots rather than a continuous clock, and its probabilistic structure (arbitrary i.i.d. arrival increments rather than Poisson arrivals) is different enough that the fluid-limit compactness argument has to be redone, even though — as the book's own text notes — "the proof mimics that of Theorem 6.2."

Formalizing it. A live prior-art check (GET /theorems?q=packet+network, q=discrete-time+Markov+chain, q=slotted+time) finds no relevant hits. A dedicated further check for MarkovMixing, a different mission's own corpus offering a PositiveRecurrent predicate for Markov chains, found a representationally distinct formalization (a row-function transition kernel rather than this series' own PMF-based jump-chain convention); this mission restates positive recurrence and irreducibility locally instead, consistent with every mission in this series since mission I. Everything else — the packet network model, schedules and configurations, the subcritical region, Markovian policies, and the discrete-time fluid-limit apparatus — is formalized from scratch.

Difficulty

The most consequential decision in this mission is representational, not mathematical: Theorem 12.10's fluid-limit-stability hypothesis quantifies over all fluid limit paths, which are themselves scaling limits of a genuinely stochastic discrete-time process — reconstructing that process from Chapter 2/4's own primitive stochastic elements (arrival processes, phase-type service mechanics) would require rebuilding infrastructure this chapter's own numbered results do not supply. Following mission III's own precedent for its structurally identical Theorem 6.5, this mission instead takes the raw, per-initial-state schedule-usage and arrival processes as given data satisfying only the recap properties the chapter's own proofs actually cite (Eq. 12.9-12.10's system equation, monotonicity), and fixes a single sample point together with an explicit SLLN hypothesis rather than a bare "for almost all $\omega$" quantifier — matching the book's own statement of Theorem 12.13, which itself begins "Fix an $\omega\in\Omega_1$" rather than quantifying almost everywhere within the theorem itself. A second difficulty is genuinely combinatorial: Proposition 12.6's proof constructs an explicit product-form probability distribution over schedules (one independent randomization per link) whose mean recovers an arbitrary point of the polytope $\{Ax\le c\}$ — a real combinatorial argument, not a formal consequence of the convex-hull operator's definition.

Formalization scope

Classes, activities, links, schedules and configurations are represented via Fin-indexed types throughout; S and C are taken as Finsets (every worked example in the book has finitely many configurations, and the link usage matrix's single-1-per-column structure bounds each schedule component, so this is a genuine, if implicit, standing feature of the model rather than an added restriction). IsScheduleSetAt/IsScheduleSet characterize $S_c$/$S$ via an ↔ against the underlying capacity constraint, rather than constructing them, matching how set-valued model data is handled elsewhere in this series. The formalization does not admit a trivializing reading: the ambient chain's positive recurrence (PositiveRecurrent) is the same non-vacuous renewal-theoretic notion used throughout this series, PacketFluidLimitStable quantifies over every genuine fluid limit path (not a hand-picked one), and subcriticalRegion is stated with a strict inequality ($A\hat s < \hat c$) exactly as Eq. 12.24 requires, not weakened to $\le$. Contributions completing the eight by sorry proofs are welcome, particularly Proposition 12.6's explicit randomized- schedule construction and Theorem 12.13's compactness argument (mirroring mission III's own Theorem 6.5 proof, adapted to discrete time).

Selected references

• J. G. Dai and J. Michael Harrison, Processing Networks: Fluid Models and Stability, Cambridge University Press (forthcoming), pre-publication draft 2020-4-2. http://spnbook.org

Motivation

Mission IX (09-proportional-fairness-core) formalized proportional fairness (PF) as a control policy and proved the technical core of its stability theory: under a load condition, the PF fluid model is stable (Theorem 10.5), via an entropy Lyapunov function that is genuinely not Lipschitz continuous — a departure from every other stability argument in J. G. Dai and J. Michael Harrison's Processing Networks: Fluid Models and Stability (Cambridge University Press, forthcoming; cited here from the authors' pre-publication draft, 2020-4-2, http://spnbook.org). A stability theorem for one fixed arrival-rate vector is, on its own, a narrower claim than practitioners actually want: real systems see load that changes over time, and a control policy worth adopting should not need re-tuning every time the mix of traffic shifts. This mission completes Theorem 10.5's proof and turns it into exactly that stronger guarantee — proportional fairness is maximally stable: stable throughout the entire region where any policy could be stable, without knowing the arrival rates in advance — and specializes the result to two concrete network families, bandwidth-sharing networks and queueing networks under head-of-line proportional processor sharing (HLPPS), that were already familiar from earlier in the book under different control policies.

Setting

Fix a unitary network operating under PF control with mean service times $m > 0$, routing matrix $P$, a partition of job classes into demand groups $\{\mathcal I(\ell), \ell \in \mathcal L\}$, and a reduced allocation set $\tilde{\mathcal A} \subset \mathbb R^{\mathcal L}_+$ (all restated from mission IX, Definition 10.3). The entropy Lyapunov function $\varphi(t) := \sum_i Z_i(t)\log(\dot D_i(t)/\alpha_i)$ (Eq. 10.38, mission IX) admits an alternative decomposition $\varphi = \sum_\ell \varphi_\ell$ in terms of the within-group entropy term

$$f(t) := \sum_{\ell\in\mathcal L}\sum_{i\in\mathcal I(\ell)} Z_i(t)\log\!\left(\frac{Z_i(t)}{Y_\ell(t)}\right), \qquad Y(t) := GZ(t) \quad \text{(Eq. 10.50)},$$

with the convention that the term for class $i$ is $0$ when $Z_i(t)=0$. Here $D^+$/$D^-$ denote the upper-right/upper-left Dini derivatives (Appendix A.4, Eqs. A.9-A.10): at a point where the ordinary derivative may not exist, these one-sided $\limsup$s still let a Lyapunov-drift argument go through. A control policy is maximally stable (Section 5.7) for a network if its implementation does not depend on the arrival-rate vector $\lambda$ and it is stable for every $\lambda$ in the network's stability region $\Lambda^*$ — the largest region any policy could possibly stabilize.

Formalization targets

Goal: Corollary 10.16 — maximal stability of PF control for a unitary network

$$\text{IsMaximallyStable}\Big(\lambda \mapsto \text{PFFluidStable}(\lambda, m, P, \mathrm{grp}, \tilde{\mathcal A})\Big)$$

for the single PF policy value (its implementation never depends on $\lambda$). This is the applied payoff of Theorem 10.5 (mission IX): combined with Theorem 6.2 (mission III, fluid stability implies SPN stability) and Corollary 5.6 (mission II, a $\lambda$-independent policy stable throughout the subcritical region is automatically maximally stable), it upgrades a single-$\lambda$ stability statement to the strongest form the book's own framework can express.

Supporting milestones

Lemmas 10.11-10.15 supply the remaining technical content of Theorem 10.5's proof that mission IX's own milestones left open: Lemma 10.11 is the uniform negative-drift bound $\sum_i \dot Z_i(t)\log(\dot D_i(t)/\alpha_i) \le -\varepsilon$ at every regular point with $Z(t) \ne 0$; Lemmas 10.12-10.14 establish continuity and two successively sharper Dini-derivative bounds on the within-group entropy term $f$; Lemma 10.15 shows that positivity of the departure rate $\dot D_i(t)$ propagates from occupied classes to every class. Corollaries 10.17 and 10.18 specialize the goal to bandwidth-sharing networks and to HLPPS-controlled queueing networks, respectively.

Significance

The result itself. A stability theorem tied to one fixed $\lambda$ is of limited practical use: it would need to be re-verified every time the arrival-rate vector changes, which real traffic does constantly. Maximal stability removes that dependency entirely — a single policy, implemented without any knowledge of $\lambda$, is guaranteed stable throughout the full region any control could stabilize. Corollaries 10.17 and 10.18 make this concrete for two network families with independent histories in the literature: bandwidth-sharing networks (the original motivation for proportional fairness, Kelly 1997) and queueing networks under HLPPS, connecting PF's static, utility-theoretic motivation to a scheduling rule that predates it.

Formalizing it. A live prior-art check (GET /theorems?q=proportional%20fairness, q=maximal%20stability, q=bandwidth%20sharing) finds no relevant hits on the platform. This mission formalizes the remaining entropy-Lyapunov lemmas, the within-group entropy term, the BWS and HLPPS network models (restated locally, since no other drafted chunk covers Sections 4.5-4.6), and the maximal-stability predicate, reusing only Mathlib's general real-analysis substrate (Dini derivatives via Filter.limsup) and definitions restated from missions II, III, V, and IX under this series' restate-not-import convention for concurrently-drafted chunks.

Difficulty

The obvious approach to Corollary 10.16 — restate "maximally stable" with an explicit $\lambda$-dependent policy family and add "the policy doesn't actually depend on $\lambda$" as a side hypothesis — obscures the point: a policy that is definitionally independent of $\lambda$ is a stronger and cleaner claim than one that happens to satisfy an extra equation. This formalization instead instantiates the abstract policy type at Unit, so $\lambda$-independence holds by construction rather than as a hypothesis to verify, matching the book's own reading of Section 5.7's definition. A second difficulty is Lemma 10.13's Dini-derivative inequality (Eq. 10.56): the plain-text extraction of this display equation loses bracket and subscript structure that changes its meaning, so the exact grouping was confirmed against the PDF page directly and cross-checked against the book's own re-derivation of the same bracketed expression inside Lemma 10.14's proof. A third is Corollary 10.17's proof, which genuinely depends on three facts outside this chunk's own chapter portion (Proposition 4.4 and the Section 4.4 model translation, Proposition 5.1, Theorem 5.2); rather than silently assuming them or re-deriving their proofs from scratch, they are stated as explicit hypotheses of the milestone itself, so the item's actual content — deriving the two-sided conclusion — is exactly what remains to be proved.

Formalization scope

RestatedCore, RestatedFluidModel, and the maximal-stability predicate MaximalStability are restated verbatim (or, for MaximalStability, in shape) from missions IX, III/V, and II respectively, since concurrently-drafted chunks in this series do not import one another's Lean files even when they share a sub-namespace. New to this chunk: diniUpperLeft (Eq. A.10, needed alongside mission IX's diniUpperRight for Lemma 10.13's two-sided bound), the within-group entropy term withinGroupEntropy (Eq. 10.50, deliberately not named f, since the book's own f already denotes the unrelated PF optimization objective of Eq. 10.2 within the same chapter), and BWSNetworkData/QueueingNetworkDataHL/HLPPSFluidStable (Sections 4.5-4.6, restated locally since no drafted chunk's BRIEF.md covers them). The formalization does not admit a trivializing reading: IsMaximallyStable is instantiated with the genuine, non-vacuous predicate PFFluidStable/HLPPSFluidStable — the same predicate whose stability Theorem 10.5 (mission IX) already establishes on the load-condition region — never with a policy type or stability predicate engineered to make the maximal-stability claim vacuous. Contributions completing the eight by sorry proofs are welcome, particularly Lemma 10.13's Dini-derivative estimate (Section B.4's preliminary results) and Lemma 10.15's connectivity argument (Appendix B.9/B.17).

Selected references

• J. G. Dai and J. Michael Harrison, Processing Networks: Fluid Models and Stability, Cambridge University Press (forthcoming), pre-publication draft 2020-4-2. http://spnbook.org
• F. P. Kelly, "Charging and rate control for elastic traffic," European Transactions on Telecommunications 8 (1997), 33–37.
• F. P. Kelly, A. K. Maulloo, and D. K. H. Tan, "Rate control for communication networks: shadow prices, proportional fairness and stability," Journal of the Operational Research Society 49 (1998), 237–252.