Regularity and optimization

Transcription

1 Regularity and optimization Bruno Gaujal INRIA SDA2, 2007 B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

2 Majorization x = (x 1,, x n ) R n, x is a permutation of x with decreasing coordinates. { k x y si i=1 x i k i=1 y i k, 1 k n 1, n i=1 x = n i=1 y Marshall Olkin, 1979 i i. B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

3 Majorization x = (x 1,, x n ) R n, x is a permutation of x with decreasing coordinates. { k x y si i=1 x i k i=1 y i k, 1 k n 1, n i=1 x = n i=1 y i Marshall Olkin, 1979 If x = (0, 1, 3, 1) and y = (3, 1, 8, 1), x = (1, 0, 1, 3), y = (3, 1, 1, 8), partial sums verify : = , which implies x y. i. B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

4 Majorization (II) If P is a doubly stochastic matrix, y = x P y x. Using Birkhoff Theorem (doubly stochastic matrices are convex combination of permutation matrices). B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

5 Majorization (II) If P is a doubly stochastic matrix, y = x P y x. Using Birkhoff Theorem (doubly stochastic matrices are convex combination of permutation matrices). x e 3 e 1 e 2 B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

6 Schur convexity f : R n R is Schur-convex if x y f (x) f (y). B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

7 Schur convexity f : R n R is Schur-convex if x y f (x) f (y). Theorem If f (x) f (y) for all Schur convex functions f, then x y. B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

8 Schur convexity f : R n R is Schur-convex if x y f (x) f (y). Theorem If f (x) f (y) for all Schur convex functions f, then x y. Let φ : R n R and g : R R such that ψ : R n R is such that ψ(x 1,..., x n ) = φ(g(x 1 ),..., g(x n )). If φ is increasing and Schur convex and g is convex, then ψ est Schur-decreasing. B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

9 Some applications Polygons with n vertices inside a disk. A 3 A 4 A 2 θ 4 θ 3 O θ 2 θ 1 θ 5 θ 6 A 1 A 5 A 6 The surface is 1/2 n i=1 sin(θ i), which is Schur-concave. (θ 1,... θ n ) (α 1,... α n ) implies that the surfaces S θ S α. The polygon with the largest surface S is regular. B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

10 Some applications Polygon with n vertices outside a disk, with the smallest k-th moment, for all k. h r A 2 θ 2 θ 1 C(r) A 3 θ 3 θ 4 θ 5 A 4 A 5 A 1 h r = r n i=1 max(0, θ i 2 cos 1 (1/r)). is Schur convex. The polygon with the smallest k-th moment is regular, for all k. B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

11 Application in Networks Bandwidth allocation in Networks Eitan Altman Using Shannon SINR Formula, The total throughput is a sum of a convex function of the emited power, so that it is maximized by small (w.r.t.majorization)power allocation. B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

12 Application in Networks Bandwidth allocation in Networks Eitan Altman Using Shannon SINR Formula, The total throughput is a sum of a convex function of the emited power, so that it is maximized by small (w.r.t.majorization)power allocation. Structure of optimal control Ger Koole B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

13 Routing applications Majorization and Schur convexity cannot be used when the problem depends on the actual sequence and is not invariant up to permutations. Actually, the following applications will be invariant up to cyclic permutations. B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

14 Mechanical words This talk will now focus on finite or periodic mechanical words. 1 Extremal properties of rational mechanical words for scheduling problems 2 Factorizations of mechanical words using Continued Fractions and their consequences on computational problems 3 Compute the extremal points in the scheduling problems. B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

15 Mechanical words This talk will now focus on finite or periodic mechanical words. 1 Extremal properties of rational mechanical words for scheduling problems 2 Factorizations of mechanical words using Continued Fractions and their consequences on computational problems 3 Compute the extremal points in the scheduling problems. There exist many ways to link continued fraction decompositions with mechanical words. Here, we will exhibit a link which has some algorithmic and computational interest. B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

16 Mechanical words Definition (Mechanical words) A mechanical word with slope (density) 0 α 1 and intercept ρ is the infinite word w α,ρ whose letters are n 0, w α,ρ (n) = (n + 1)α + ρ nα + ρ, or n 0, w α,ρ (n) = (n + 1)α + ρ nα + ρ B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

17 Mechanical words Definition (Mechanical words) A mechanical word with slope (density) 0 α 1 and intercept ρ is the infinite word w α,ρ whose letters are n 0, w α,ρ (n) = (n + 1)α + ρ nα + ρ, or n 0, w α,ρ (n) = (n + 1)α + ρ nα + ρ When α is a rational number (α = p/q), then w α,ρ is periodic of period q (i.e. n 0, w α,ρ (n) = w α,ρ (n + q)) Definition (Finite mechanical words) A finite word m is mechanical if w = m is a mechanical word. B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

18 Optimal properties of rational mechanical words λ w µ 1 µ λ 1 λ 2 w µ 2 The arrival rate(s) λ (λ 1 and λ 2 ) are arbitrary positive real numbers, the service rates µ 1, µ 2 (µ) are also arbitrary. They verify the stability property : µ 1 + µ 2 λ (µ > λ 1 + λ 2 ). B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

19 Optimal properties of rational mechanical words µ 1 µ λ w λ 1 λ 2 w µ 2 The arrival rate(s) λ (λ 1 and λ 2 ) are arbitrary positive real numbers, the service rates µ 1, µ 2 (µ) are also arbitrary. They verify the stability property : µ 1 + µ 2 λ (µ > λ 1 + λ 2 ). Theorem G.-Hyon 03, G.-Hordijk-Van der Laan 06 The routing (allocation) sequence minimizing the workload is a mechanical word with a rational density α. B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

20 Workload If σ n is the size of packet n, δ n the time between the nth arrival and the n 1th arrival, and a n {0, 1}, the routing decision to Queue 1, then the workload verifies : W n = max (W n 1 + a n 1 σ n 1 δ n, 0). a a a a a a 6 δ 1 W(t) δ δ δ δ δ σ 4 σ 2 σ 3 σ 5 σ 1 σ t B. Gaujal (INRIA ) d 1 Regularity andd Optimization 2 d3 d4 d5 SDA2, / 39

21 Optimality (II) Under routing sequence w, let W 1 n (w) be the workload in queue 1 at the n-th decision and W 1 (w) = lim sup 1 N N Wn 1 (w). n=1 Theorem (AGH) Altman, G., Hordijk, 2000 Over all sequences w of density at least α, W 1 (w) is minimal when w = m α, the mechanical sequence of density α. Based on multimodularity properties. B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

22 Convexity of α αw 1 (m α ) Let α 1 = p 1 q 1 and α 2 = p 2 Let m = m q 2 α 1 m q 1 α 2. q 2 in the stability interval,j = [1 µ 2, µ 1 ]. Let N(m) be the average number of packets in queue 1 under the routing sequence m. B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

23 Convexity of α αw 1 (m α ) Let α 1 = p 1 q 1 and α 2 = p 2 Let m = m q 2 α 1 m q 1 α 2. q 2 in the stability interval,j = [1 µ 2, µ 1 ]. Let N(m) be the average number of packets in queue 1 under the routing sequence m. N(m) = q 1q 2 N(m 1 )+q 2 q 1 N(m 2 ) q 1 q 2 +q 2 q 1. B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

24 Convexity of α αw 1 (m α ) Let α 1 = p 1 q 1 and α 2 = p 2 Let m = m q 2 α 1 m q 1 α 2. q 2 in the stability interval,j = [1 µ 2, µ 1 ]. Let N(m) be the average number of packets in queue 1 under the routing sequence m. N(m) = q 1q 2 N(m 1 )+q 2 q 1 N(m 2 ) q 1 q 2 +q 2 q 1. Let α = p 1q 2 +p 2 q 1 q 1 q 2 +q 2 q 1 (note that α = α 1 /2 + α 2 /2 is the density of m). B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

25 Convexity of α αw 1 (m α ) Let α 1 = p 1 q 1 and α 2 = p 2 Let m = m q 2 α 1 m q 1 α 2. q 2 in the stability interval,j = [1 µ 2, µ 1 ]. Let N(m) be the average number of packets in queue 1 under the routing sequence m. N(m) = q 1q 2 N(m 1 )+q 2 q 1 N(m 2 ) q 1 q 2 +q 2 q 1. Let α = p 1q 2 +p 2 q 1 q 1 q 2 +q 2 q 1 (note that α = α 1 /2 + α 2 /2 is the density of m). Using Theorem [AGH], N(m α ) N(m) = 1/2N(m α1 ) + 1/2N(m α2 ), and λ [0, 1] N(m λα1 +(1 λ)α 2 ) λn(m α1 ) + (1 λ)n(m α2 ). B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

26 Convexity of α αw 1 (m α ) Let α 1 = p 1 q 1 and α 2 = p 2 Let m = m q 2 α 1 m q 1 α 2. q 2 in the stability interval,j = [1 µ 2, µ 1 ]. Let N(m) be the average number of packets in queue 1 under the routing sequence m. N(m) = q 1q 2 N(m 1 )+q 2 q 1 N(m 2 ) q 1 q 2 +q 2 q 1. Let α = p 1q 2 +p 2 q 1 q 1 q 2 +q 2 q 1 (note that α = α 1 /2 + α 2 /2 is the density of m). Using Theorem [AGH], N(m α ) N(m) = 1/2N(m α1 ) + 1/2N(m α2 ), and λ [0, 1] N(m λα1 +(1 λ)α 2 ) λn(m α1 ) + (1 λ)n(m α2 ). Using Little s Formula, N(m α ) = α(w 1 (m α ) + 1/µ). B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

27 Linearity over Farey intervals The interval [d 1, d 2 ] (d i = p i q i ) is called a Farey interval if q 1 p 2 p 1 q 2 = 1. B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

28 Linearity over Farey intervals The interval [d 1, d 2 ] (d i = p i q i ) is called a Farey interval if q 1 p 2 p 1 q 2 = 1. Proposition Let I = [α 1, α 2 ] be a Farey interval, then for all α I, m α can be factorized into m α1 and m α2. B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

29 Linearity over Farey intervals The interval [d 1, d 2 ] (d i = p i q i ) is called a Farey interval if q 1 p 2 p 1 q 2 = 1. Proposition Let I = [α 1, α 2 ] be a Farey interval, then for all α I, m α can be factorized into m α1 and m α2. Using the greatest integer continued fraction expansion of α and the associated factorization of m α (below) one can show that the queue is empty at the end of each period of a mechanical word with intercept 0. B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

30 Linearity over Farey interals(ii) B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

31 Linearity over Farey interals(ii) Corollary The function α αw 1 (m α ) is linear over any Farey interal I J. B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

32 Construction of the best control sequence The global cost function in both queues is C(α) = αw 1 (m α ) + (1 α)w 2 (m 1 α ). Inside the stability interval J = [1 µ 2, µ 1 ], let us consider the successive convergents of the lower continued fraction decomposition of µ 1 and of the upper continued fraction decomposition of 1 µ 2. They form Farey intervals over which the function C(α) is linear. B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

33 Construction of the best control sequence The global cost function in both queues is C(α) = αw 1 (m α ) + (1 α)w 2 (m 1 α ). Inside the stability interval J = [1 µ 2, µ 1 ], let us consider the successive convergents of the lower continued fraction decomposition of µ 1 and of the upper continued fraction decomposition of 1 µ 2. They form Farey intervals over which the function C(α) is linear. By convexity and piecewise linearity, the function C(α) is minimal at one of these convergents. B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

34 Construction of the best control sequence The global cost function in both queues is C(α) = αw 1 (m α ) + (1 α)w 2 (m 1 α ). Inside the stability interval J = [1 µ 2, µ 1 ], let us consider the successive convergents of the lower continued fraction decomposition of µ 1 and of the upper continued fraction decomposition of 1 µ 2. They form Farey intervals over which the function C(α) is linear. By convexity and piecewise linearity, the function C(α) is minimal at one of these convergents. Furthermore, using the recursive factorisation of mechanical words shown below (Theorem 7) one can compute the function C at all those points. B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

35 Numerical computations(i) g 0.2 B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39 α

36 Numerical computations(ii) 2 1/S α = 0 opt α = 1/4 opt α = 1/3 opt α opt= 1/2 1/S 2 = c 1/S 1 α =1 opt Instability domain 1 1/S B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

37 Numerical computations(iii) 2 1/S Double cusp optimal Instability domain (0,0) 1 1/S The number of convergents browsed by the algorithm before reaching the minimum point is small (as show in the figure). More numerical evidences show that for 99% of the stability polytope, the optimal sequence has a period less than 10. B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

38 Nearest Integer Continued fraction A semi regular continued fraction (SRCF)-expansion of a number s [0, 1) is a finite or infinite fraction s = 0 + ɛ 1 ɛ 2 b 1 + b ɛ n. b n +... where, b n N and ɛ n { 1, 1}. (b n + ε n 1, b n + ε n+1 1 and b n + ε n+1 2 infinitely often.), B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

39 Nearest Integer Continued fraction A semi regular continued fraction (SRCF)-expansion of a number s [0, 1) is a finite or infinite fraction s = 0 + ɛ 1 ɛ 2 b 1 + b ɛ n. b n +... where, b n N and ɛ n { 1, 1}. (b n + ε n 1, b n + ε n+1 1 and b n + ε n+1 2 infinitely often.) Let U γ : [0, 1] [0, 1], U γ (s) = γ if s 0 and U γ (0) = 0 otherwise. s s The coefficients of the SRCF-expansion are ( ) ɛ n (s) = sign (s) ; b n (s) = U n 1 γ Uγ n 1, γ. (s) B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

40 Nearest Integer Continued fraction (II) Some SRCFs are quite remarkable. γ = 1 corresponds to the regular continued fraction expansion (RCF). γ = 0 corresponds to an expansion where, for all n, ɛ n = 1. It appears in the litterature under the name greatest integer continued fraction (GICF). γ = 1/2 is the Nearest Integer Continued Fraction (NICF), on which we will mainly focus. B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

41 Nearest Integer Continued fraction (III) Let α = 13/31. U 1/2 (13/31) = = 5 13, b 1 = = 2, ɛ 2 = 1. Continuing the expansion gives U 2 1/2 = 2/5, b 2 = 3, ɛ 3 = 1 and U 3 1/2 = 1/2, b 3 = 3, ɛ 4 = 1 finally b 4 = 2. The NICF expansion of 13/31 is 2, 3, 3, 2. The sequence of convergents is 2 =1/2, 2, 3 =3/7, 2, 3, 3 =8/19 and 2, 3, 3, 2 =13/31. B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

42 NICF and word factorization Let α (0, 1/2) such that its partial NICF of order n + 1 is 0, b 1, ɛ 2 b 2,..., ɛ n b n, ɛ n+1 b n+1 +ɛ n+2 α n+1. Let I(n) = n+1 i=2 1 {ɛ i =+1}. Theorem The mechanical word m α can be factorized only using two factors x n and y n for all n defined by x 0 = 1, y 0 = 0 and for n 1, ( ) bn 1 ( ) bn x n = x n 1 yn 1, yn = x n 1 yn 1 if I(n) even ɛn+2 =+1, ( ) bn 2 ( ) bn 1 x n = x n 1 yn 1, yn = x n 1 yn 1 if I(n) even ɛ n+2 = 1, x n = ( ) bn x n 1 y n 1, y n = ( ) bn 1 x n 1 yn 1 if I(n) odd ɛ n+2 = +1, x n = ( ) bn 1 x n 1 yn 1, y n = ( ) bn 2 x n 1 yn 1 if I(n) odd ɛ n+2 = 1. B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

43 example Let α be equal to 13/31, its partial remainders are 5/13, 2/5 and 1/2 and its convergents are 1/2, 3/7 and 8/19. It comes x 3 { }} {{ }} { x 1 x 1 y 1 x 1 x 1 y 1 x 1 y 1 x 1 x 1 y 1 x 1 y 1 {}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{ m 13/31 = } {{ 100 }} {{ 100 }} 10 {{ 100 }} {{ 100 }} 10 {{ 100 }, } x 2 x 2 y 2 {{ x 2 y 2 } x 4 since Φ(m 13/31 ) = m 1 5/13, Φ(m 8/13 ) = m 1 2/5, Φ(m 3/5 ) = m 1/2 and Φ(m 1/2 ) = m 1. y 3 B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

44 Algorithmic considerations B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

45 Algorithmic considerations Result Let α = p/q 1/2 be a rational number. The number of iterations to reach the complete expansion in NICF is upper bounded by log η (1 + q 8) with η = B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

46 Algorithmic considerations Result Let α = p/q 1/2 be a rational number. The number of iterations to reach the complete expansion in NICF is upper bounded by log η (1 + q 8) with η = Corollary The maximal number of regular reductions used to test if a word w is balanced is bounded by log η (1 + w 8) with η = B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

47 Extension 1 : The stochastic case The cost is now the Cesaro limit of the expected worloads. The optimal polling between two queues can be computed when the service is exponential (with rate µ i in queue i) and the inter-arrivals are exponential (with rate λ in both queues). B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

48 Extension 1 : The stochastic case The cost is now the Cesaro limit of the expected worloads. The optimal polling between two queues can be computed when the service is exponential (with rate µ i in queue i) and the inter-arrivals are exponential (with rate λ in both queues). The continuous time Markov chain X t is a quasi birth and death process whose generator Q is given by C A A 2 A 1 A Q = 0 A 2 A 1 A 0..., 0 0 A 2 A B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

49 Extension 1 : The stochastic case The cost is now the Cesaro limit of the expected worloads. The optimal polling between two queues can be computed when the service is exponential (with rate µ i in queue i) and the inter-arrivals are exponential (with rate λ in both queues). The continuous time Markov chain X t is a quasi birth and death process whose generator Q is given by C A A 2 A 1 A Q = 0 A 2 A 1 A 0..., 0 0 A 2 A The stationary measure can be computed using the Kernel method : Π(z)K (z) = π 0 µ(1 z)m. B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

50 The stochastic case(ii) B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

51 EXtension 2 : Multidimensional routing problems A natural extension of the routing problem is to consider a routing to N > 2 queues. B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

52 EXtension 2 : Multidimensional routing problems A natural extension of the routing problem is to consider a routing to N > 2 queues. The control sequence can be seen as a word over N letters. B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

53 EXtension 2 : Multidimensional routing problems A natural extension of the routing problem is to consider a routing to N > 2 queues. The control sequence can be seen as a word over N letters. The optimization problem is to minimize the average waiting time of all incoming packets : W (a) = N i=1 p iw i (1 i (a)). B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

54 EXtension 2 : Multidimensional routing problems A natural extension of the routing problem is to consider a routing to N > 2 queues. The control sequence can be seen as a word over N letters. The optimization problem is to minimize the average waiting time of all incoming packets : W (a) = N i=1 p iw i (1 i (a)). The problem is still open. One of the main difficulties : the sequences 1 i (a) cannot be all balanced (Fraenkel Conjecture Altman, G.,Hordijk, 2000 ; Tijdeman 2004). B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

55 The multimodular ordering D (n + 1) n of rank N s. t. the rows s s n = 0. The Mesh M D is the set {a 0 s a n s n, a i Z}. D M D B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

56 The multimodular ordering D (n + 1) n of rank N s. t. the rows s s n = 0. The Mesh M D is the set {a 0 s a n s n, a i Z}. D M D Definition (AGH, 2003) A function f is D-multimodular if a Z n and i j, f (a + s i ) + f (a + s j ) f (a) + f (a + s i + s k ) B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

57 The multimodular ordering D (n + 1) n of rank N s. t. the rows s s n = 0. The Mesh M D is the set {a 0 s a n s n, a i Z}. D M D Definition (AGH, 2003) A function f is D-multimodular if a Z n and i j, f (a + s i ) + f (a + s j ) f (a) + f (a + s i + s k ) f is D-mm iff its linear interpolation over the atoms of M D is convex. B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

58 The multimodular ordering (II) Let F(D, r) be the set of all D-mm functions minimum at r M D. Definition x mm y if f (x) f (y) for all f F(D, r). B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

59 The multimodular ordering (II) Let F(D, r) be the set of all D-mm functions minimum at r M D. Definition x mm y if f (x) f (y) for all f F(D, r). Theorem x mm y iff there exists a path over M D from x to y, away from r. B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

60 The multimodular ordering (II) Let F(D, r) be the set of all D-mm functions minimum at r M D. Definition x mm y if f (x) f (y) for all f F(D, r). Theorem x mm y iff there exists a path over M D from x to y, away from r. y x r D M D B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

61 Multimodular order and shifts Let P(T, S) be the set of all integer sequences of size T that sum to S. D is the mesh induced from D over the hyperplane i a i = S. P(T, S) is the set of conjugacy classes w.r.t. cyclic permutations. B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

62 Multimodular order and shifts Let P(T, S) be the set of all integer sequences of size T that sum to S. D is the mesh induced from D over the hyperplane i a i = S. P(T, S) is the set of conjugacy classes w.r.t. cyclic permutations. Definition x mms y in P(T, S), if f (x) f (y) for all D -mm function f which is invariant by cyclic permutations. B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

63 Multimodular order and shifts Let P(T, S) be the set of all integer sequences of size T that sum to S. D is the mesh induced from D over the hyperplane i a i = S. P(T, S) is the set of conjugacy classes w.r.t. cyclic permutations. Definition x mms y in P(T, S), if f (x) f (y) for all D -mm function f which is invariant by cyclic permutations. Theorem The balanced word w(t, S) is minimal over P(T, S) for the mms-order. B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

64 Multimodular order and shifts Let P(T, S) be the set of all integer sequences of size T that sum to S. D is the mesh induced from D over the hyperplane i a i = S. P(T, S) is the set of conjugacy classes w.r.t. cyclic permutations. Definition x mms y in P(T, S), if f (x) f (y) for all D -mm function f which is invariant by cyclic permutations. Theorem The balanced word w(t, S) is minimal over P(T, S) for the mms-order. Corollary x mms y if the cone -distance to w(t, S) over mesh M D is larger for y. B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

65 Unbalance in P(T, S). Partial sums : k s (n) = s i + + s n Discrepancy : φ s (n) = k s (n) ns/t. Theorem If T, S are co-prime, among all cyclic permutations of s, only one verifies n, φ s (n) 0 and only one verifies n, φ s (n) 0, called s and s respectively. The discrepancy also induces orders over P(T, S) : Definition u g v if n, φ u (n) φ v (n). u g v if n, φ u (n) φ v (n). u g v if u g v and u g v. Theorem w = w(t, S) is minimal over P(T, S) for the orders g, g, g. B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

66 Unbalance(II) The upper unbalance of a sequence u in P(T, S) is I(u) = 1 T T k u (n) k w (n) n=1 The lower unbalance of a sequence u in P(T, S) is I(u) = 1 T T k u (n) k w (n) n= unbalance: 15/ B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

67 Links between the orders In Z n : u mm v Path u v away from r In P(T, S), Path u v away from w u mms v Path u v away from w u g u g u mms v B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

68 Links between the orders In Z n : u mm v Path u v away from r In P(T, S), Path u v away from w u mms v Path u v away from w u g u g u mms v Therefore, the g-order does not look very useful. However, the unbalance provides the good measure of the gap with an optimal sequence w.r.t. waiting times. B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

69 Unbalance and waiting times If W (u) is the long run average waiting time in a queue, where interarrival times and service times and iid sequences. then B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

70 Unbalance and waiting times If W (u) is the long run average waiting time in a queue, where interarrival times and service times and iid sequences. then Theorem W is D -mm over P(T, S) when D is the set of left shifts. W is invariant by cyclic permutations. B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

71 Unbalance and waiting times If W (u) is the long run average waiting time in a queue, where interarrival times and service times and iid sequences. then Theorem W is D -mm over P(T, S) when D is the set of left shifts. W is invariant by cyclic permutations. Therefore, u mms v implies W (u) W (v). B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

72 Unbalance and waiting times If W (u) is the long run average waiting time in a queue, where interarrival times and service times and iid sequences. then Theorem W is D -mm over P(T, S) when D is the set of left shifts. W is invariant by cyclic permutations. Therefore, u mms v implies W (u) W (v). The density of u P(T, S) is d = S/T. Theorem W (u) W (w) + E(δ) d I(u). Moreover, if the system is deterministic and δ = dσ, then W (u) = W (w) + δ d I(u). B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

73 Routing to parallel queues We consider all routing sequences a where the frequency of routing in queue i is d i. The admission sequence in queue i, u i = 1 i (a) is a binary sequence with density d i. B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

74 Routing to parallel queues We consider all routing sequences a where the frequency of routing in queue i is d i. The admission sequence in queue i, u i = 1 i (a) is a binary sequence with density d i. The average waiting time of all packets can be written as C(a) = d i W (u i ). B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

75 Routing to parallel queues We consider all routing sequences a where the frequency of routing in queue i is d i. The admission sequence in queue i, u i = 1 i (a) is a binary sequence with density d i. The average waiting time of all packets can be written as C(a) = d i W (u i ). Theorem Let L = n i=1 d iw (w(d i )), then L C(a) L + E(δ) n I(u i ). i=1 B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

76 Routing to parallel queues We consider all routing sequences a where the frequency of routing in queue i is d i. The admission sequence in queue i, u i = 1 i (a) is a binary sequence with density d i. The average waiting time of all packets can be written as C(a) = d i W (u i ). Theorem Let L = n i=1 d iw (w(d i )), then L C(a) L + E(δ) n I(u i ). i=1 Moreover, in the deterministic case, under load one, C(a) = L + E(δ) n i=1 I(u i). B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

77 Billiard sequences m = B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

78 Billiards and unbalance In dimension n, there exists an optimal starting point for rational billiard sequences in terms of unbalance Van der Laan, 2004 B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

79 Billiards and unbalance In dimension n, there exists an optimal starting point for rational billiard sequences in terms of unbalance Van der Laan, 2004 The optimal starting point can be computed using an integer program. The optimal unbalance of a rational billiard sequence is n/2 1. B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39

80 Billiards and unbalance In dimension n, there exists an optimal starting point for rational billiard sequences in terms of unbalance Van der Laan, 2004 The optimal starting point can be computed using an integer program. The optimal unbalance of a rational billiard sequence is n/2 1. This provides in polynomial time a routing sequence which cost is at most d i W (w(d i )) + E(δ)n/2 1. B. Gaujal (INRIA ) Regularity and Optimization SDA2, / 39