Some trajectories in Rn arising in statistics, scheduling ...€¦ · Some trajectories in Rn arising in statistics, scheduling and queues Yoni Nazarathy The University of Queensland

Some trajectories in Rn arising instatistics, scheduling and queues

Yoni NazarathyThe University of Queensland

Presented at the Maths Seminar Series,QUT, Sep 11, 2015.

1

Outline

1 Moment matching of truncated distributions

2 Cyclic queueing systems

3 Scheduling with linear slowdown

4 Overflow fluid buffer networks

2

Moment matching of truncated distributionsJoint work with Benoit Liquet

3

Problem: Matching moments of truncated distributions

Moment matching∫x ig(x ; θ) dx = m∗i , i = 1, . . . , n

Truncated distributions

ga,b(x ; θ) =f (x ; θ)∫ b

a f (u ; θ) du

Equations for θ (given [a, b] and m∗) are “hard”

Exponential: Normal:

m∗1 = θ−1 (bθ+1)eaθ−(aθ+1)ebθ

eaθ−ebθ

m∗1 = θ1 − θ2

φ( b−θ1

θ2

)−φ

( a−θ1θ2

)Φ( b−θ1

θ2

)−Φ

( a−θ1θ2

)

m∗2 = θ2

1 + θ22 − θ2

(θ1+b)φ( b−θ1

θ2

)−(θ1+a)φ

( a−θ1θ2

)Φ( b−θ1

θ2

)−Φ

( a−θ1θ2

)

4




ga,b(x ; θ) =f (x ; θ)∫ b

a f (u ; θ) du




eaθ−ebθ

m∗1 = θ1 − θ2

φ( b−θ1

θ2

)−φ

( a−θ1θ2

)Φ( b−θ1

θ2

)−Φ

( a−θ1θ2

)

m∗2 = θ2

1 + θ22 − θ2

(θ1+b)φ( b−θ1

θ2

)−(θ1+a)φ

( a−θ1θ2

)Φ( b−θ1

θ2

)−Φ

( a−θ1θ2

)

4




ga,b(x ; θ) =f (x ; θ)∫ b

a f (u ; θ) du




eaθ−ebθ

m∗1 = θ1 − θ2

φ( b−θ1

θ2

)−φ

( a−θ1θ2

)Φ( b−θ1

θ2

)−Φ

( a−θ1θ2

)

m∗2 = θ2

1 + θ22 − θ2

(θ1+b)φ( b−θ1

θ2

)−(θ1+a)φ

( a−θ1θ2

)Φ( b−θ1

θ2

)−Φ

( a−θ1θ2

)

4

Idea

Start with support (−∞,∞) and truncate “bit by bit”

z ∈ (0, 1] is level of truncation, e.g.(a− 1− z

z, b +

1− z

z

)θ(z) is the solution for each z

Derive expression for F (·, ·) in the ODE:

d

dzθ(z) = F (θ(z), z)

In most cases θ(0+) has a simple closed form

Find the trajectory θ(z) numerically

5

f (x ; θ) = θ exp(−θx), m∗1 = 2.4, [a, b] = [0, 5]

0.0 0.2 0.4 0.6 0.8 1.0z

0.1

0.2

0.3

0.4

0.5

ΘHzL

7

11

2

5

6

f (x ; θ) = 1θ2√

2πexp

(− (x−θ1)2

2θ22

), m∗

1 = 0.1,√m∗

2 − (m∗1 )2 = 0.6, [a, b] = [−0.9, 1.35]

-0.4 -0.2 0.0 0.2 0.4Θ1

0.2

0.4

0.6

0.8

1.0

1.2Θ2

-1.15 1.6

-1.01 1.46

-0.95 1.4

-0.9 1.35

z=Ε

z=0.8

z=0.9

z=0.95

z=1

7

The ODE

d

dzθ(z) =

1

z2B(z , θ(z)

)−1c(z , θ(z)

)

ci (z , θ(z)) =((b +

1− z

z)i −m∗

i

)f(b +

1− z

z; θ(z)

)+((a− 1− z

z)i −m∗

i

)f(a− 1− z

z; θ(z)

)

Bi,j (z , θ(z)) =

∫ b+ 1−zz

a− 1−zz

(x i −m∗i )hj

(x , θ(z)

)dx

hj

(x ,(θ1, . . . , θn

))=

d

d θj

f (x ; θ1, . . . , θn)

8

Cyclic queueing systemsJoint work with Matthieu Jonckheere and Leonardo Rojas-Nandayapa

9

Problem: Performance of queues in cyclic environments

5 10 15 20t

10

20

30

40

Want to evaluate:

F (y) = limt→∞

1

t

∫ t

01{X (u) ≤ y} du

10

Two variations

T0=0 T1 T2 T3 T4

L1 L2 L1 L2 L1

Hysteresis control

Tn = inf{t > Tn−1 : X (t) =

{`2 for n odd,

`1 for n even.

}

Fixed cycles

Tn − Tn−1 =

{τ1 for n odd,

τ2 for n even.

11

Basic idea: Approximate the random trajectory with switched ODE

Hysteresis control

Hysteresis Control

Fixed cyclesFixed Cycles

12

Use the ODE to construct an approximation for F (·)

m2

{1

{2

m1

Τ1 Τ2

F (y) =1

τ1 + τ2

(τ1(y) + (τ2 − τ2(y)

)Hysteresis control: `1, `2 given ⇒ find τ1, τ2

Fixed Cycles: τ1, τ2 given ⇒ find `1, `2

x1∣∣(τ1)

x1(0)=`1

= `2, x2∣∣(τ2)

x2(0)=`2

= `1

13

Example: Infinite server case, ddtxi(t) = λi − µi xi(t)

ODE based approximation for F (·)

F (y) =

∫ y

−∞f (u)du, f (u) =

(µ1−µ2)u+(λ2−λ1)(µ1u−λ1)(µ2u−λ2)

log(µ1`1−λ1µ1`2−λ1

) 1µ1(µ2`2−λ2µ2`1−λ2

) 1µ2

1{`1 ≤ u ≤ `2}

For fixed cycles set: `i =(eτi µi−1)

λiµi

+(eτ

iµ

i−1)λ

iµ

ieτi µi

eτi µi +τ

iµ

i−1

Approximation becomes exact when accelerating the arrival rates:

λ(N)i = Nλi with N →∞

14

0.0 0.5 1.0 1.5 2.0y

0.2

0.4

0.6

0.8

1.0N = 1

LimitFHyLHysteresis Control

PHXN H¥L�N £ y LFixed Cycles

PHXN H¥L�N £ y L

15

0.0 0.5 1.0 1.5 2.0y

0.2

0.4

0.6

0.8

1.0N = 5




16

0.0 0.5 1.0 1.5 2.0y

0.2

0.4

0.6

0.8

1.0N = 10




17

0.0 0.5 1.0 1.5 2.0y

0.2

0.4

0.6

0.8

1.0N = 50




18

0.0 0.5 1.0 1.5 2.0y

0.2

0.4

0.6

0.8

1.0N = 100




19

0.0 0.5 1.0 1.5 2.0y

0.2

0.4

0.6

0.8

1.0N = 500




20

Scheduling with linear slowdownJoint work with Liron Ravner

21

Problem: Optimisation with rush hour traffic

Processor sharing scheduling for n users

1 =

∫ di

ai

v(q(t)

)dt q(t) =

n∑j=1

1{t ∈ [aj , dj ]}

Linear slowdown

v(q(t)

)= β − α

(q(t)− 1

)(assume n < β/α + 1)

Objective

mina∈Rn

c(a) =n∑

i=1

ci

(ai , di (a)

), ci (ai , di ) = (di−d∗i )2 +γ (di−ai )

22



1 =

∫ di

ai

v(q(t)

)dt q(t) =

n∑j=1

1{t ∈ [aj , dj ]}

Linear slowdown

v(q(t)

)= β − α

(q(t)− 1

)

(assume n < β/α + 1)

Objective

mina∈Rn

c(a) =n∑

i=1

ci

(ai , di (a)


22



1 =

∫ di

ai

v(q(t)

)dt q(t) =

n∑j=1

1{t ∈ [aj , dj ]}

Linear slowdown

v(q(t)

)= β − α

(q(t)− 1


Objective

mina∈Rn

c(a) =n∑

i=1

ci

(ai , di (a)


22



1 =

∫ di

ai

v(q(t)

)dt q(t) =

n∑j=1

1{t ∈ [aj , dj ]}

Linear slowdown

v(q(t)

)= β − α

(q(t)− 1


Objective

mina∈Rn

c(a) =n∑

i=1

ci

(ai , di (a)


22

Piecewise affine relationship between a and d

E.g. with n = 3, β = 1/2, α = 1/6:

t

worka1 = 0.00

a2 = 1.00 a3 = 3.00

d1 = 2.50 d2 = 3.75 d3 = 5.25

23

Key attributes and algorithms

An exponential number of convex quadratic programs

The objective function is piecewise quadratic with number ofregions equal to, (2n

n

)n + 1

∼ 4n

n3/2√π,

and with explicit expressions for describing each of the QPs.

Algorithms:

1 Calculate d based on a or vice versa2 Exhaustive search3 Neighbour search4 Efficient trajectory calculation for d(ai )5 Global search by means of CPI6 Combined global-local search heuristic

Not known if problem is NP–complete

24




n

)n + 1

∼ 4n

n3/2√π,


Algorithms:

1 Calculate d based on a or vice versa

2 Exhaustive search3 Neighbour search4 Efficient trajectory calculation for d(ai )5 Global search by means of CPI6 Combined global-local search heuristic


24




n

)n + 1

∼ 4n

n3/2√π,


Algorithms:

1 Calculate d based on a or vice versa2 Exhaustive search

3 Neighbour search4 Efficient trajectory calculation for d(ai )5 Global search by means of CPI6 Combined global-local search heuristic


24




n

)n + 1

∼ 4n

n3/2√π,


Algorithms:

1 Calculate d based on a or vice versa2 Exhaustive search3 Neighbour search

4 Efficient trajectory calculation for d(ai )5 Global search by means of CPI6 Combined global-local search heuristic


24




n

)n + 1

∼ 4n

n3/2√π,


Algorithms:

1 Calculate d based on a or vice versa2 Exhaustive search3 Neighbour search4 Efficient trajectory calculation for d(ai )

5 Global search by means of CPI6 Combined global-local search heuristic


24




n

)n + 1

∼ 4n

n3/2√π,


Algorithms:

1 Calculate d based on a or vice versa2 Exhaustive search3 Neighbour search4 Efficient trajectory calculation for d(ai )5 Global search by means of CPI

6 Combined global-local search heuristic


24




n

)n + 1

∼ 4n

n3/2√π,


Algorithms:



24




n

)n + 1

∼ 4n

n3/2√π,


Algorithms:


Not known if problem is NP–complete24

Trajectory of d when changing a1 (α = 1.5, β = 5 and d∗ = 0)

x = a1-0.5 0 0.5

0

1

a1

d1

a2

d2a3

d3

a4

d4

x = a1-0.5 0 0.51

2

∑ni=1 ci (a)

25

Optimal dynamics n = 15 (α = 0.8n

, β = 1 and d∗ quantiles of Normal(0, 14

) )

{di = •d∗i = ?

ai = •

-3 -2 -1 0 1 2 3

•••••••••• • • • • •

• •••••••••• • •• •???????????????γ = 0.1

{di = •d∗i = ?

ai = •

-3 -2 -1 0 1 2 3

•••• •• • • • • • • • • •

• ••• •• • • •• • • • • •???????????????γ = 1

{di = •d∗i = ?

ai = •

-3 -2 -1 0 1 2 3

• • • • • • • • • • • • • • •

• • • • • • • • • • • • • • •???????????????γ = 20

26

Heuristic dynamics n = 50 (α = 0.8n

, β = 1 and d∗ quantiles of Normal(0, 14

) )

{di = •d∗i = ?

ai = •

-3 -2 -1 0 1 2 3

••••••••••••••••••••••••••••••••••••••••••••••••• •

••••••••••••••••••••••••••••••••••••••••••••••••••??????????????????????????????????????????????????γ = 0.1

{di = •d∗i = ?

ai = •

-3 -2 -1 0 1 2 3

••••••••••••••••••••••••••••••••••••••••••••••••••

••••••••••••••••••••••••••••••••••••••••••••••••••??????????????????????????????????????????????????γ = 1

{di = •d∗i = ?

ai = •

-3 -2 -1 0 1 2 3

•• •• •• •••••••••••••••••••••••••••••• ••••• •••• •• • • •

•• •• •• •••••••••••••••••••••••••••••• ••••• •••• •• • • •??????????????????????????????????????????????????γ = 20

27

Overflow fluid buffer networksJoint work with Stijn Fleuren and Erjen Lefeber

28

Problem: Modelling complex manufacturing systems

µ1

µ2

µ3

α1

α2

α3

29

An overflow fluid buffer model

α1, . . . , αn – exogenous arrival rates

µ1, . . . , µn – service rates

P =(pi ,j

)– routing matrix (sub-stochastic or stochastic)

K1, . . . ,Kn – buffer sizes (can also be ∞)

Q =(qi ,j

)– overflow matrix (strictly sub-stochastic)

µ1

µ2

µ3

α1

α2

α3

30

λi : The effective arrival rate to buffer i

Basic Jackson:

λi = αi +n∑

j=1

λjpj ,i

Evolution of equations:

Basic Jackson, 1950’s: λ = α + λP

Goodman and Massey, 1984: λ = α + (λ ∧ µ)P

Our steady state flow equations: λ = α + (λ ∧ µ)P + (λ− µ)+Q

λi = αi +n∑

j=1

(λj ∧ µj )pj ,i +n∑

j=1

(λj − µj )+qj ,i

31


Basic Jackson:

λi = αi +n∑

j=1

λjpj ,i





λi = αi +n∑

j=1


j=1


31


Basic Jackson:

λi = αi +n∑

j=1

λjpj ,i





λi = αi +n∑

j=1


j=1


31


Basic Jackson:

λi = αi +n∑

j=1

λjpj ,i





λi = αi +n∑

j=1


j=1


31

Buffer trajectories, X (t) ∈ Rn

Modes

E(t) = {i : Xi (t) = 0} , F(t) := {i : Xi (t) = Ki}

Equations based on mode

λ = α + (λ ∧ µ)PE + µPE + (λ− µ)+QF

Trajectories

X (t) = X (0) +

∫ t

0∆(E(u), F(u)

)du

with,

∆i (E ,F) =

λi (E ,F)− λi (E ,F) ∧ µi , i ∈ E ,λi (E ,F)− µi , i 6∈ E , i 6∈ F ,λi (E ,F)− λi (E ,F) ∨ µi , i ∈ F .

32

33

Properties and results

Conditions for uniqueness and existence

Efficient algorithm for solving the equations

Trajectories are non-cycling

Example of an exponential number of mode changes

34

References

B. Liquet and Y. Nazarathy, “A dynamic view to moment matching oftruncated distributions”, Statistics and Probability Letters, 2015.

M. Jonckheere, Y. Nazarathy and L. Rojas-Nandayapa, “Scaling Approximationsfor Cyclic Queues”, in preparation (draft available upon demand).

L. Ravner and Y. Nazarathy, “Scheduling for a Processor Sharing System withLinear Slowdown”, arXiv preprint arXiv:1508.03136, 2015.

S. Fleuren, E. Lefeber and Y. Nazarathy, “Single Class Queueing Networks withOverflows: The Deterministic Fluid Case”, in preparation (draft available upondemand).

35

Documents

Some trajectories in Rn arising in statistics, scheduling ...€¦ · Some trajectories in Rn arising in statistics, scheduling and queues Yoni Nazarathy The University of Queensland