some trajectories in rn arising in statistics, scheduling ...€¦ · some trajectories in rn...
TRANSCRIPT
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Some trajectories in Rn arising instatistics, scheduling and queues
Yoni NazarathyThe University of Queensland
Presented at the Maths Seminar Series,QUT, Sep 11, 2015.
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Outline
1 Moment matching of truncated distributions
2 Cyclic queueing systems
3 Scheduling with linear slowdown
4 Overflow fluid buffer networks
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Moment matching of truncated distributionsJoint work with Benoit Liquet
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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
)
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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
)
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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
)
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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
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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
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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
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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)
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Cyclic queueing systemsJoint work with Matthieu Jonckheere and Leonardo Rojas-Nandayapa
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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
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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.
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Basic idea: Approximate the random trajectory with switched ODE
Hysteresis control
Hysteresis Control
Fixed cyclesFixed Cycles
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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
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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 →∞
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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
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0.0 0.5 1.0 1.5 2.0y
0.2
0.4
0.6
0.8
1.0N = 5
LimitFHyLHysteresis Control
PHXN H¥L�N £ y LFixed Cycles
PHXN H¥L�N £ y L
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0.0 0.5 1.0 1.5 2.0y
0.2
0.4
0.6
0.8
1.0N = 10
LimitFHyLHysteresis Control
PHXN H¥L�N £ y LFixed Cycles
PHXN H¥L�N £ y L
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0.0 0.5 1.0 1.5 2.0y
0.2
0.4
0.6
0.8
1.0N = 50
LimitFHyLHysteresis Control
PHXN H¥L�N £ y LFixed Cycles
PHXN H¥L�N £ y L
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0.0 0.5 1.0 1.5 2.0y
0.2
0.4
0.6
0.8
1.0N = 100
LimitFHyLHysteresis Control
PHXN H¥L�N £ y LFixed Cycles
PHXN H¥L�N £ y L
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0.0 0.5 1.0 1.5 2.0y
0.2
0.4
0.6
0.8
1.0N = 500
LimitFHyLHysteresis Control
PHXN H¥L�N £ y LFixed Cycles
PHXN H¥L�N £ y L
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Scheduling with linear slowdownJoint work with Liron Ravner
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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 )
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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 )
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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 )
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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 )
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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
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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
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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 versa
2 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
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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 search
3 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
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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 search
4 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
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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
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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 CPI
6 Combined global-local search heuristic
Not known if problem is NP–complete
24
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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
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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–complete24
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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)
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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
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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
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Overflow fluid buffer networksJoint work with Stijn Fleuren and Erjen Lefeber
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Problem: Modelling complex manufacturing systems
µ1
µ2
µ3
α1
α2
α3
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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
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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
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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
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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
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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
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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 .
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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
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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).
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