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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.
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
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
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
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
LimitFHyLHysteresis Control
PHXN H¥L�N £ y LFixed Cycles
PHXN H¥L�N £ y L
16
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
17
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
18
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
19
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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