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Improved Queue-Size Scaling for Input-QueuedSwitches via Graph Factorization
Jiaming Xu
The Fuqua School of BusinessDuke University
Joint work withYuan Zhong (Chicago Booth)
Mostly OM Workshop, June 2, 2019
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Data Center Switches
Jiaming Xu (Duke) Queue-Size Scaling for Input-Queued Switches 2
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Data Center Switches
Switch
Inputs
OutputsHP Data Center Switch
Jiaming Xu (Duke) Queue-Size Scaling for Input-Queued Switches 3
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Input-Queued Switch
Output 1 Output 2
Input 1
Input 2
• n× n input-queued switch: n inputs and n outputs
• unit-sized packets
• n2 queues: (input, output) ↔ queue
Jiaming Xu (Duke) Queue-Size Scaling for Input-Queued Switches 4
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Input-Queued Switch
Output 1 Output 2
Input 1
Input 2
Matching constraints (2n resource constraints):
• each input can connect to at most one output
• each output can connect to at most one input
Jiaming Xu (Duke) Queue-Size Scaling for Input-Queued Switches 4
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Input-Queued Switch
Output 1 Output 2
Input 1
Input 2
Not allowed
0 10 1
!
"#
$
%&
Matching constraints (2n resource constraints):
• each input can connect to at most one output
• each output can connect to at most one input
Jiaming Xu (Duke) Queue-Size Scaling for Input-Queued Switches 4
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Input-Queued Switch
AllowedInput 1
Input 2
Output 1 Output 2
1 00 1
!
"#
$
%&
Matching constraints (2n resource constraints):
• each input can connect to at most one output
• each output can connect to at most one input
Jiaming Xu (Duke) Queue-Size Scaling for Input-Queued Switches 4
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Input-Queued Switch
Output 1 Output 2
Input 1
Input 21 00 1
!
"#
$
%&
Allowed
Matching constraints (2n resource constraints):
• each input can connect to at most one output
• each output can connect to at most one input
Jiaming Xu (Duke) Queue-Size Scaling for Input-Queued Switches 4
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Input-Queued Switch
Input 1
Input 2
Output 1 Output 2
0 11 0
!
"#
$
%&
Allowed
Matching constraints (2n resource constraints):
• each input can connect to at most one output
• each output can connect to at most one input
Jiaming Xu (Duke) Queue-Size Scaling for Input-Queued Switches 4
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Input-Queued Switch
Output 1 Output 2
Input 1
Input 20 11 0
!
"#
$
%&
Allowed
Matching constraints (2n resource constraints):
• each input can connect to at most one output
• each output can connect to at most one input
Jiaming Xu (Duke) Queue-Size Scaling for Input-Queued Switches 4
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Queueing Dynamics
Input 1
Input 2
Output 1 Output 2
Q = 2 34 2
!
"#
$
%&
Q12
• Independent Bernoulli arrivals with rate λij• Λ = [λij ] is admissible if∑
i
λij < 1 and∑j
λij < 1
• Focus on uniform arrival rates: λij = ρ/n and
ρ =∑i
λij =∑j
λij
Jiaming Xu (Duke) Queue-Size Scaling for Input-Queued Switches 5
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• Input-queued switches extensively studied
• Throughput and stability well understood
• Refiner metrics (moments of queue sizes/delay) less understood, butbecome increasingly important in the big data era
Focus of this talk:
How∑
ij E [Qij ] scales with n (large system) and 1− ρ (heavy traffic)?
Outline of the remainder
1 A universal lower bound
2 Previously best-known and our improved upper bounds
3 Our policy via batching + graph factorization
4 Summary and concluding remarks
Jiaming Xu (Duke) Queue-Size Scaling for Input-Queued Switches 6
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• Input-queued switches extensively studied
• Throughput and stability well understood
• Refiner metrics (moments of queue sizes/delay) less understood, butbecome increasingly important in the big data era
Focus of this talk:
How∑
ij E [Qij ] scales with n (large system) and 1− ρ (heavy traffic)?
Outline of the remainder
1 A universal lower bound
2 Previously best-known and our improved upper bounds
3 Our policy via batching + graph factorization
4 Summary and concluding remarks
Jiaming Xu (Duke) Queue-Size Scaling for Input-Queued Switches 6
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• Input-queued switches extensively studied
• Throughput and stability well understood
• Refiner metrics (moments of queue sizes/delay) less understood, butbecome increasingly important in the big data era
Focus of this talk:
How∑
ij E [Qij ] scales with n (large system) and 1− ρ (heavy traffic)?
Outline of the remainder
1 A universal lower bound
2 Previously best-known and our improved upper bounds
3 Our policy via batching + graph factorization
4 Summary and concluding remarks
Jiaming Xu (Duke) Queue-Size Scaling for Input-Queued Switches 6
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A Universal Lower Bound
Input 1
Input 2
Output 1 Output 2
𝜌
𝜌
• Decouples into n independent components• Expected total queue size scales as
n
1− ρ• A universal lower bound for any policy
Jiaming Xu (Duke) Queue-Size Scaling for Input-Queued Switches 7
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A Universal Lower Bound
𝜌
𝜌
1
1
• Decouples into n independent components
• Expected total queue size scales as
n
1− ρ
• A universal lower bound for any policy
Jiaming Xu (Duke) Queue-Size Scaling for Input-Queued Switches 7
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Previously Best-known Upper Bounds
nn2
n
11−ρ
Universal Lower bound: n1−ρ
Jiaming Xu (Duke) Queue-Size Scaling for Input-Queued Switches 8
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Previously Best-known Upper Bounds
n
n log n(1−ρ)2[NMC’07]
n2
n
11−ρ
Universal Lower bound: n1−ρ
Jiaming Xu (Duke) Queue-Size Scaling for Input-Queued Switches 8
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Previously Best-known Upper Bounds
n
n log n(1−ρ)2[NMC’07]
n2
n1−ρ
[SWZ’11]
n
11−ρ
Universal Lower bound: n1−ρ
Jiaming Xu (Duke) Queue-Size Scaling for Input-Queued Switches 8
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Previously Best-known Upper Bounds
n
n log n(1−ρ)2[NMC’07]
n2
n1−ρ
[SWZ’11]n1.5 log n
1−ρ[STZ’16]
n
11−ρ
Universal Lower bound: n1−ρ
Jiaming Xu (Duke) Queue-Size Scaling for Input-Queued Switches 8
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Our Improved Upper Bound
n
n log n(1−ρ)2
n2
n1−ρ
n1.5 log n1−ρ
n1.5 n
n
11−ρ
Improvements:
• 11−ρ < n: n logn
(1−ρ)2 −→n logn
(1−ρ)4/3
• 11−ρ = n: n2.5 log n −→ n7/3 log n
• n < 11−ρ ≤ n
1.5: n1.5 logn1−ρ −→ n logn
(1−ρ)4/3
Jiaming Xu (Duke) Queue-Size Scaling for Input-Queued Switches 9
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Our Improved Upper Bound
nn2
n1−ρ
n1.5 n
n1.5 logn1−ρ
n log n
(1−ρ)4/3
n
11−ρ
Improvements:
• 11−ρ < n: n logn
(1−ρ)2 −→n logn
(1−ρ)4/3
• 11−ρ = n: n2.5 log n −→ n7/3 log n
• n < 11−ρ ≤ n
1.5: n1.5 logn1−ρ −→ n logn
(1−ρ)4/3
Jiaming Xu (Duke) Queue-Size Scaling for Input-Queued Switches 9
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Main Theorem
Theorem (X. and Zhong ’19)
Consider an n× n input-queued switch for which the n2 arrival streamsform independent Bernoulli processes with a common arrival rate ρ/n,where ρ ∈ (0, 1). There exists a scheduling policy under which
E
n∑i,j=1
Qij(τ)
≤ c n
(1− ρ)4/3log
n
1− ρ, ∀τ ∈ N
Remarks
• A multiplicative factor 1(1−ρ)1/3 log
n1−ρ away from the lower bound
• Computational complexity per slot is at most polynomial in n
Jiaming Xu (Duke) Queue-Size Scaling for Input-Queued Switches 10
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Main Theorem
Theorem (X. and Zhong ’19)
Consider an n× n input-queued switch for which the n2 arrival streamsform independent Bernoulli processes with a common arrival rate ρ/n,where ρ ∈ (0, 1). There exists a scheduling policy under which
E
n∑i,j=1
Qij(τ)
≤ c n
(1− ρ)4/3log
n
1− ρ, ∀τ ∈ N
Remarks
• A multiplicative factor 1(1−ρ)1/3 log
n1−ρ away from the lower bound
• Computational complexity per slot is at most polynomial in n
Jiaming Xu (Duke) Queue-Size Scaling for Input-Queued Switches 10
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Key innovation: efficient scheduling via graph factorization
Question
Given a queue matrix q = (qij)ni,j=1, how to deplete the packets as much
as possible without wasting service opportunities?
k∗ = maxk,g
k
s.t.∑i
gij =∑j
gij = k
gij ≤ qij no service waste
gij ∈ N
aa
aa
• g can be viewed as a k-factor (spanning k-regular graph) of abipartite multigraph q
• A simple upper bound: k∗ ≤ min{mini
∑j qij , minj
∑i qij
}• Is the upper bound tight?
Jiaming Xu (Duke) Queue-Size Scaling for Input-Queued Switches 11
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Key innovation: efficient scheduling via graph factorization
Question
Given a queue matrix q = (qij)ni,j=1, how to deplete the packets as much
as possible without wasting service opportunities?
k∗ = maxk,g
k
s.t.∑i
gij =∑j
gij = k
gij ≤ qij no service waste
gij ∈ N
aa
aa
• g can be viewed as a k-factor (spanning k-regular graph) of abipartite multigraph q
• A simple upper bound: k∗ ≤ min{mini
∑j qij , minj
∑i qij
}• Is the upper bound tight?
Jiaming Xu (Duke) Queue-Size Scaling for Input-Queued Switches 11
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Key innovation: efficient scheduling via graph factorization
Question
Given a queue matrix q = (qij)ni,j=1, how to deplete the packets as much
as possible without wasting service opportunities?
k∗ = maxk,g
k
s.t.∑i
gij =∑j
gij = k
gij ≤ qij no service waste
gij ∈ N
aa
aa
• g can be viewed as a k-factor (spanning k-regular graph) of abipartite multigraph q
• A simple upper bound: k∗ ≤ min{mini
∑j qij , minj
∑i qij
}• Is the upper bound tight?
Jiaming Xu (Duke) Queue-Size Scaling for Input-Queued Switches 11
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Key innovation: efficient scheduling via graph factorization
Question
Given a queue matrix q = (qij)ni,j=1, how to deplete the packets as much
as possible without wasting service opportunities?
k∗ = maxk,g
k
s.t.∑i
gij =∑j
gij = k
gij ≤ qij no service waste
gij ∈ N
aa
aa
• g can be viewed as a k-factor (spanning k-regular graph) of abipartite multigraph q
• A simple upper bound: k∗ ≤ min{mini
∑j qij , minj
∑i qij
}• Is the upper bound tight?
Jiaming Xu (Duke) Queue-Size Scaling for Input-Queued Switches 11
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Key innovation: efficient scheduling via graph factorization
Question
Given a queue matrix q = (qij)ni,j=1, how to deplete the packets as much
as possible without wasting service opportunities?
k∗ = maxk,g
k
s.t.∑i
gij =∑j
gij = k
gij ≤ qij no service waste
gij ∈ N
aa
aa
• g can be viewed as a k-factor (spanning k-regular graph) of abipartite multigraph q
• A simple upper bound: k∗ ≤ min{mini
∑j qij , minj
∑i qij
}• Is the upper bound tight?
Jiaming Xu (Duke) Queue-Size Scaling for Input-Queued Switches 11
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Largest k-factor of random queue matrices (multigraphs)
Theorem (X. and Zhong ’19)
Let q = (qij)ni,j=1 be an n× n queue matrix with qij
i.i.d.∼ Binom(m, p).
With probability 1− n−16, q has a k-factor with
k ≥ pmn−√
304pmn log n.
• Matches the upper bound up to a constant factor:
k∗ ≤ min
mini
∑j
qij , minj
∑i
qij
≤ pmn−√pmn log n
• Proof based on Gale-Ryser Theorem (extension of max-flow min-cut)+ Large deviation analysis
Jiaming Xu (Duke) Queue-Size Scaling for Input-Queued Switches 12
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Largest k-factor of random queue matrices (multigraphs)
Theorem (X. and Zhong ’19)
Let q = (qij)ni,j=1 be an n× n queue matrix with qij
i.i.d.∼ Binom(m, p).
With probability 1− n−16, q has a k-factor with
k ≥ pmn−√
304pmn log n.
• Matches the upper bound up to a constant factor:
k∗ ≤ min
mini
∑j
qij , minj
∑i
qij
≤ pmn−√pmn log n
• Proof based on Gale-Ryser Theorem (extension of max-flow min-cut)+ Large deviation analysis
Jiaming Xu (Duke) Queue-Size Scaling for Input-Queued Switches 12
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A Standard Batching Policy [NMC’07]
0 T 2T 3T
Batch 2
Serve batch 1
Batch 1
• No. of arrivals to any input/output port in time T∼ Binom(nT, ρ/n)
• Max no. of arrivals to any input/output port in time T≈ ρT +
√T log n
• Finishing serving a batch in time T needs
T ≥ ρT +√T log n ⇐⇒ T ≥ log n
(1− ρ)2
• Expected total queue size ≈ nT ≥ n logn(1−ρ)2
Jiaming Xu (Duke) Queue-Size Scaling for Input-Queued Switches 13
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A Standard Batching Policy [NMC’07]
0 T 2T 3T
Batch 2
Serve batch 1
Batch 1
• No. of arrivals to any input/output port in time T∼ Binom(nT, ρ/n)
• Max no. of arrivals to any input/output port in time T≈ ρT +
√T log n
• Finishing serving a batch in time T needs
T ≥ ρT +√T log n ⇐⇒ T ≥ log n
(1− ρ)2
• Expected total queue size ≈ nT ≥ n logn(1−ρ)2
Jiaming Xu (Duke) Queue-Size Scaling for Input-Queued Switches 13
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A Standard Batching Policy [NMC’07]
0 T 2T 3T
Batch 2
Serve batch 1
Batch 1
• No. of arrivals to any input/output port in time T∼ Binom(nT, ρ/n)
• Max no. of arrivals to any input/output port in time T≈ ρT +
√T log n
• Finishing serving a batch in time T needs
T ≥ ρT +√T log n ⇐⇒ T ≥ log n
(1− ρ)2
• Expected total queue size ≈ nT ≥ n logn(1−ρ)2
Jiaming Xu (Duke) Queue-Size Scaling for Input-Queued Switches 13
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An Impatient Batching Policy [STZ’16]
0 TS
Round robin
T+S
Clear all pktsduring [0, T]
2T
Arrival batch 1 Arrival batch 2
Serve batch 1
• Start serving before the arrival of entire batch1 Wait for S time slots2 Simple round-robin for T − S time slots
• Need to ensure no waste of service opportunities during round-robin
T − Sn≤ ρT
n−√T log n
n⇐⇒ S ≥ (1− ρ)T +
√nT log n
• T � logn(1−ρ)2 ⇒ Expected total queue size ≈ nS ≥ n1.5 logn
1−ρ
Jiaming Xu (Duke) Queue-Size Scaling for Input-Queued Switches 14
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An Impatient Batching Policy [STZ’16]
0 TS
Round robin
T+S
Clear all pktsduring [0, T]
2T
Arrival batch 1 Arrival batch 2
Serve batch 1
• Start serving before the arrival of entire batch1 Wait for S time slots2 Simple round-robin for T − S time slots
• Need to ensure no waste of service opportunities during round-robin
T − Sn≤ ρT
n−√T log n
n⇐⇒ S ≥ (1− ρ)T +
√nT log n
• T � logn(1−ρ)2 ⇒ Expected total queue size ≈ nS ≥ n1.5 logn
1−ρ
Jiaming Xu (Duke) Queue-Size Scaling for Input-Queued Switches 14
![Page 37: Improved Queue-Size Scaling for Input-Queued Switches via ...jx77/jiaming_MostlyOM19.pdf · Switches via Graph Factorization Jiaming Xu The Fuqua School of Business Duke University](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b7b2de3036143c15977ba/html5/thumbnails/37.jpg)
Our Improved Batching Policy
𝐼" 𝐼#
𝐼#
𝐼$
𝐼$
Arrival Period of T𝐼ℓ
𝐼ℓ
𝐼ℓ&#⋯
⋯
Service PeriodFactorization Clearing
• Start serving even earlier1 Wait for I0 time slots2 Serve packets via factorization for T − I0 time slots:
Iu serves arrivals in Iu−1 for 1 ≤ u ≤ `
• To ensure no waste of service opportunities during factorizationIu ≤ ρIu−1 −
√Iu log n
I0 ≥ I1 ≥ · · · ≥ I` � I0I0 + I1 + · · ·+ I` = T
⇐⇒ I0 � T 2/3 log1/3 n
• T � logn(1−ρ)2 ⇒ Expected total queue size ≈ nI0 � n logn
(1−ρ)4/3
Jiaming Xu (Duke) Queue-Size Scaling for Input-Queued Switches 15
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Our Improved Batching Policy
𝐼" 𝐼#
𝐼#
𝐼$
𝐼$
Arrival Period of T𝐼ℓ
𝐼ℓ
𝐼ℓ&#⋯
⋯
Service PeriodFactorization Clearing
• Start serving even earlier1 Wait for I0 time slots2 Serve packets via factorization for T − I0 time slots:
Iu serves arrivals in Iu−1 for 1 ≤ u ≤ `
• To ensure no waste of service opportunities during factorizationIu ≤ ρIu−1 −
√Iu log n
I0 ≥ I1 ≥ · · · ≥ I` � I0I0 + I1 + · · ·+ I` = T
⇐⇒ I0 � T 2/3 log1/3 n
• T � logn(1−ρ)2 ⇒ Expected total queue size ≈ nI0 � n logn
(1−ρ)4/3
Jiaming Xu (Duke) Queue-Size Scaling for Input-Queued Switches 15
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Conclusion and remarks
nn2
n1−ρ
n1.5
n1.5 logn1−ρ
n log n
(1−ρ)4/3[X.-Zhong ’19]
n
11−ρ
1 Improved queue-size scalingsvia graph factorization
2 A tight characterization of thelargest k-factor in randombipartite multigraphs
Open problem
• Achieving the universal lower bound n1−ρ?
References
• X. & Yuan Zhong (2019). Improved Queue-Size Scaling forInput-Queued Switches via Graph Factorization, ACM SIGMETRICS2019, arXiv:1903.00398.
Jiaming Xu (Duke) Queue-Size Scaling for Input-Queued Switches 16
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Conclusion and remarks
nn2
n1−ρ
n1.5
n1.5 logn1−ρ
n log n
(1−ρ)4/3[X.-Zhong ’19]
n
11−ρ
1 Improved queue-size scalingsvia graph factorization
2 A tight characterization of thelargest k-factor in randombipartite multigraphs
Open problem
• Achieving the universal lower bound n1−ρ?
References
• X. & Yuan Zhong (2019). Improved Queue-Size Scaling forInput-Queued Switches via Graph Factorization, ACM SIGMETRICS2019, arXiv:1903.00398.
Jiaming Xu (Duke) Queue-Size Scaling for Input-Queued Switches 16