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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
Data Center Switches
Jiaming Xu (Duke) Queue-Size Scaling for Input-Queued Switches 2
Data Center Switches
Switch
Inputs
OutputsHP Data Center Switch
Jiaming Xu (Duke) Queue-Size Scaling for Input-Queued Switches 3
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
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
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
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
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
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
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
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
• 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
• 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
• 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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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