On the Memory Requirements of Block Interleaver for Batched Network - - PowerPoint PPT Presentation

On the Memory Requirements of Block Interleaver for Batched Network Codes

Hoover H. F. Yin, Ka Hei Ng, Xishi Wang, Qi Cao, and Lucien K. L. Ng

The Chinese University of Hong Kong

June, 2020

Hoover H. F. Yin (CUHK)

• Mem. Req. of Block Interleaver for BNC

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Background

Hoover H. F. Yin (CUHK)

• Mem. Req. of Block Interleaver for BNC

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Background

Packet loss

Hoover H. F. Yin (CUHK)

• Mem. Req. of Block Interleaver for BNC

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Background

Packet loss No feedback

Hoover H. F. Yin (CUHK)

• Mem. Req. of Block Interleaver for BNC

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Batched Network Coding

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• Mem. Req. of Block Interleaver for BNC

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Batched Network Coding

Hoover H. F. Yin (CUHK)

• Mem. Req. of Block Interleaver for BNC

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Batched Network Coding

Burst loss?

Hoover H. F. Yin (CUHK)

• Mem. Req. of Block Interleaver for BNC

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Batched Network Coding

Burst loss? Depth of interleaver = # of batches in a block

Hoover H. F. Yin (CUHK)

• Mem. Req. of Block Interleaver for BNC

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Block Interleaved Batched Network Coding

batches

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

blocks

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

interleaved blocks

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 Hoover H. F. Yin (CUHK)

• Mem. Req. of Block Interleaver for BNC

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Block Interleaved Batched Network Coding

batches

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

blocks

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

interleaved blocks

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

Reinterleave

Hoover H. F. Yin (CUHK)

• Mem. Req. of Block Interleaver for BNC

June, 2020 4 / 13

Block Interleaved Batched Network Coding

batches

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

blocks

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

interleaved blocks

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

Reinterleave Temporary storage for the batches

Hoover H. F. Yin (CUHK)

• Mem. Req. of Block Interleaver for BNC

June, 2020 4 / 13

Block Interleaved Batched Network Coding

batches

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

blocks

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

interleaved blocks

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

Reinterleave Temporary storage for the batches Lower bound on the memory requirement? Tight bound?

Hoover H. F. Yin (CUHK)

• Mem. Req. of Block Interleaver for BNC

June, 2020 4 / 13

Batch Buffer

T1 +L1 T1 +L1 T1 +L1 T1 +L1 T1 +L1 −L2 T2 −L2 T2 change in buffer size incoming blocks

• utgoing blocks

change in buffer size · · · · · ·

Hoover H. F. Yin (CUHK)

• Mem. Req. of Block Interleaver for BNC

June, 2020 5 / 13

Batch Buffer

T1 +L1 T1 +L1 T1 +L1 T1 +L1 T1 +L1 −L2 T2 −L2 T2 change in buffer size incoming blocks

• utgoing blocks

change in buffer size · · · · · ·

Enqueue event: put L1 batches (after deinterleaving) Dequeue event: fetch L2 batches (for recoding and reinterleaving)

Hoover H. F. Yin (CUHK)

• Mem. Req. of Block Interleaver for BNC

June, 2020 5 / 13

Batch Buffer

T1 +L1 T1 +L1 T1 +L1 T1 +L1 T1 +L1 −L2 T2 −L2 T2 change in buffer size incoming blocks

• utgoing blocks

change in buffer size · · · · · ·

Enqueue event: put L1 batches (after deinterleaving) Dequeue event: fetch L2 batches (for recoding and reinterleaving) More terminologies: Total delay Delay-bounded enqueuer Valid dequeuing scheme

Hoover H. F. Yin (CUHK)

• Mem. Req. of Block Interleaver for BNC

June, 2020 5 / 13

Batch Buffer

T1 +L1 T1 +L1 T1 +L1 T1 +L1 T1 +L1 −L2 T2 −L2 T2 change in buffer size incoming blocks

• utgoing blocks

change in buffer size · · · · · ·

Enqueue event: put L1 batches (after deinterleaving) Dequeue event: fetch L2 batches (for recoding and reinterleaving) More terminologies: Total delay Delay-bounded enqueuer Valid dequeuing scheme

Hoover H. F. Yin (CUHK)

• Mem. Req. of Block Interleaver for BNC

June, 2020 5 / 13

Batch Buffer

T1 +L1 T1 +L1 T1 +L1 T1 +L1 T1 +L1 −L2 T2 −L2 T2 change in buffer size incoming blocks

• utgoing blocks

change in buffer size · · · · · ·

Enqueue event: put L1 batches (after deinterleaving) Dequeue event: fetch L2 batches (for recoding and reinterleaving) More terminologies: Total delay Delay-bounded enqueuer Valid dequeuing scheme

Hoover H. F. Yin (CUHK)

• Mem. Req. of Block Interleaver for BNC

June, 2020 5 / 13

Buffer Sizes

Put or fetch first? +L1 −L2

Hoover H. F. Yin (CUHK)

• Mem. Req. of Block Interleaver for BNC

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Buffer Sizes

Put or fetch first? Put first +L1 −L2

Hoover H. F. Yin (CUHK)

• Mem. Req. of Block Interleaver for BNC

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Buffer Sizes

Put or fetch first? Put first Biqueue event +L1 −L2

Hoover H. F. Yin (CUHK)

• Mem. Req. of Block Interleaver for BNC

June, 2020 6 / 13

Buffer Sizes

Put or fetch first? Put first Biqueue event +L1 −L2 Before fetching: prior buffer size After fetching: buffer size

Hoover H. F. Yin (CUHK)

• Mem. Req. of Block Interleaver for BNC

June, 2020 6 / 13

Buffer Sizes

Put or fetch first? Put first Biqueue event +L1 −L2 Before fetching: prior buffer size

Memory required

After fetching: buffer size

Causality

Hoover H. F. Yin (CUHK)

• Mem. Req. of Block Interleaver for BNC

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Bounded Buffer Size and Finite Total Delay

Incoming rate > Outgoing rate ?

Hoover H. F. Yin (CUHK)

• Mem. Req. of Block Interleaver for BNC

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Bounded Buffer Size and Finite Total Delay

Incoming rate > Outgoing rate ? Outgoing rate > Incoming rate ?

Hoover H. F. Yin (CUHK)

• Mem. Req. of Block Interleaver for BNC

June, 2020 7 / 13

Bounded Buffer Size and Finite Total Delay

Incoming rate > Outgoing rate ? Outgoing rate > Incoming rate ? L1/T1 = L2/T2

Hoover H. F. Yin (CUHK)

• Mem. Req. of Block Interleaver for BNC

June, 2020 7 / 13

Bounded Buffer Size and Finite Total Delay

Incoming rate > Outgoing rate ? Outgoing rate > Incoming rate ? L1/T1 = L2/T2 T1 = L1ω and T2 = L2ω

Hoover H. F. Yin (CUHK)

• Mem. Req. of Block Interleaver for BNC

June, 2020 7 / 13

Bounded Buffer Size and Finite Total Delay

Incoming rate > Outgoing rate ? Outgoing rate > Incoming rate ? L1/T1 = L2/T2 T1 = L1ω and T2 = L2ω d timeslots = total delay d′ timeslots = total incoming delay

Hoover H. F. Yin (CUHK)

• Mem. Req. of Block Interleaver for BNC

June, 2020 7 / 13

Bounded Buffer Size and Finite Total Delay

Incoming rate > Outgoing rate ? Outgoing rate > Incoming rate ? L1/T1 = L2/T2 T1 = L1ω and T2 = L2ω d timeslots = total delay d′ timeslots = total incoming delay

Theorem

The minimum total delay among all valid dequeuing schemes is L1 + L2 − gcd(L1, L2) + d′ timeslots.

Hoover H. F. Yin (CUHK)

• Mem. Req. of Block Interleaver for BNC

June, 2020 7 / 13

Periodic Structure: L1 = 8, L2 = 6

in s1 s2 52 76 100 124 148 12 16 10 t prior buffer size s1 s2 Hoover H. F. Yin (CUHK)

• Mem. Req. of Block Interleaver for BNC

June, 2020 8 / 13

Periodic Structure: L1 = 8, L2 = 6

in s1 s2 52 76 100 124 148 12 16 10 t prior buffer size s1 s2

Exists scheme with periods Exists scheme achieving min total delay and lowest memory requirement

Hoover H. F. Yin (CUHK)

• Mem. Req. of Block Interleaver for BNC

June, 2020 8 / 13

Periodic Structure: L1 = 8, L2 = 6

in s1 s2 52 76 100 124 148 12 16 10 t prior buffer size s1 s2

Exists scheme with periods Exists scheme achieving min total delay and lowest memory requirement Same total delay

• =

⇒ same memory requirement

Hoover H. F. Yin (CUHK)

• Mem. Req. of Block Interleaver for BNC

June, 2020 8 / 13

Periodic Structure: L1 = 8, L2 = 6

in s3 s4 52 76 100 124 148 12 16 t prior buffer size s3 s4 Hoover H. F. Yin (CUHK)

• Mem. Req. of Block Interleaver for BNC

June, 2020 9 / 13

Periodic Structure: L1 = 8, L2 = 6

in s3 s4 52 76 100 124 148 12 16 t prior buffer size s3 s4

Exists scheme with no period and no biqueue event in a long run Lowest memory requirement

• =

⇒ min total delay

Hoover H. F. Yin (CUHK)

• Mem. Req. of Block Interleaver for BNC

June, 2020 9 / 13

Periodic Structure: L1 = 8, L2 = 6

in s3 s4 52 76 100 124 148 12 16 t prior buffer size s3 s4

Exists scheme with no period and no biqueue event in a long run Lowest memory requirement

• =

⇒ min total delay Exists period

• =

⇒ exists biqueue event in the period Peak is not always unique

Hoover H. F. Yin (CUHK)

• Mem. Req. of Block Interleaver for BNC

June, 2020 9 / 13

Periodic Structure: L1 = 8, L2 = 6

in s5 s6 52 76 100 124 148 12 14 16 2 t prior buffer size s5 s6 Hoover H. F. Yin (CUHK)

• Mem. Req. of Block Interleaver for BNC

June, 2020 10 / 13

Periodic Structure: L1 = 8, L2 = 6

in s5 s6 52 76 100 124 148 12 14 16 2 t prior buffer size s5 s6

Same total delay

• =

⇒ same lower bound on the memory requirement

Hoover H. F. Yin (CUHK)

• Mem. Req. of Block Interleaver for BNC

June, 2020 10 / 13

Lower Bound

d timeslots = total delay d′ timeslots = total incoming delay

Theorem

When there are finitely many delay blocks, the limit superior of the prior buffer size is d − d′ − (d − d′) mod gcd(L1, L2) batches. ✶

Hoover H. F. Yin (CUHK)

• Mem. Req. of Block Interleaver for BNC

June, 2020 11 / 13

Lower Bound

d timeslots = total delay d′ timeslots = total incoming delay

Theorem

When there are finitely many delay blocks, the limit superior of the prior buffer size is d − d′ − (d − d′) mod gcd(L1, L2) batches.

y x gcd(L1, L2) d − d′ xi

✶

Hoover H. F. Yin (CUHK)

• Mem. Req. of Block Interleaver for BNC

June, 2020 11 / 13

Lower Bound

d timeslots = total delay d′ timeslots = total incoming delay

Theorem

When there are finitely many delay blocks, the limit superior of the prior buffer size is d − d′ − (d − d′) mod gcd(L1, L2) batches.

y x gcd(L1, L2) d − d′ xi

Corollary

When there are infinitely many delay blocks, the limit superior of the prior buffer size is d − d′ − (d − d′) mod gcd(L1, L2) − gcd(L1, L2)✶gcd(L1,L2)Z(d − d′) batches.

Hoover H. F. Yin (CUHK)

• Mem. Req. of Block Interleaver for BNC

June, 2020 11 / 13

Lower Bound

Minimum total delay = L1 + L2 − gcd(L1, L2) + d′

Hoover H. F. Yin (CUHK)

• Mem. Req. of Block Interleaver for BNC

June, 2020 12 / 13

Lower Bound

Minimum total delay = L1 + L2 − gcd(L1, L2) + d′

Theorem

The memory requirement is lower bounded by L1 + L2 − gcd(L1, L2) batches.

Hoover H. F. Yin (CUHK)

• Mem. Req. of Block Interleaver for BNC

June, 2020 12 / 13

Lower Bound

Minimum total delay = L1 + L2 − gcd(L1, L2) + d′

Theorem

The memory requirement is lower bounded by L1 + L2 − gcd(L1, L2) batches. Natural Dequeuing Scheme (NDS): Dequeue once there are L2 batches in the buffer

Hoover H. F. Yin (CUHK)

• Mem. Req. of Block Interleaver for BNC

June, 2020 12 / 13

Lower Bound

Minimum total delay = L1 + L2 − gcd(L1, L2) + d′

Theorem

The memory requirement is lower bounded by L1 + L2 − gcd(L1, L2) batches. Natural Dequeuing Scheme (NDS): Dequeue once there are L2 batches in the buffer

Theorem

The natural dequeuing scheme achieves the minimum total delay and lowest memory requirement.

Hoover H. F. Yin (CUHK)

• Mem. Req. of Block Interleaver for BNC

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∼END∼

Hoover H. F. Yin (CUHK)

• Mem. Req. of Block Interleaver for BNC

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