Fundamental rate delay tradeoffs in multipath routed and network coded networks John Walsh and Steven Weber Drexel University, Dept of ECE Philadelphia, PA 94 {jwalsh,sweber}@ecedrexeledu
IP networks subject packets to delay, reordering, loss Delay 3 2 3 2 3 2 2 3 2 3 Reordering 2 3 3 2 3 2 2 Loss We will focus on delay and reordering
are ill-suited to uncoded transfer of ordered content s, s N s s 2 s 3 s N Ordered media frames IP network: reordering and delay? Required network: No reordering Constant delay s, s N IN t s N s 3 s 2 s OUT IP network: Variable delays, possible reordering s, s N IN t s N s s 3 2 OUT 2
Competing costs of encoding rate and decoding delay Ordered media frames s s 2 s 3 s N Encoder Encoding rate x x 2 x M K Tx'd coded packets IP network: reordering and delay π t,, t M τ,, τ M x M x x 3 x 2 t M t t 3 t 2 t τ M τ 3 τ 2 τ Rx'd coded packets x π(m) x π(3) x π(2) x π() Decoder s N s 2 s t s N s N s 3 s 2 D 2 D s T s Rx'd Decoding media delays frames Encoding rate: cost of adding too much redundancy Decoding delay: cost of not adding enough redundancy 3
Outline Degraded broadcast channels Tradeoff between rate and delay Relaxation to a calculus of variations problem 4
Reordering and delay as a degraded broadcast channel received packets upon each reception y = (π(), x π() ) decoded frames Media source source frames s,, s N Encoder packets x,, x M reordering and delay p(t,, t M ) y 2 = (π(), x π(), π(2), x π(2) ) y h(ki ) = (π(), x π(),, π(h(k i )), x π(h(ki ))) y M = (π(), x π(),, π(m), x π(m) ) ŝ, ŝ 2,, ŝ ki ŝ, ŝ 2,, ŝ N Network Decoders Each receiver is the cumulative set of arrived packets at each arrival time The channel is degraded since y y M form a Markov chain Each packet is labeled, hence upon the first packet arrival the receiver knows (π(), x π() ) 5
Capacity region of the degraded broadcast channel y x y 2 u u 2 u M x y M y 2 y p(y x) y M Theorem (Cover 972, El Gamal 978): The capacity region of the degraded broadcast channel is the closure of the convex hull of the region R of rates satisfying R I(y ; u ) R 2 I(y 2 ; u 2 u ) R M I(y M ; x u,, u M ) for u u 2 u M x dummy rvs with bounded support 6
Capacity region of the permutation broadcast channel y = (π(), x π() ) y 2 = (π(), x π(), π(2), x π(2) ) x,, x M p(π) y k = (π(), x π(),, π(k), x π(k) ) y M = (π(), x π(),, π(m), x π(m) ) Proposition The capacity region of the degraded broadcast channel is the closure of the convex hull of the region R of rates satisfying R p(π) ( H(x π() ) H(x π() u ) ) π R k π p(π) ( H(x π(),, x π(k) u k ) H(x π(),, x π(k) u ) u k ) R M H(x π(),, x π(m) u M ) for u u 2 u M x dummy rvs with bounded support 7
Capacity region for M = 2 packet permutation BC R p =, p = /8, 7/8 p = /4, 3/4 p = 3/8, 5/8 p = /2 2 R 2 Packets arrive in order (, 2) wp p, or out of order (2, ) wp p Achievable (R, R 2 ) means we may be sure of receiving R K bits upon first arrival and R 2 K new bits upon second arrival R =, R 2 = always achievable by setting x = x 2 R =, R 2 = 2 always achievable by not coding at all When p = /2, increasing R by bit requires decreasing R 2 by 2 bits, since that bit must be added to both packets 8
Capacity region for two specific permutation BCs R R p = R + R 2 2 2R + R 2 2 p = 2 2 R 2 Uniform channel: if p(π) = /M! then R is given by M i= R i/i Single permutation channel: if p(π) = for some π then R is given by p i= R i p for p =,, M These two cases give lower and upper bounds on the region for general p(π) 9
Point to point random linear network coded networks t g N, y N s,, s N s,, s N g 2, y 2 g, y D Each node transmits a random linear combination of the combinations it has received Each packet contains both the linear combination (y k ) and the corresponding encoding vector (g k ) All N packets are decodable as soon as N linearly independent combinations are received Efficiency requires N be large potentially large delay
Random linear network coded networks with delay mitigating outer codes t s N s x s 2 s 3 x 2 g M, y M g 2, y 2 s x M s N K Delay-mitigating encoder Random linear network coded network g, y Outer delay mitigating code and inner random linear network coded network Redundancy of outer code permits frames to be decoded before receiving a full-rank matrix of encoding vectors Redundancy reduces code rate, but reduces decoding delay
Broadcast channel for a random linear network coded network y := (g, g T x) collective received packets upon each innovation packets x,, x M combining and delay p(g) Random linear network coded network y 2 := (g, g T x, g 2, g T 2 x) y h(ki ) = (g, g T x,, g h(ki ), g T h(k i ) x) y M := (g, g T x,, g M, g T M x) Proposition The capacity region of the degraded broadcast channel is the closure of the convex hull of the region R of rates satisfying R G G p(g ) (H(G x) H(G x u )) R k G k G k p(g k ) (H(G k x u k ) H(G k x u k )) R M H(x u M ) 2
Equal capacity regions for the uniform case x,, x M Uniform permutation channel y = (π(), x π() ) M i= R i i p(π)= M! y M = (π(), x π(),, π(m), x π(m) ) x,, x M Uniform random linear network coded network p(g)= G y = (g, g T x) y M = (g, g T x,, g M, g T M x) M i= R i i 3
Outline Degraded broadcast channels Tradeoff between rate and delay Relaxation to a calculus of variations problem 4
Defining rate and delay R M Actual decoding schedule y M τ M Decoding delays s N t Desired decoding schedule s N T s s N Bits per frame s 5 s 4 s 3 Guaranteed # of new bits in each arrival R 3 R 3 R 2 R y 3 y 2 y τ 3 τ 2 τ s 2 s D 2 D s 2 T s s NT s s 2 s R 2 R Rate: ρ(r) = ( M k= R k)/(nt s ) (bps) Packet arrival for frame decoding: g(i)=inf{n : n k= R k i j= s j } Delay: D(R) = N i= (τ g(i) it s ) + (total decoding delay violation) 5
Tradeoff between rate and delay ρ (d) = max {ρ(r) E[D(R)] d} R R Maximizing rate equivalent to maximizing the resolution of the source Total number of bits in frames is limited by the rate: N s i i= M k= R k NT s N s i ρ(r) i= Fundamental tension for reordering and delay: Costs less to get bits in later frames than earlier ones (2R + R 2 2) Getting bits later causes decoding delays, sufficiently high delay violates delay bound 6
Tradeoff between rate and delay is a combinatorial optimization problem ρ (d) = max {ρ({h(i)}) E[D({h(i)})] d} {h(i)} H where {h(i)} H is the set of all non-decreasing sequences from [N] [M] giving the number of packet receptions required to decode the first i frames Proof Decompose the optimization: max {ρ(r) E[D(R)] d} = max R R {h(i)} : E[D({h(i)})] d max ρ(r) R R : {g(i,r)}={h(i)} Evaluation of the sequence space will be prohibitively difficult for moderate to large M, N 7
Outline Degraded broadcast channels Tradeoff between rate and delay Relaxation to a calculus of variations problem 8
Relaxation of the integrality of the sequence space h(i) h(i)/m z[x] M = 3 2 2/3 2/3 /3 /3 2 3 4 N = 5 i /5 2/5 3/5 4/5 i/n /5 2/5 3/5 4/5 x Combinatorial optimization problem is over H, the set of all nondecreasing sequences {h(i)} from [N] [M] Relaxation of the (normalized) integrality constraint using a zero-order hold yields a piecewise continuous function z : [, ] [, ]: z[i/n] = h(i), i =,, N, M z[x] = N k= h(x) M k N x N k, x Optimization becomes a calculus of variations problem over all piecewise continuous z : [, ] [, ] 9
Sample result: constant arrival times (τ i = it s, N = M) 8 ρ ( d) M = 2 M = 8 z [x] d = 75 6 M = 2 6 4 2 d = 25 d = 75 d 4 2 d = 25 2 4 6 8 x Theorem Under constant arrival times, the optimal rate delay tradeoff and the optimal decoding deadline function are ρ ( d) = ( log ζ), z [x] = { ζ, w ζ w, w > ζ, ζ = M M d 2
Const arr times: relaxation accurate for moderate M Normalized Average Delay 9 8 7 6 5 4 3 Rate vs Avg Delay Tradeoff (K=,T s =) M=N=3 M=N=7 Asymptotic tradeoff 2 2 4 6 8 Normalized Rate 2
Sample result: Exp iid arrival times Asymptotic Rate vs Avg Delay Tradeoff (N=5,K=,λ=) 8 Normalized Rate 6 4 2 6 5 4 log (M) 3 2 2 8 6 4 Normalized Average Delay Significant delay bounds achievable at moderate rate costs 22
Practical delay mitigating codes Priority encoded transmission codes (Albanese, Blomer, Edmonds, Luby, Sudan, 996): time sharing MDS codes to grant different levels of error protection to different parts of a stream Our analysis helps the user identify the time sharing scheme and protection levels for a PET code to achieve the maximum rate subject to a delay bound Feedback (eg, ARQ, TCP) won t affect tradeoff, since feedback doesn t change the capacity region (El Gamal, 978) Packet loss can be integrated into the model (current work) 23