Novel Latency Bounds for Distributed Coded Storage Jean-Francois Chamberland Parimal Parag Electrical and Computer Engineering Texas A&M University Electrical Communication Engineering Indian Institute of Science Information Theory and Applications Feb 13, 2018 1/ 17
Building a Stronger Cloud Cloud Readiness Characteristics ◮ Network access and broadband ubiquity ◮ Download and upload speeds ◮ Delays experienced by users are due to high network and server latencies Reducing delay in delivering packets to and from the cloud is crucial to delivering advanced services 2/ 17
Inspirational Prior Work � � x x Redundancy- d Systems Power of 2 Choices ◮ FIFO; Info – d queues ◮ FIFO; Info – none ◮ 1 copy w/o feedback ◮ d copies w cancellation ◮ Exponential gain, d = 2 ◮ Exact queue distribution e.g.: Karp, Luby, Meyer auf der Heide, (1992); e.g.: Gardner, Zbarsky, Doroudi, Harchol-Balter, Adler, Chakrabarti, Mitzenmacher, Rasmussen (1995); Hyyti¨ a, Scheller-Wolf (2015); Vvedenskaya, Dobrushin, Karpelevich (1996); Gardner, Harchol-Balter, Scheller-Wolf, Velednitsky, Mitzenmacher (2001) Zbarsky (2016) 3/ 17
Duplication versus MDS Coding • • • • • • • • • • • • ∗ ∗ ∗ ∗ • • ⋆ ⋆ ⋆ ⋆ ⋆ ⋄ ⋄ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ Queueing Analysis Canonical Example ◮ Minimize expected delay ◮ Four servers ◮ MDS outperforms ◮ Two distinct pieces of Repetition information ◮ Elusive exact expression ◮ Find bounds e.g.: Joshi, Liu, Soljanin (2012, 2014), Shah, Lee, Ramchandran (2013), Joshi, Soljanin, Wornell (2015), Sun, Zheng, Koksal, Kim, Shroff (2015), Kadhe, Soljanin, Sprintson (2016), Li, Ramamoorthy, Srikant (2016) 4/ 17
Model Variations for Distributed Storage Centralized MDS Queue without Replication • • • • • ∗ ∗ ∗ ∗ ∗ ∗ � ⋄ ⋄ ⋄ ⋆ ⋄ ⋄ ⋆ ⋆ ⋆ ⋆ ⋆ Distributed ( n , k ) Fork-Join Model with MDS Coding • • • • ∗ ∗ ∗ ∗ ∗ � ⋄ ⋆ ⋄ ⋄ ⋄ ⋆ ⋆ ⋆ ⋆ e.g.: Lee, Shah, Huang, Ramchandran (2017) 5/ 17
Mean Sojourn Time Mean Sojourn Time for (9 , 3) Repetition Code Mean Sojourn Time for (9 , 3) MDS Code Block Service Block Service 10 1 . 5 10 1 . 5 QBD Reservation-3 QBD Reservation-3 Mean Sojourn Time W Upper Bound Mean Sojourn Time W Upper Bound Simulation Simulation Approximation Approximation 10 1 10 1 Lower Bound Lower Bound QBD Violation-3 QBD Violation-3 10 0 . 5 10 0 . 5 10 0 10 0 0 0 . 2 0 . 4 0 . 6 0 . 8 1 0 0 . 2 0 . 4 0 . 6 0 . 8 1 Arrival Rate λ (Load) Arrival Rate λ (Load) ◮ MDS coding significantly outperforms replication ◮ Bounding techniques are only meaningful under light loads ◮ Approximation is accurate over range of loads 6/ 17
Adopted Model: Priority Policy with MDS Coding • • • • ∗ ∗ ∗ ∗ ∗ • � ⋄ ⋄ ⋄ ⋄ ⋄ ⋆ ⋆ ⋆ ⋆ ⋆ Challenges Assumptions ◮ Intricate QBD Markov ◮ FIFO, k out of n copies process ◮ Information: global loads ◮ Infinite states in n ◮ Feedback: cancellation dimensions ◮ MDS or replication ◮ Tightly coupled transitions Parimal Parag, JFC (ITA 2013, ITA 2018), Parimal Parag, Archana Bura, JFC (ITA 2017, INFOCOM 2017) gratias: Kannan Ramchandran, Salim El Rouayheb 7/ 17
Establishing Lower and Upper Bounds • • • • • • • ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ MDS-Reservation( t ) MDS-Violation( t ) ◮ Restriction on depth of ◮ Unconstrained servers scheduler ◮ Equivalent to resource ◮ Reduces dimension of chain pooling without coding ◮ Upper bound on E [ T ] ◮ Lower bound on E [ T ] Shah, Lee, Ramchandran (2013), Lee, Shah, Huang, Ramchandran (2017) 8/ 17
Aggregate System – Level Abstraction A 2 C 2 A 2 A 2 A 0 A 0 A 0 A 0 0 1 2 3 Transition Operator C 1 C 2 0 0 0 · · · ◮ Block partitioning far A 0 A 1 A 2 0 0 · · · more important than 0 A 0 A 1 A 2 0 · · · entries of submatrices 0 0 A 0 A 1 A 2 · · · ◮ C 1 and C 2 account for . . . . . ... . . . . . . . . . . boundary conditions 9/ 17
Aggregate System – Stationary Distribution C 2 A 2 A 2 ∗ ∗ ∗ ∗ ∗ ∗ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋆ ⋆ ⋆ A 0 A 0 A 0 0 1 2 Chapman-Kolmogorov Equations Stationary distribution, denoted π = ( π 0 , π 1 , π 2 , . . . , ) with � � π q = Pr( s 1 , q ) , . . . , Pr( s k , q ) is unique solution to balance equations π q = π q − 1 A 2 + π q A 1 + π q + 1 A 0 10/ 17
The Cautionary Tale of Braess’s Paradox B w/ 80 min N / 100 min w/o 65 min 45 min Start Destination 45 min N = 4000 N / 100 min A “For each point of a road network, let there be given the number of cars starting from it and the destination of the cars. Under these conditions, one wishes to estimate the distribution of traffic flow. [...] If every driver takes the path that looks most favorable to them, the resultant running times need not be minimal. Furthermore, it is indicated by an example that an extension of the road network may cause a redistribution of the traffic that results in longer individual running times.” 11/ 17
Sample Path Failure of Eviction/Violation Bound Regular Distributed Coded Storage • • ∗ ∗ ⋄ ⋄ ⋆ ⋆ Eviction/Violation Lower Bound • • ∗ ∗ ⋄ ⋄ ⋆ ⋆ 12/ 17
Sample Path Failure of Eviction/Violation Bound Regular Distributed Coded Storage • • ∗ ∗ ⋄ ⋄ ⋆ ⋆ Eviction/Violation Lower Bound • • ∗ ∗ ⋄ ⋄ ⋆ ⋆ 12/ 17
Sample Path Failure of Eviction/Violation Bound Regular Distributed Coded Storage • • • ∗ ∗ ∗ ⋄ ⋄ ⋄ × + ⋆ ⋆ ⋆ Eviction/Violation Lower Bound • • • ∗ ∗ ∗ ⋄ ⋄ ⋄ × + ⋆ ⋆ ⋆ 12/ 17
Sample Path Failure of Eviction/Violation Bound Regular Distributed Coded Storage • • ∗ ∗ ∗ ⋄ ⋄ ⋄ ⋆ ⋆ ⋆ Eviction/Violation Lower Bound • • ∗ ∗ ∗ ⋄ ⋄ ⋄ ⋆ ⋆ ⋆ 12/ 17
Sample Path Failure of Eviction/Violation Bound Regular Distributed Coded Storage • • ∗ ∗ ∗ ⋄ ⋄ ⋄ ⋆ ⋆ ⋆ Eviction/Violation Lower Bound • • ∗ ∗ ∗ ⋄ ⋄ ⋄ ⋆ ⋆ ⋆ 12/ 17
Sample Path Failure of Eviction/Violation Bound Regular Distributed Coded Storage • • • • ∗ ∗ ∗ ∗ ∗ ⋄ ⋄ ⋄ ⋄ ⋄ × + ⋆ ⋆ ⋆ ⋆ ⋆ Eviction/Violation Lower Bound • • • • ∗ ∗ ∗ ∗ ∗ ⋄ ⋄ ⋄ ⋄ ⋄ × + ⋆ ⋆ ⋆ ⋆ ⋆ 12/ 17
Sample Path Failure of Eviction/Violation Bound Regular Distributed Coded Storage • • • ∗ ∗ ∗ ∗ ⋄ ⋄ ⋄ ⋄ ⋆ ⋆ ⋆ ⋆ Eviction/Violation Lower Bound • • • ∗ ∗ ∗ ∗ ⋄ ⋄ ⋄ ⋄ ⋆ ⋆ ⋆ ⋆ 12/ 17
Sample Path Failure of Eviction/Violation Bound Regular Distributed Coded Storage • • • • • ∗ ∗ ∗ ∗ ∗ ∗ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ × + ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ Eviction/Violation Lower Bound • • • ∗ ∗ ∗ ∗ ⋄ ⋄ ⋄ ⋄ ⋆ ⋆ ⋆ ⋆ 12/ 17
Sample Path Failure of Eviction/Violation Bound Regular Distributed Coded Storage • • • • ∗ ∗ ∗ ∗ ∗ ⋄ ⋄ ⋄ ⋄ ⋄ ⋆ ⋆ ⋆ ⋆ ⋆ Eviction/Violation Lower Bound • • • ∗ ∗ ∗ ∗ ⋄ ⋄ ⋄ ⋄ ⋆ ⋆ ⋆ ⋆ 12/ 17
Sample Path Failure of Eviction/Violation Bound Regular Distributed Coded Storage • • • • ∗ ∗ ∗ ∗ ∗ ⋄ ⋄ ⋄ ⋄ ⋄ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ Eviction/Violation Lower Bound • • • ∗ ∗ ∗ ∗ ⋄ ⋄ ⋄ ⋄ ⋆ ⋆ ⋆ ⋆ ⋆ 12/ 17
Sample Path Failure of Eviction/Violation Bound Regular Distributed Coded Storage • • • • ∗ ∗ ∗ ∗ ∗ ⋄ ⋄ ⋄ ⋄ ⋄ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ Eviction/Violation Lower Bound • • • ∗ ∗ ∗ ∗ ⋄ ⋄ ⋄ ⋄ ⋆ ⋆ ⋆ ⋆ ⋆ 12/ 17
Sample Path Failure of Eviction/Violation Bound Regular Distributed Coded Storage • • • • ∗ ∗ ∗ ∗ ∗ ⋄ ⋄ ⋄ ⋄ ⋄ ⋆ ⋆ ⋆ ⋆ ⋆ Eviction/Violation Lower Bound • • • ∗ ∗ ∗ ∗ ⋄ ⋄ ⋄ ⋄ ⋆ ⋆ ⋆ ⋆ 12/ 17
Sample Path Failure of Eviction/Violation Bound Regular Distributed Coded Storage • • • • ∗ ∗ ∗ ∗ ∗ ⋄ ⋄ ⋄ ⋄ ⋄ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ Eviction/Violation Lower Bound • • • ∗ ∗ ∗ ∗ ⋄ ⋄ ⋄ ⋄ ⋆ ⋆ ⋆ ⋆ ⋆ 12/ 17
Sample Path Failure of Eviction/Violation Bound Regular Distributed Coded Storage • • • • ∗ ∗ ∗ ∗ ∗ ⋆ ⋄ ⋄ ⋄ ⋄ ⋄ ⋆ ⋆ ⋆ ⋆ ⋆ Eviction/Violation Lower Bound • • • ∗ ∗ ∗ ∗ ⋆ ⋄ ⋄ ⋄ ⋄ ⋆ ⋆ ⋆ ⋆ 12/ 17
Sample Path Failure of Eviction/Violation Bound Regular Distributed Coded Storage • • • • ∗ ∗ ∗ ∗ ∗ ⋆ ⋄ ⋄ ⋄ ⋄ ⋄ ⋆ ⋆ ⋆ ⋆ Eviction/Violation Lower Bound • • • ∗ ∗ ∗ ∗ ⋆ ⋄ ⋄ ⋄ ⋄ ⋆ ⋆ ⋆ 12/ 17
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