Exact Analysis of TTL Cache Networks

Transcription

1 Exact Analysis of TTL Cache Networks Daniel S. Berger, Philipp Gland, Sahil Singla, and Florin Ciucu IFIP WG 7.3 Performance October 7, 2014

2 Classes of Caching

3 Classes of Caches capacity-driven eviction timer-based eviction Least-Recently-Used (LRU) First-In-First-Out (FIFO) Random Eviction (RND) Time-To-Live models TTL-R TTL-Σ TTL-min(Σ, R) Exact Analysis of TTL Cache Networks 1

4 Notation for a TTL Caching Node Exact Analysis of TTL Cache Networks 2 inter-arrival TTL inter-miss inter-arrival TTL cache model cache model { R, Σ, min(σ, R) } examples: G-G-R or M-D-Σ

5 Some TTL Policies Exact Analysis of TTL Cache Networks 3 Policy R: Reset TTL with every request. miss Evict object upon TTL expiry. Policy Σ: Reset TTL only at the times of miss unsuccessful requests (misses). Evict object upon TTL expiry. hit hit T 1 T 2 T 3 hit T 1 hit miss miss

6 Some TTL Policies Exact Analysis of TTL Cache Networks 3 Policy R: Reset TTL with every request. miss Evict object upon TTL expiry. Policy Σ: Reset TTL only at the times of miss unsuccessful requests (misses). Evict object upon TTL expiry. hit hit T 1 T 2 T 3 hit T 1 hit miss miss Policy min(σ, R): Two parallel TTLs, with R and Σ reset policies, respectively. Evict object upon expiry of either TTL.

7 Some TTL Policies Exact Analysis of TTL Cache Networks 3 Policy min(σ, R): Two parallel TTLs, with R and Σ reset policies, respectively. Evict object upon expiry of either TTL.

8 Some TTL Policies Exact Analysis of TTL Cache Networks 3 Policy min(σ, R): Two parallel TTLs, with R and Σ reset policies, respectively. Evict object upon expiry of either TTL.

9 Some TTL Policies Exact Analysis of TTL Cache Networks 3 Policy min(σ, R): Two parallel TTLs, with R and Σ reset policies, respectively. Evict object upon expiry of either TTL.

10 Some TTL Policies Exact Analysis of TTL Cache Networks 3 Policy min(σ, R): Two parallel TTLs, with R and Σ reset policies, respectively. Evict object upon expiry of either TTL.

11 LRU, FIFO, RND TTL Exact Analysis of TTL Cache Networks 4 plots omitted copyright [Martina et al., A unified approach to the performance analysis of caching systems, IEEE IN- FOCOM 2014] LRU M-D-R FIFO M-D-Σ RND M-M-Σ plots c [Martina et al, IEEE Infocom 2014]

12 Exact Analysis of TTL Cache Networks 5 Contribution of this work I) unifies the analysis of R and Σ [Choungmo Fofack et al., VALUETOOLS 2012 and 2013] II) first analysis of min(σ, R) III) first exact analysis of feedforward networks superposition input-output

13 Outline Exact Analysis of TTL Cache Networks 5 1 Introduction and Contribution 2 Analysis Line Networks (Renewal Request Model) Feedforward Networks (MAP Request Model) 3 Accuracy of the Poisson Approximation 4 Model Comparison and Conclusions

14 Line Networks Exact Analysis of TTL Cache Networks 6 {X t } t 1 and {T t } t 1 : independent renewal processes {X t } {?} {T t }

15 Line Networks Exact Analysis of TTL Cache Networks 6 {X t } t 1 and {T t } t 1 : independent renewal processes {X t } {?} {T t }

16 Stopping Time Formalization Exact Analysis of TTL Cache Networks 6 {X t } t 1 and {T t } t 1 : independent renewal processes the miss process: Sτ := X Xτ R : τ := min{t : X t > T t } hit hit miss X 1 X 2 X 3 T 1 T 2 T 3

17 Stopping Time Formalization Exact Analysis of TTL Cache Networks 6 {X t } t 1 and {T t } t 1 : independent renewal processes the miss process: Sτ := X Xτ R : τ := min{t : X t > T t } t Σ : τ := min{t : X s > T 1 } s=1 hit hit miss X 1 X 2 X 3 T 1

18 Stopping Time Formalization Exact Analysis of TTL Cache Networks 6 {X t } t 1 and {T t } t 1 : independent renewal processes the miss process: Sτ := X Xτ R : τ := min{t : X t > T t } t Σ : τ := min{t : X s > T 1 } min(σ, R) : τ := min { s=1 min{t : t X s > T1 Σ }, s=1 min{t : X t > T R t } }

19 Laplace Transform of Miss Process Exact Analysis of TTL Cache Networks 7 Find [ (LS τ )(ω) =E e ωs ] τ =E [e ω(x X τ ) ]

20 Laplace Transform of Miss Process Exact Analysis of TTL Cache Networks 7 Find [ (LS τ )(ω) =E e ωs ] τ =E [e ω(x X τ ) ] Challenge dependencies τ X i

21 Exact Analysis of TTL Cache Networks 8 (Fractional) Change of Measure P P Theorem [ t P e ωs t ] t (F ) := E L(X ) t 1 F F F t = σ((x 1, T 1 ),..., (X t, T t )) For an a.s. finite stopping time τ : (LS τ ) = Ẽ [ (LX ) τ ]

22 Exact Analysis of TTL Cache Networks 8 (Fractional) Change of Measure P P Theorem [ t P e ωs t ] t (F ) := E L(X ) t 1 F F F t = σ((x 1, T 1 ),..., (X t, T t )) For an a.s. finite stopping time τ : (LS τ ) = Ẽ [ (LX ) τ ] constant

23 Exact Analysis of TTL Cache Networks 8 (Fractional) Change of Measure P P Theorem [ t P e ωs t ] t (F ) := E L(X ) t 1 F F F t = σ((x 1, T 1 ),..., (X t, T t )) For an a.s. finite stopping time τ : (LS τ ) = Ẽ [ (LX ) τ ] P(τ = k) constant

24 From Renewals to MAPs Exact Analysis of TTL Cache Networks 9 renewals input-output input-output

25 From Renewals to MAPs Exact Analysis of TTL Cache Networks 9 renewals input-output input-output

26 From Renewals to MAPs Exact Analysis of TTL Cache Networks 9 renewals input-output input-output

27 From Renewals to MAPs Exact Analysis of TTL Cache Networks 9 renewals input-output superposition splitting input-output superposition splitting

28 From Renewals to MAPs Exact Analysis of TTL Cache Networks 9 renewals input-output superposition splitting input-output superposition splitting

29 From Renewals to MAPs Exact Analysis of TTL Cache Networks 9 renewals MAPs input-output superposition splitting input-output superposition splitting

30 From Renewals to MAPs Exact Analysis of TTL Cache Networks 9 renewals MAPs input-output superposition splitting input-output superposition splitting

31 From Renewals to MAPs Exact Analysis of TTL Cache Networks 9 renewals MAPs? input-output superposition splitting input-output superposition splitting

32 Markov Arrival Processes (MAPs) Exact Analysis of TTL Cache Networks 10 example MMPP in MAP representation 0, λ 1 0, λ 2 a, b, 0

33 Markov Arrival Processes (MAPs) Exact Analysis of TTL Cache Networks 10 example MMPP in MAP representation active transition 0, λ 1 0, λ 2 a, b, 0 passive transition

34 Exact Analysis of TTL Cache Networks 11 MMPP-M-Σ: Cache-Miss MAP arrival MAP: 0, λ 1 0, λ 2 a, 0 1 b, 0 2 phase-type TTL: 1 µ 0

35 Exact Analysis of TTL Cache Networks 11 MMPP-M-Σ: Cache-Miss MAP arrival MAP: cache-miss MAP: 0, λ 1 0, λ 2 a, 0 a, 0 C1 1 C b, 0 b, 0 0, λ 1 phase-type TTL: µ, 0 µ, 0 0, λ 2 1 µ a, 0 C1 0 C2 0 b, 0 0

36 MMPP-E 2 -Σ: Cache-Miss MAP Exact Analysis of TTL Cache Networks 12 phase-type TTL: cache-miss MAP: a, 0 2 C1 2 b, 0 C2 2 µ µ, 0 µ, 0 µ a, 0 1 C1 1 b, 0 C , λ 1 0, λ 2 µ, 0 a, 0 µ, 0 C1 0 C2 0 b, 0

37 General Theorem for MAP-PH-Σ Cache Exact Analysis of TTL Cache Networks 13 arrival MAP (D 0, D 1 ) phase-type TTL (S, π) with generator P the miss process is a MAP (D 0, D 1 ) given by D 0 =(P D 0 ) + 1 D 1 ( ) D 0n n πd 1 = 1 0 m n 0 n m similar for MAP-PH-R and MAP-PH-min(Σ, R)

38 Accuracy of the Poisson Approximation

39 Exact Analysis of TTL Cache Networks 14 Accuracy of the Poisson Approximation a cache s miss process Poisson relative error in a binary tree M-M-Σ at lowest level 1000 objects & Zipf(0.85)

40 Accuracy of the Poisson Approximation relative error [%] a cache s miss process Poisson relative error in a binary tree M-M-Σ at lowest level 1000 objects & Zipf(0.85) object rank # 1 object rank # level in cache hierarchy Exact Analysis of TTL Cache Networks

41 Summary and Conclusion

42 Conclusions Exact Analysis of TTL Cache Networks 15 three practical cache models R, Σ, min(σ,r) renewal request model computationally efficient limited to line networks MAP request model broadly applicable exponential space complexity numerical methods future work: performance bounds and tool support

43 Exact Analysis of TTL Cache Networks Daniel S. Berger, Philipp Gland, Sahil Singla, and Florin Ciucu IFIP WG 7.3 Performance October 7, 2014

44 Backup Slides

45 Model Comparison and Summary Exact Analysis of TTL Cache Networks 1 request model timer assumptions Renewals IID assumption renewals & deterministic MAPs dense in point processes phase-type ( deterministic) superposition approximations exact analysis splitting p-thinning p-thinning & Markovian type of solution explicit/ numerical Laplace inversion explicit (<100 states) numerical (<10e5 states)

46 Caching MAP: MMPP-E 2 -R Exact Analysis of TTL Cache Networks 2 phase-type TTL: caching MAP: a, 0 2 C1 2 b, 0 C2 2 µ λ 1, 0 λ 2, 0 µ 1 µ, 0 µ, 0 a, 0 C1 1 C2 1 b, 0 0 0, λ 1 0, λ 2 µ, 0 a, 0 µ, 0 C1 0 C2 0 b, 0

47 1 Introduction and Contribution 2 Analysis Line Networks (Renewal Request Model) Feedforward Networks (MAP Request Model) 3 Accuracy of the Poisson Approximation 4 Model Comparison and Conclusions