Window Flow Control Systems with Random Service Alireza Shekaramiz Joint work with Prof. J¨ org Liebeherr and Prof. Almut Burchard April 6, 2016 1 / 20
Content Introduction 1 Related work 2 State-of-the-art 3 Results: Stochastic service analysis of feedback systems 4 Results: Variable bit rate server with feedback system 5 Results: Markov-modulated On-Off server with feedback system 6 Numerical results 7 2 / 20
Feedback system Feedback system: ≤ window throttle arrivals departures Network delay For the analysis we use network calculus methodology Network calculus has analyzed feedback systems under deterministic assumptions Open problem in network calculus Analysis of feedback systems with probabilistic assumptions 3 / 20
Related work Performance bonds for flow control protocols 1 - Deterministic analysis - Min-plus algebra - Window flow control model A min,+ system theory for constrained traffic regulation and dynamic service guarantees 2 - Deterministic analysis - Min-plus algebra - Window flow control model TCP is max-plus linear 3 - Deterministic service process - Max-plus algebra - TCP Tahoe and TCP Reno 1 R. Agrawal et al. “Performance bonds for flow control protocols”. In: IEEE/ACM Transactions on Networking 7.3 (1999), pp. 310–323. 2 C.-S Chang et al. “A min,+ system theory for constrained traffic regulation and dynamic service guarantees”. In: IEEE/ACM Transactions on Networking 10.6 (2002), pp. 805–817. 3 F. Baccelli and D. Hong. “TCP is max-plus linear and what it tells us on its throughput”. In: ACM SIGCOMM 30.4 (2000), pp. 219–230. 4 / 20
Related work TCP congestion avoidance 4 - Deterministic analysis - Min-plus algebra - Window flow control model - TCP Vegas and Fast TCP Window flow control in stochastic network calculus 5 - Stochastic analysis - Min-plus algebra - Window flow control model 4 M. Chen et al. “TCP congestion avoidance: A network calculus interpretation and performance improvements”. In: IEEE INFOCOM . vol. 2. 2005, pp. 914–925. 5 M. Beck and J. Schmitt. “Window flow control in stochastic network calculus - The general service case”. In: ACM VALUETOOLS . Jan. 2016. 5 / 20
Bivariate network calculus ( f ∧ g ) ( s , t ) = min { f ( s , t ) , g ( s , t ) } ( f ⊗ g ) ( s , t ) = min s ≤ τ ≤ t { f ( s , τ ) + g ( τ, t ) } ( f ⊗ g )( s , t ) � = ( g ⊗ f )( s , t ) ( ∧ , ⊗ ) operations form a non-commutative dioid over non-negative non-decreasing bivariate functions discrete-time domain ( t = 0 , 1 , 2 , ... ) Sub-additive closure: ∞ f ∗ � δ ∧ f ∧ f (2) ∧ f (3) ∧ . . . = f ( n ) � n =0 where f ( n +1) = f ( n ) ⊗ f for n ≥ 1 , f (0) = δ , and f (1) = f 6 / 20
Moment-generating function network calculus 6 Moment-generating function of a random variable X : � e θ X � M X ( θ ) = E Moment-generating function of operations ⊗ and ⊘ : t � M f ⊗ g ( − θ, s , t ) ≤ M f ( − θ, s , τ ) M g ( − θ, τ, t ) τ = s s � M f ⊘ g ( θ, s , t ) ≤ M f ( θ, τ, t ) M g ( − θ, τ, s ) τ =0 � � S ( s , t ) ≤ S ε ( s , t ) For Pr ≤ ε , statistical service bound 1 � � S ε ( s , t ) = max log ε − log M S ( − θ, s , t ) θ θ> 0 6 M. Fidler. “An end-to-end probabilistic network calculus with moment generating functions”. In: IEEE IWQoS . 2006, pp. 261–270. 7 / 20
State-of-the-art: Window flow control S win ≤ window A 0 A D min S throttle arrivals departures Network D 0 δ + w A ′ = min � A , D ′ � A ′ − D = min { A , D + w } − D � w s ≥ t , δ + w ( s , t ) = ≤ D + w − D ∞ s < t = w D ′ = D ⊗ δ + w = D + w 8 / 20
State-of-the-art: Window flow control Delay element represent feedback delay: δ d ( s , t ) = δ ( s , t − d ) Equivalent feedback service: S ⊗ δ d ⊗ δ + w � ∗ ⊗ S � S win = A 0 A D min S A D S win ≡ D 0 δ + w δ d 9 / 20
Results: Exact result S win Feedback system with w > 0 , d ≥ 0 A 0 and with an additive service process A D min S t − 1 � D 0 S ( s , t ) = c k k = s δ + w δ d c k ’s are arbitrary sequence of non-negative random variables If feedback delay is one ( d = 1 ), t − 1 � S win ( s , t ) = min { c k , w } k = s 10 / 20
Results: Upper and lower bounds For the equivalent service process S win of a general feedback system with window size w > 0 , and feedback delay d ≥ 0 , we have Upper and lower bounds: � w � t − s S ′ � S ( s , t ) , � win ( s , t ) < S win ( s , t ) < min d S ′ win ( s , t ) is the equivalent service process of the feedback system with window size w ′ = w / d and feedback delay d ′ = 1 The lower bound corresponds to the exact result 11 / 20
Results: Equivalent service Feedback system with window size w > 0 and delay d ≥ 0 : S win A 0 D A min S D 0 δ + w δ d ⌈ t − s d ⌉ � n � � � � � � � S win ( s , t ) = min S ( τ i − 1 , τ i − d ) + S ( τ n , t ) + nw C n ( s , t ) n =0 i =1 where C n ( s , t ) is given as � ∀ i = 0 , . . . , n τ i − τ i − 1 ≥ d � s = τ o ≤ · · · ≤ τ n ≤ t � � C n ( s , t ) = 12 / 20
Results: Feedback system with VBR Variable Bit Rate (VBR) server t − 1 � S ( s , t ) = c k k = s where c k ’s are independent and identically distributed random variables For a feedback system with VBR server with window size w > 0 and delay d ≥ 0 : M c ( − θ ) d + de − θ w � ⌊ t − s d ⌋ � M S win ( − θ, s , t ) ≤ M c ( θ ) is the moment-generating function of c k , � � e θ c k M c ( θ ) = E 13 / 20
Results: Feedback system with MMOO Markov-modulated On-Off (MMOO) server operates in two states: ON (state 1): The server transmits a constant amount of P > 0 units of traffic per time slot, c k = P OFF (state 0): The server does not transmit, c k = 0 The MMOO server offers an additive service process t − 1 � S ( s , t ) = c k k = s For a feedback system with MMOO server with window size w > 0 and delay d ≥ 0 , if p 01 + p 10 < 1 : m + ( − θ ) d + de − θ w � ⌊ t − s d ⌋ � M S win ( − θ, s , t ) ≤ m + ( θ ) is the larger eigenvalue of the matrix � � � � p 00 p 01 1 0 L ( θ ) = e θ P 0 p 10 p 11 14 / 20
Numerical results: Statistical service bounds 1 � � S ε win ( s , t ) = max log ε − log M S win ( − θ, s , t ) θ θ> 0 MMOO server with VBR server with exponential c k p 00 = 0 . 2 , p 11 = 0 . 9 , P = 1 . 125 Mb S ε S ε win win d = 400 ms S ε Upper bound upper d = 400 ms w = 200 Mb S ε lower and S ε Lower bound and S ε win for d = 1 ms win for d = 1 ms 20 20 w = 200 Mb d = 100 ms d = 100 ms w = 50 Mb w = 50 Mb Service (Mb) Service (Mb) 15 15 d = 20 ms w = 10 Mb d = 10 ms 10 10 w = 5 Mb d = 20 ms w = 10 Mb d = 5 ms d = 10 ms w = 2 . 5 Mb 5 5 w = 5 Mb d = 5 ms d = 1 ms w = 2 . 5 Mb d = 1 ms w = 500 Kb w = 500 Kb 0 0 0 10 20 30 40 50 0 10 20 30 40 50 Time (ms) Time (ms) Average rate = 1 Gbps, time unit = 1 ms, w / d = 500 Mbps, ε = 10 − 6 15 / 20
Numerical results: Effective capacity t →∞ − 1 γ S win ( − θ ) = lim θ t log M S win ( − θ, 0 , t ) MMOO server with VBR server with exponential c k p 00 = 0 . 2 , p 11 = 0 . 9 , P = 1 . 125 Mb d = 100 ms Lower bounds for γ S win ( − θ ) Lower bounds of γ S win ( − θ ) w = 50 Mb 500 Lower bound Lower bound 500 Upper bound Upper bound d = 20 ms 450 w = 10 Mb E ff ective capacity (Mbps) E ff ective capacity (Mbps) 400 400 d = 10 ms w = 5 Mb 350 300 500 d = 10 ms 400 200 w = 5 Mb d = 5 ms d = 20 ms w = 2 . 5 Mb d = 1 ms 300 d = 1 ms w = 10 Mb w = 500 Kb d = 5 ms w = 500 Kb 100 w = 2 . 5 Mb d = 100 ms 200 w = 50 Mb 100 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 θ ( × 10 − 5 ) θ ( × 10 − 5 ) Average rate = 1 Gbps, w / d = 500 Mbps 16 / 20
Numerical results: Backlog and delay bounds t − 1 � A ( s , t ) = a k with exponential a k and average rate λ k = s Backlog bound Delay bound 40 250 ε = 10 − 9 ε = 10 − 9 ε = 10 − 6 35 ε = 10 − 6 ε = 10 − 3 ε = 10 − 3 200 Sim. ε = 10 − 6 Sim. ε = 10 − 6 30 Backlog bound (Mb) Delay bound (ms) 25 150 20 w/d = 500 Mbps 100 15 10 50 w/d = 100 Mbps w/d = 100 Mbps w/d = 500 Mbps 5 0 0 0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 400 Arrival rate λ (Mbps) Arrival rate λ (Mbps) Exponential VBR, time unit = 1 ms, feedback delay d = 1 ms 17 / 20
Conclusions S win A 0 D A min S D 0 δ + w δ d Results: Exact results Upper and lower service bounds Equivalent service of the feedback system Bounds for a feedback system with VBR server Bounds for a feedback system with MMOO server Backlog and delay bounds 18 / 20
Technical Report - July 2016 A. Shekaramiz, J. Liebeherr, and A. Burchard. Window Flow Control Systems with Random Service. arXiv:1507.04631, July 2015. 19 / 20
Thank you Q & A 20 / 20
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