Recent Progress on a Statistical Network Calculus

1. Recent Progress on a Statistical Network Calculus Jorg Liebeherr Department of Computer Science University of Virginia

2. Collaborators Almut Burchard Robert Boorstyn Chaiwat Oottamakorn Stephen Patek Chengzhi Li

3. Contents • R. Boorstyn, A. Burchard, J. Liebeherr, C. Oottamakorn. “Statistical Service Assurances for Packet Scheduling Algorithms”, IEEE Journal on Selected Areas in Communications. Special Issue on Internet QoS, Vol. 18, No. 12, pp. 2651-2664, December 2000. • A. Burchard, J. Liebeherr, and S. D. Patek. “A Calculus for End–to–end Statistical Service Guarantees.” (2nd revised version), Technical Report, May 2002. • J. Liebeherr, A. Burchard, and S. D. Patek , “Statistical Per-Flow Service Bounds in a Network with Aggregate Provisioning”, Infocom 2003. • C. Li, A. Burchard, J. Liebeherr, “Calculus with Effective Bandwidth”, July 2002.

4. Service Guarantees Switch Receiver Sender • A deterministic service gives worst-case guarantees • Delay  d • A statistical service provides probabilistic guarantees • Pr[ Delay  d ]   or Pr[ Loss  l ]  

5. Multiplexing Gain Sources of multiplexing gain: • Traffic Conditioning (Policing, Shaping) • Scheduling • Statistical Multiplexing of Traffic

6. Scheduling • Scheduling algorithm determines the order in which traffic is transmitted

7. Expected case Probable worst-case Deterministic worst-case

8. Earliest-Deadline-First GPS Static Priority FCFS Peak Rate Multiple Buckets Token Bucket Peak Rate Allocation Deterministic service Statistical service Average Rate Designing Networks for Multiplexing Gain Scheduling Traffic Characterization/ Conditioning Service /Admission Control

9. Earliest-Deadline-First GPS Static Priority FCFS Token Bucket Multiple Buckets Peak Rate Peak Rate Allocation Deterministic service Statistical service Average Rate Designing Networks for Multiplexing Gain Scheduling Traffic Characterization/ Conditioning Service /Admission Control

10. Earliest-Deadline-First GPS Static Priority FCFS Peak Rate Token Bucket Peak Rate Allocation Statistical service Average Rate Designing Networks for Multiplexing Gain By now: The design space for determi-nistic guarantees is well understood. Scheduling Deterministic service Multiple Buckets Traffic Characterization/ Conditioning Service /Admission Control

11. Earliest-Deadline-First GPS Static Priority FCFS Peak Rate Token Bucket Peak Rate Allocation Deterministic service Statistical service Average Rate Designing Networks for Multiplexing Gain Still open: Is there an elegant framework to reason about statistical guarantees?  Statistical Network Calculus Scheduling ? Multiple Buckets Traffic Characterization/ Conditioning Service /Admission Control

12. 1985 1990 1995 2000 Related Work (small subset) Deterministic network calculusCruz, 1991 Effective bandwidth in network calculusChang 94 Effective Bandwidth: J. Hui ’88Guerin et.al. ’91Kelly `91Gibbens, Hunt `91 (min,+) algebra for det. networks: Agrawal et.al. 99Chang 98LeBoudec 98 ServiceCurvesCruz 95 • Motivation for our work on statistical network calculus: • Maintain elegance of deterministic calculus • (2) Exploit know-how of statistical multiplexing

13. Source Assumptions • Arrivals Aj(t,t+) are random processes • Deterministic Calculus: • (A1) Additivity: For any t1 < t2 < t3, we have: • (A2) Subadditive Bounds: Traffic Aj is constrained by a subadditive deterministic envelope A*j as follows • with

14. Source Assumptions Statistical Calculus: (A1) +(A2) (A3) Stationarity: The Aj are stationary random variables (A4) Independence: The Ai and Aj (ij) are stochastically independent (No assumptions on arrival distribution!)

15. Aggregating Arrivals Flow 1 . . . C Flow N Buffer with Scheduler Regulated arrivals Regulator Arrivals from multiple flows: Deterministic Calculus: Worst-case of multiple flows is sum of the worst-case of each flow

16. 2000 Aggregating Arrivals • Statistical Calculus: • To bound aggregate arrivals we define a function that is a bound on the sum of multiple flows with high probability “Effective Envelope” • Effective envelopes are non-random functions • effective envelope : • strong effective envelope :

17. Obtaining Effective Envelopes with with

18. Effective vs. Deterministic Envelope Envelopes A*=min (Pt, s+rt) Type 1 flows: P =1.5 Mbps r = .15 Mbps s =95400 bits Type 2 flows: P = 6 Mbps r = .15 Mbps s = 10345 bits Type 1 flows

19. Effective vs. Deterministic Envelope Envelopes Traffic rate at t = 50 msType 1 flows

20. Scheduling Algorithms Arrivals from class p • with • FIFO: • SP: • EDF:

21. 2000 Admission Control for Scheduling Algorithms with Deterministic Envelopes: with Effective Envelopes: with Strong Effective Envelopes:

22. Statistical multiplexing makes a big difference Scheduling has small impact Effective vs. Deterministic Envelope Envelope C= 45 Mbps, e = 10-6Delay bounds: Type 1: d1=100 ms, Type 2: d2=10 ms,

23. Effective Envelopes and Effective Bandwidth 2002 Effective Bandwidth (Kelly, Chang) Given a(s,t), an effective envelope is given by

24. Effective Envelopes and Effective Bandwidth Now, we can calculate statistical service guarantees for schedulers and traffic types Schedulers: SP- Static PriorityEDF – Earliest Deadline FirstGPS – Generalized Processor Sharing Traffic: Regulated – leaky bucketOn-Off – On-off sourceFBM – Fractional Brownian Motion C= 100 Mbps, e = 10-6

25. Statistical Network Calculus with Min-Plus Algebra D(t) A(t) S(t)

26. Convolution and Deconvolution operators • Convolution operation: • Deconvolution operation • Impulse function:

27. Service Curves (Cruz 1995) • A (minimum) service curve for a flow is a function S such that: • Examples: • Constant rate service curve: • Service curve with delay guarantees:

28. Network Calculus Main Results (Cruz, Chang, LeBoudec) • Output Envelope:is an envelope for the departures: • Backlog bound:is an upper bound for the backlog B • Delay bound:An upper bound for the delay is

29. Snet Network Service Curve(Cruz, Chang, LeBoudec) Traffic Conditioning S3 S1 Receiver S2 Sender Network Service Curve: If S1, S2 and S3 are service curves for a flow at nodes, then Snet = S1 * S2 * S3 is a service curve for the entire network.

30. 2001 Statistical Network Calculus A (minimum) service curve for a flow is a function S such that: A (minimum) effective service curve for a flow is a function S such that:

31. 2001 Statistical Network Calculus Theorems • Output Envelope:is an envelope for the departures: • Backlog bound:is an upper bound for the backlog • Delay bound:A probabilistic upper bound for the delay , i.e.,

32. Unfortunately, this network service is not very useful! 2002 Effective Network Service Curve Network Service Curve: If S1,, S2 , … SH , are effective service curves for a flow at nodes, then . A “good” network service curve can be obtained by working with a modified service curve definition

33. What is the cause of the problem with the network effective service curve? A1 D1 = A2 D2 S1, S2, Sender Receiver In the convolution the range [0,t] where the infimum is taken is a random variable that does not have an a priori bound.

34. Sender Receiver 2002 Statistical Per-Flow Service Bounds Service available to aggregate SC • Given: • Service guarantee to aggregate (SC ) is known • Total Traffic is known • What is a lower bound on the service seen by a single flow?

35. Sender Receiver 2002 Statistical Per-Flow Service Bounds Service available to aggregate SC Can show: is an effective service curve for a flow where is a strong effective envelope and is a probabilistic bound on the busy period

36. 2002 Number of flows that can be admitted Type 1 flows: Goal: probabilisticdelay bound d=10ms

37. Conclusions • Convergence of deterministic and statistical analysis with new constructs: • Effective envelopes • Effective service curves • Preserves much (but not all) of the deterministic calculus • Open issues: • So far: Often need bound on busy period or other bound on “relevant time scale”. • Many problems still open for multi-node calculus