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 Sender ReceiverSwitch 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. Deterministic worst-case Expected case Probable worst- case

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

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

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

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

12. Related Work (small subset) 1985199019952000 Deterministic network calculus Cruz, 1991 (min,+) algebra for det. networks: Agrawal et.al. 99 Chang 98 LeBoudec 98 Effective bandwidth in network calculus Chang 94 Effective Bandwidth: J. Hui ’88 Guerin et.al. ’91 Kelly `91 Gibbens, Hunt `91 Motivation for our work on statistical network calculus: (1) Maintain elegance of deterministic calculus (2) Exploit know-how of statistical multiplexing Service Curves Cruz 95

13. Source Assumptions Arrivals A j (t,t+  ) are random processes Deterministic Calculus: (A1) Additivity: For any t 1 < t 2 < t 3, we have: (A2) Subadditive Bounds: Traffic A j is constrained by a subadditive deterministic envelope A * j as follows with

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

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

16. 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 : 2000

17. Obtaining Effective Envelopes with

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

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

20. Scheduling Algorithms Consider class- q arrival at t with t + d q : Tagged arrival has no delay bound violation if Class-p arrivals from class p which Are transmitted before tagged arrival. Arrivals from class p Tagged arrival Limit (Scheduler Dependent) Deadline of Tagged arrival Consider a work-conserving scheduler with rate R

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

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

23. Effective vs. Deterministic Envelope Envelope Statistical multiplexing makes a big difference Scheduling has small impact C= 45 Mbps,  = 10 -6 Delay bounds: Type 1: d 1 =100 ms, Type 2: d 2 =10 ms,

24. Effective Envelopes and Effective Bandwidth Effective Bandwidth (Kelly, Chang) Given  (s,  ), an effective envelope is given by 2002

25. Effective Envelopes and Effective Bandwidth Now, we can calculate statistical service guarantees for schedulers and traffic types Schedulers: SP- Static Priority EDF – Earliest Deadline First GPS – Generalized Processor Sharing Traffic: Regulated – leaky bucket On-Off – On-off source FBM – Fractional Brownian Motion C= 100 Mbps,  = 10 -6

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

27. Convolution operation: Deconvolution operation Impulse function: Convolution and Deconvolution operators

28. 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:

29. Network Calculus Main Results (Cruz, Chang, LeBoudec) 1.Output Envelope:is an envelope for the departures: 2.Backlog bound: is an upper bound for the backlog B 3.Delay bound: An upper bound for the delay is

30. Network Service Curve (Cruz, Chang, LeBoudec) Sender Receiver Traffic Conditioning S3S3 S1S1 S2S2 Network Service Curve: If S 1, S 2 and S 3 are service curves for a flow at nodes, then S net = S 1 * S 2 * S 3 is a service curve for the entire network. S net

31. 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: Statistical Network Calculus 2001

32. 1.Output Envelope: is an envelope for the departures: 2.Backlog bound: is an upper bound for the backlog 3.Delay bound: A probabilistic upper bound for the delay, i.e., Statistical Network Calculus Theorems 2001

33. Effective Network Service Curve Network Service Curve: If S 1, , S 2,  … S H,  are effective service curves for a flow at nodes, then. Unfortunately, this network service is not very useful! A “good” network service curve can be obtained by working with a modified service curve definition 2002

34. What is the cause of the problem with the network effective service curve? In the convolution the range [0,t] where the infimum is taken is a random variable that does not have an a priori bound. SenderReceiver S 2,  S 1,  D 1 = A 2 A1A1 D2D2

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

36. Statistical Per-Flow Service Bounds 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 2002 Service available to aggregate S C Sender Receiver

37. Effective service curve of a single flow Type 1 flows: Bandwidth needed by a per- flow allocation to meet a delay bound of d=10ms

38. Number of flows that can be admitted Type 1 flows: Goal: probabilistic delay bound d=10ms 2002

39. 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 Conclusions

40. Adaptive service curves Modified convolution operation

41. Adaptive service curves adaptive service curve: Many service curves are adaptive (  Cruz/Okino, LeBoudec) Obtain service curve with t 0 =0 l -adaptive service curve: l -adaptive effective service curve: strong (l -adaptive) effective service curve:

42. Effective Network Service Curve Sender Receiver Traffic Conditioning T3,l,T3,l, T1,l,T1,l, T2,l,T2,l, A* Network Service Curve: If T 1,l, , T 2,l, , and T 3,l, , are strong effective service curves for a flow at nodes, then T net,l, 3  = T 1,l,  * T 2,l,  * T 3,l,  is a service curve for the entire network. T net,l, 3 

43. Recover original effective setwork curve Given a strong effective service curve T l, . If the backlog clears in any time interval of length l with probability  1, i.e, Thenis an effective service curve