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Delays in Packet Networks

Delays in Packet Networks. Packet Switch. Fixed-capacity links Variable delay due to waiting time in buffers Delay depends on Traffic Scheduling. Traffic Arrivals. Peak rate. Frame size. Mean rate. Frame number. First-In-First-Out. Static Priority (SP). Blind Multiplexing (BMux):

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Delays in Packet Networks

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  1. Delays in Packet Networks

  2. Packet Switch • Fixed-capacity links • Variable delay due to waiting time in buffers • Delay depends on • Traffic • Scheduling

  3. Traffic Arrivals Peak rate Frame size Mean rate Frame number

  4. First-In-First-Out

  5. Static Priority (SP) • Blind Multiplexing (BMux): All “other traffic” has higher priority

  6. Earliest Deadline First (EDF) Benchmark scheduling algorithm for meeting delay requirements

  7. Network

  8. Disclaimer • This talk makes a few simplifications

  9. Traffic Description • Traffic arrivals in time interval [s,t) is • Burstiness can be reduced by “shaping” traffic Cumulative arrivals A

  10. Shaped Arrivals Flow 1 . . . C Flow N Flows areshaped Buffered Link Regulated arrivals Traffic is shaped by an envelope such that: Popular envelope: “token bucket”

  11. What is the maximum number of shaped flows with delay requirements that can be put on a single buffered link? • Link capacity C • Each flows j has • arrival function Aj • envelope Ej • delay requirement dj

  12. Delay Analysis of Schedulers • Consider a link scheduler with rate C • Consider arrival from flow i at t with t+di: Arrivals from flow j Deadline of Tagged arrival Tagged arrival Limit (Scheduler Dependent) • Tagged arrival departs by if

  13. Delay Analysis of Schedulers Arrivals from flow j • with • FIFO: • Static Priority: • EDF:

  14. Schedulability Condition We have: Therefore: An arrival from class i never has a delay bound violation if Condition is tight, when Ej is concave

  15. Numerical Result (Sigmetrics 1995) C = 45 Mbps MPEG 1 traces: Lecture: d = 30 msec Movie (Jurassic Park): d = 50 msec EDF Static Priority (SP) Peak Rate strong effective envelopes Type 1 flows

  16. Expected case Probable worst-case Deterministic worst-case

  17. Worst-casebacklog Backlog Backlog Statistical Multiplexing Gain Worst-case arrivals Arrivals Flow 1 Flow 2 Flow 3 Time With statistical multiplexing Arrivals Flow 1 Flow 2 Flow 3 Time Backlog

  18. Statistical Multiplexing Gain Statistical multiplexing gain is the raison d’être for packet networks.

  19. What is the maximum number of flows with delay requirements that can be put on a buffered link and considering statistical multiplexing? • Arrivalsare random processes • Stationarity: is stationary random processes • Independence: Any two flows and are stochastically independent

  20. Envelopes for random arrivals Statistical envelope bounds arrival from flow j with high certainty • Statistical envelope : • Statistical sample path envelope : Statistical envelopes are non-random functions

  21. Arrivals from group of flows: with deterministic envelopes: with statistical envelopes: Aggregating arrivals

  22. Statistical envelope for group of indepenent (shaped) flows • Exploit independence and extract statistical multiplexing gain when calculating • For example, using the Chernoff Bound, we can obtain

  23. Statistical vs. Deterministic Envelope Envelopes (JSAC 2000) 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 statistical envelopes Type 1 flows

  24. Statistical vs. Deterministic Envelope Envelopes (JSAC 2000) Traffic rate at t = 50 msType 1 flows

  25. Deterministic Service Never a delay bound violation if: Scheduling Algorithms • Work-conserving scheduler that serves Q classes • Class-q has delay bound dq • D-scheduling algorithm . . . Scheduler Statistical Service Delay bound violation with if:

  26. Statistical multiplexing makes a big difference Scheduling has small impact Statistical Multiplexing vs. Scheduling (JSAC 2000) Example: MPEG videos with delay constraints at C= 622 Mbps Deterministic service vs. statistical service (e = 10-6) dterminator=100 ms dlamb=10 ms Thick lines: EDF SchedulingDashed lines: SP scheduling

  27. Peak rate effectivebandwidth Mean rate More interesting traffic types • So far: Traffic of each flow was shaped • Next: • On-Off traffic • Fraction Brownian Motion (FBM) traffic Approach: • Exploit literature on Effective Bandwidth • Derived for many traffic types

  28. Statistical Envelopes and Effective Bandwidth Effective Bandwidth (Kelly 1996) Given , an effective envelope is given by

  29. Different Traffic Types (ToN 2007) Comparisons of statistical service guarantees for different 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

  30. Delays on a path with multiple nodes: • Impact of Statistical Multiplexing • Role of Scheduling • How do delays scale with path length? • Does scheduling still matter in a large network?

  31. Deterministic Network Calculus (1/3) • Systems theory for networksin (min,+) algebra developed by Rene Cruz, C. S. Chang, JY LeBoudec (1990’s) • Service curve Scharacterizes node • Used to obtain worst-case bounds on delay and backlog

  32. Deterministic Network Calculus (2/3) • Worst-case view of • arrivals: • service : • Implies worst-case bounds • backlog: • delay : • (min,+) algebra operators • Convolution: • Deconvolution:

  33. S S network network D (t) D (t) D (t) D (t) network network network network A (t) A (t) A (t) A (t) network network network network Deterministic Network Calculus (3/3) • Main result: If describes the service at each node, then describes the service given by the network as a whole. Sender Receiver Sender Receiver S3 S1 S3 S2 S1 S2 S = S * S * S network 1 2 3 S = S * S * S network 1 2 3 Receiver Sender Receiver Sender

  34. Stochastic Network Calculus • Probabilistic view on arrivals and service • Statistical Sample Path Envelope • Statistical Service Curve • Results on performance bounds carry over, e.g.: • Backlog Bound

  35. Stochastic Network Calculus • Hard problem: Find so that • Technical difficulty: is a random variable!

  36. Statistical Network Service Curve (Sigmetrics 2005) • Notation: • Theorem: If are statistical service curves, then for any : • is a statistical network service curve with some finite violation probability.

  37. EBB model • Traffic with Exponentially Bounded Burstiness (EBB) • Sample path statistical envelope obtained via union bound

  38. Example: Scaling of Delay Bounds • Traffic is Markov Modulated On-Off Traffic (EBB model) • All links have capacity C • Same cross-traffic (not independent!) at each node • Through flow has lower priority:

  39. Example: Scaling of Delay Bounds • Two methods to compute delay bounds: • Add per-node bounds: Compute delay bounds at each node and sum up • Network service curve: Compute single-node delay bound with statistical network service curve

  40. Example: Scaling of Delay Bounds (Sigmetrics 2005) • C = 100 Mbps • Cross traffic = through traffic • e = 10-9 • Peak rate: P = 1.5 MbpsAverage rate: r = 0.15 Mbps • T= 1/m + 1/l = 10 msec • Addition of per-node bounds grows O(H3) • Network service curve bounds grow O(H log H)

  41. Result: Lower Bound on E2E Delay (ToN 2011) • M/M/1 queues with identical exponential service at each node Theorem: E2E delay satisfies for all Corollary: -quantile of satisfies

  42. Numerical examples • Tandem network without cross traffic • Node capacity: • Arrivals are compound Poisson process • Packets arrival rate: • Packet size: • Utilization:

  43. Upper and Lower Bounds on E2E Delays (ToN 2011)

  44. Superlinear Scaling of Network Delays • For traffic satisfying “Exponential Bounded Burstiness”, E2E delays follow a scaling law of • This is different than predicted by … worst-case analysis … networks satisfying “Kleinrock’s independence assumption”

  45. Back to scheduling … So far: Through traffic has lowest priority and gets leftover capacity  Leftover Service or Blind Multiplexing BMux C How do end-to-end delay bounds look like for different schedulers? Does link scheduling matter on long paths?

  46. Service curves vs. schedulers (JSAC 2011) • How well can a service curve describe a scheduler? • For schedulers considered earlier, the following is ideal: with indicator function and parameter

  47. Example: End-to-End Bounds • Traffic is Markov Modulated On-Off Traffic (EBB model) • Fixed capacity link

  48. Example: Deterministic E2E Delays (Infocom ‘11) • C = 100 Mbps • Peak rate: P = 1.5 MbpsAverage rate: r = 0.15 Mbps BMUX EDF(delay-tolerant) FIFO EDF(delay intolerant

  49. Example: Statistical E2E Delays (Infocom`11) • C = 100 Mbps • e = 10-9 • Peak rate: P = 1.5 MbpsAverage rate: r = 0.15 Mbps

  50. Example: Statistical Output Burstiness (Infocom ‘11) • C = 100 Mbps • e = 10-9 • Peak rate: P = 1.5 MbpsAverage rate: r = 0.15 Mbps

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