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Perspectives on Network Calculus No Free Lunch but Still Good Value - - PowerPoint PPT Presentation

Apr 17, 2023 141 likes •472 views

ACM Sigcomm 2012 Perspectives on Network Calculus No Free Lunch but Still Good Value Florin Ciucu Jens Schmitt T-Labs / TU Berlin TU Kaiserslautern Outline Network Calculus (NC): A Theory for System Performance Analysis Classic

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Perspectives on Network Calculus – No Free Lunch but Still Good Value

ACM Sigcomm 2012

Jens Schmitt

TU Kaiserslautern

Florin Ciucu

T-Labs / TU Berlin

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Outline

  • Network Calculus (NC): A Theory for System Performance Analysis
  • Classic Queueing Theory
  • NC for Bellcore Traces
  • NC Key Concepts: Envelopes + Service Processes
  • Bounds Tightness
  • Conclusions

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The Problem. System Performance Analysis

  • Examples

− The system: a network, a data center, the power grid − The resources: bandwidth, processors, batteries − The load: bits, jobs, energy demand/supply − The performance: reliable transmission, completion time, matching

  • Problem formulations

− Load + resources  performance − Load + performance  resources

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Load System

(with resources)

Output (Input) Performance?

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Case Study

  • Smart Grid context …

Problem1: given the descriptions of both energy supply (wind + PV panels) and energy demand find the battery size such that …

1Wang/Ciucu/Low/Lin, JSAC 2012

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Highly Variable Energy Supply/Demand

1Wang/Ciucu/Low/Lin, JSAC 2012

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Formalizing “the System”: A Queueing Model

  • Input

− statistical descriptions on the load and server, e.g., How do customers arrive? How quickly are they served? − other factors, e.g., queue size, scheduling

  • Output

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The Invention of Q. T. (A. K. Erlang, 1910’s)

Remote Village (Customers) Telephone Lines (Server) Regional Office

Problem: given the number of phones and a target probability for getting a busy tone, determine the number of required telephone lines.

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Erlang’s Fundamental Contributions

  • Modeling human activity: exponential distribution for both

− Inter-arrival calling times (… or Poisson arrival process) − Calls duration

  • Blocking probability formula (…)

− Still used nowadays − Yields “economies of scale” (# of lines << # of customers)

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  • Q. T. for the Internet. The Rise (60’s)
  • Packet switching technology: all flows share the available bandwidth

by interleaving packets

  • Raison d‟être: statistical multiplexing gain1

                  flow 1 for service support to needed Bandwidth N flows N for service support to needed Bandwidth

1Liebeherr et al., 2001

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Modeling Internet Traffic (60’s)

  • Alike the Telephone Network traffic

− Packet arrivals: Poisson process − Packet sizes: exponential

  • But … packets must change their size (?!) downstream
  • This convenient assumption was numerically justified, but …

it leads to incorrect scaling laws of, e.g., e2e delays1

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…

1Burchard/Liebeherr/Ciucu, ToN 2011

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Bellcore Ethernet Traces (90’s)

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  • Q. T. for the Internet. The Decline
  • A.k.a. the failure of Poisson modeling
  • Applying classical results to modern Internet traffic can be very

misleading

  • Old and new alternative models (MAPs, heavy-tailed, self-similar,

alpha-stable) and tools

− capture the exact scaling behavior, e.g., − but inaccurate in finite regimes, mostly restricted to single-queues − … few scheduling, and overly-sophisticated (mathematically)

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A Concrete Problem: Find the Delay for

  • … the arrivals in the first N bins of a Bellcore trace
  • … and the system/queueing scenario
  • Solution 1: Simulate 
  • Solution 2: Fit a traffic model + run an analytical tool

… but which model? (Poisson, MAP, fBM?)

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Deterministic Network Calculus (DNC) Solution

  • Some quick notation
  • Plot the (empirical) envelope
  • … and the service line
  • Delay = max. horizontal distance (black and blue)

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DNC Solution (contd.)

  • Alternative: Draw a linear envelope for
  • Delay = max. horizontal distance (green and blue)

  • Advantage: reuse of the “traffic model” (e.g., flows aggregation +

scheduling, multiple utilization levels) + delay computation

  • Drawback: delay computed as a bound (improvements by piecewise

linear envelopes)

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Network Calculus

  • (Rough) ideas

− The Load/Resources are modeled with bounds − Use of inequalities whenever exact derivations are difficult − Performance measures are (inevitably) derived in terms of bounds

  • Why?

− Very broad classes of Loads/Resources − Tractable, intuitive (e.g., easier to work with “envelopes/curves” than distributions)

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Load Performance System

(with resources)

bounds bounds bounds

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Deterministic Envelope

  • Recall notation
  • Classic Deterministic Envelope
  • Notes - the envelope is tangent to and not to
  • is a random process but is not

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Why Does it Work?

  • Reich‟s equation
  • Using the envelope definition
  • … one can immediately derive backlog bound, i.e.,

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Why Does it Work?

  • Reich‟s equation
  • Using the envelope definition
  • … one can immediately derive backlog bound, i.e.,

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Stochastic Envelopes

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Stochastic Envelopes

  • In the literature

SBB: Stochastically Bounded Burstiness

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Fitting SBB

  • Input: trace with bins
  • Output: find such that
  • Solution: fit an exponential to values

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Fitting S2BB

  • Input: trace with bins
  • Output: find such that
  • Solution: fit an exponential to values

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Fitting S3BB

  • Input: trace with bins
  • Output: find such that
  • Solution: fit an exponential to a single (!?) value

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A Note on S3BB

  • Note the equivalence with
  • Example: let be the i.i.d. occurrences of a dice
  • Observe that
  • For stationary and ergodic processes, S3BB is quasi-deterministic

non-random!

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Service Modeling in NC. An Analogy

  • Consider a constant-rate server

… then according to Lindley‟s equation

  • Consider a linear and time invariant (LTI) system

… then there exists impulse-response s.t. Input System Output

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Service Process and Scheduling Abstraction

  • Consider the following system (from the perspective of )

… which is generally not (min,+) linear

  • NC transforms it to a „somewhat looking‟ (min,+) linear

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Service Processes and Convolution-Form Networks

  • Consider a concatenation of systems with known service processes
  • NC transforms it to a single system

… where is the (min,+) convolution of the others

  • This transformation proved to be quite hard

…

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On the Bounds Tightness

  • Myth: The “bounds” are not tight
  • DNC bounds

− Tight (they can happen) except for multi-node/multi-flow case − What about IntServ? The bounds almost surely don‟t happen…

  • SNC bounds

− Tight (but only if the right probabilistic methods are used) − … often that‟s not the case

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  • Problem: Find such that the delay for the aggregate input is .
  • With DNC

DNC vs. SNC Bounds

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…

SNC

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Conclusions

  • Sophisticated randomness of modern systems loads 

traditional tools have difficulties to predict system performance

  • (Stochastic) network calculus as an alternative

− Although mathematically less involved than classic tools, SNC can deliver more − Price lies in the bounding approach (“it is easier to approximate”) − Much more intuitive than classical QT

  • Why care about?

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QT NC

Problem space

Non-Poisson Multi-node Non-trivial scheduling ... but no TCP (yet)