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How Useful is Old Information?. IEEE Transactions on Parallel and Distributed Systems, 1997. MICHAEL MITZENMACHER. On the Management and Efficiency of Cloud Based Services seminar. Alexander Zlotnik. November 2010 Computer Science faculty Technion.

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How Useful is Old Information?

IEEE Transactions on Parallel and Distributed Systems, 1997

MICHAEL MITZENMACHER

On the Management and Efficiency of Cloud Based Services seminar

Alexander Zlotnik

November 2010 Computer Science faculty Technion


Amazon Elastic Load Balancing

  • Share Incoming Traffic

  • Scale Up

  • Scale Down

  • Detect &shutdown unhealthy EC2 instances

  • Balance single/multiple Availability Zones

  • Reports

  • $0.025 per hour [$18/month] + Traffic


Background

  • What is this paper not

    • Real system

    • Scale up/down system

    • Groups/AZ balancing


Background

  • Supermarket model & benefits

    • On incoming task, poll d queues for their load

    • Send the task to the shorter queue

    • For d = 2 the service time is exponentially shorter

    • Further d = 3, 4, … the improvement is linear


Background

  • Load Balancing Types

    • Centralized/Distributed

    • Static/Dynamic(adaptive)


Setting

  • n nodes

  • Tasks Arrival: Poisson (λn)

  • Task is forwarded to a node for execution

  • Execution rate is distributed Exp(µ)

    • Normalized to µ=1

  • Every T time units a bulletin board is updated with the current loads of all nodes


Policy

  • Look at drandom entries on the board, send the task to the shortest queue

  • d=1

    • M/M/1

  • d=n

    • Shortest queue

 Static


Periodic updates

0

T

2T

3T

Tt


Definitions

  • Pi,j(t) – Fraction of queues with posted load i, but have true load j

  • qi(t) - Rate of arrivals at a queue of size i

Pi,j(t)

True Load

Posted Load


Between board updates

True Load

Pi,j(t)

µ

µ

qi

qi

Posted Load


On board update

True Load

Pi,j(t)

0

0

Posted Load


  • bi (t) – Fraction of queues with load i posted

True Load

Pi,j(t)

Σ

Posted Load


  • bi (t) – Fraction of queues with posted load i

  • Arrival rate to a queue with posted load i:

  • Tasks arrival

  • Chance that d selected queues have load i or more

  • Chance that a queue of size i is selected

  • Same chance for all queues of size i


Fixed Cycle

  • Hope: Convergence to a fixed point

    • State that there is no “motivation” to exit from

  • Periodic Updates

    • “Jump” on bulletin board updates, t=kT

    • Fixed cycle: bi do not change

      • Or, for k≥k0, P(kT) = P(k0T)


Fixed Cycle

True Load

Pi,j(t)

Σ

Posted Load

Σ

If π((k-1)T)= π(kT), the next phase will be the same


Fixed Cycle – finding vector π

  • Method A

    Run the system until changes in π are small

  • Method B

    mi,j(T) – Probability for M/M/1 queue to start with i tasks and after time T to have j tasks

    Solving a system of equations:

  • Method A on truncated system of differential equations


Fixed Cycle – finding vector π

  • Method B

    mi,j(T) – Probability for M/M/1 queue to start with i tasks and after time T to have j tasks

    • Solving a system of equations:

Iterating

π – Iteration end

π – Iteration start

x

=

  • Until small changes in π


Fixed Cycle – finding vector π

  • Method B

    mi,j(T) – Probability for M/M/1 queue to start with i tasks and after time T to have j tasks

    • Iterating a system of equations:

Bessel function of the first kind


Fixed Cycle – finding vector π

  • Method A on truncated system of differential equations

    • Bound I, J

    • Iterate on

    • Until small changes in π


Results


More complex Centralized Strategies

  • Time based

    • Split T into subintervals

    • At subinterval [tk,tk+1) tasks sent to random server with load at most k


More complex Centralized Strategies

  • Record-insert

    • Upon task arrival:

      • Sent to random server of servers with lowest load

      • Server’s load is incremented on bulleting board

      • Task ending not updated (until end of phase)

    • Uses real loads, not expectation

      • Always better than time-based


More complex Centralized Strategies


Continuous Update


Continuous Updates

  • Tasks benefit on occasional accurate info

    • Supported by other graphs too

  • Partial accurate info – not helpful

    • If not known which data is up to date


Individual Updates


Conclusions

  • Load Balancing is useful even with stale information

  • Choosing least loaded of 2 nodes is better than shortest or 3 or more

  • More complex policies can further help load balancing (like time-based or record-insert)


Open Questions

  • Additional metrics

  • Different theoretical frameworks

  • More realistic arrival patterns (heavy-tailed)


Personal review

  • Powerful theoretical framework

  • Interesting and relevant results

  • Cons

    • Too few loads scenarios

    • No comparison between the amount of needed bandwidth to load balancing benefit


Relevant Information

  • More references

    • Collection of works:

      • “The power of Two Random Choices: A Survey of Techniques and Results” M. Mitzenmacher et al 2001

  • Not covered in this presentation:

    • Deviation of simulating differential equations instead of full system [Sections 3.4, 3.5]

    • Theory of some systems, but their results are

    • Competitive Scenarios [Section 6]

    • Small number of servers


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