Improving Consolidation of Virtual Machines with Risk-aware Bandwidth Oversubscription in Compute Clouds

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Improving Consolidation of Virtual Machines with
Risk-aware Bandwidth Oversubscription in Compute
Clouds
Amir Epstein
Joint work with
David Breitgand
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Motivation

• Network Bandwidth is a critical Data Center
  resource
• Network Bandwidth may become a bottleneck for
  consolidation
• Accurate and efficient network bandwidth demand
  estimation is difficult
• Common practice fully provision for peak loads
• Consequences resource waste

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Full Provisioning VS. Multiplexing

• The aggregate demand of VMs may be much smaller
  than the sum of the maximum demand of each VM

?i maxt di(t) gtgt maxt ?i di(t)
Max(VM1)Max(VM2)110
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Full Provisioning VS. Multiplexing
Max(VM1VM2)71 lt Max(VM1)Max(VM2)110
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Statistical Multiplexing

• Consider each VM dynamic bandwidth demands as a
  random variable
• Consider the aggregate bandwidth demand which is
  a sum of the random variables representing VMs
  Bandwidth demands
• As the number of VMs increases
• The ratio between standard deviation of the
  aggregate bandwidth demand and the mean decreases

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Overcommit

• Cloud provider aims at improving cost-efficiency
• Overcommit resources using statistical
  multiplexing
• Our focus is bandwidth

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Stochastic Bin Packing Problem (SBP)

• SX1,, Xn Set of items
• Xi random variable representing the size
  (bandwidth demand) of item i
• p overflow probability
• Goal Partition the set S into the smallest
  number of subsets (bins) S1,,Sk such that
• p represents a probabilistic SLA / policy

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SBP with Normal Distribution

• We assume that each item i independently follows
  normal distribution N(µi ,si2) .
• When si,0, for all i, then Xi µi and the
  problem reduces to the classical bin packing
  problem
• The focus of this work is SBP with normal
  variables

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Related Work Bin Packing

• The problem is NP-hard
• Bin packing is hard to approximate to a factor
  better than 3/2 unless PNP.
• First Fit Decreasing (FFD) has asymptotic
  approximation ratio of 11/9 and (absolute)
  approximation ratio of 3/2.
• MFFD algorithm has asymptotic approximation
  ratio of 71/60.
• AFPTAS exists.
• Online bin packing
• First Fit (FF) has competitive ratio of 17/10.
• Best upper and lower bounds are 1.58899 and
  154014, respectively.

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Related Work Stochastic Bin Packing

• -approximation for SBP
  with Bernoulli
• variables Kleinberg
  et. al 1997
• SBP with Poisson, Exponential and Bernoulli
  variables Goel and Indik 1999
• PTAS exists for Poisson and exponential
  distributions.
• Quasi-PTAS exists for Bernoulli variables.
• These results relax bin capacity and overflow
  probability constraints by a factor 1e.
• - competitive
  algorithm for SBP with normal variables Wang et.
  al 2011

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Our Results

• 2-approximation algorithm for SBP with normal
  variables
• (2e)-competitive algorithm for online SBP with
  normal variables
• Observe the existence of a dual PTAS for SBP
  with normal variables.

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Definitions

• Definition The effective load of bin j is
• where and the quantile
  function is the inverse function of the
  CDF ? of N(0,1).
• Observation A packing is feasible for a given
  overflow probability p iff for every bin j,
• 
  
• The load of bin j is normally distributed with
  mean and variance

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Simple solution approach

• Reduce the problem to the classical bin packing
  problem with item sizes
• , thus
• A feasible solution to the classical bin packing
  problem is a feasible solution SBP, since
• 
  
• The optimum for the classical bin packing
  instance with the new sizes may be significantly
  larger than the optimum for SBP.

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Effective Size

• 
  
• Thus, the effective size of item i on bin j can
  be viewed as

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Approximation Algorithm

• Algorithm 1 First Fit VMR decreasing
• Order the items in non-increasing order of VMR
• Place the next item in the first bin into which
  it can be feasibly packed
• If no such bin exists, open a new bin to pack
  this item
• Variance to Mean Ratio (VMR) is

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Approximation Algorithm

• Theorem 1 Algorithm 1 is a 2-approximation
  algorithm for SBP with normal variables.

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Integer Program for SBP

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Mathematical Program Relaxation

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Fractional Algorithm (Algorithm
2)

• Order the items in non-increasing order of VMR
• Place the next item in the bin with remaining
  capacity. If the item causes an overflow to the
  bin, assign maximum fraction of this item to the
  bin. Then, open a new bin to pack the remaining
  part of this item.
• Variance to Mean Ratio (VMR) is
  

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Analysis

• Lemma There exists a feasible solution to the MP
  with the following property. For any pair of
  items k,l and a pair of bins iltj, if xkjgt0 and
  xligt0, then dl dk.
• Observation Fractional algorithm produces a
  feasible fractional solution to the MP.
• This implies that collocating items with high VMR
  (bursy) minimizes the total effective size of the
  items
• Variance to Mean Ratio (VMR) is

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Proof Outline

• Consider a feasible solution to the MP with
  lexicographically maximal standard deviation
  (STD) vector of the bins S(S1,,Sm), where
• Assume by contradiction that the items are not
  packed into the bins according to non-increasing
  order of VMR
• Thus, there exists at least one pair of items
  that are not placed in this order (i.e., item
  with smaller VMR is packed to a bin with smaller
  index than the other item).
• We show that we can exchange fractions of these
  items between the bins, such that
• the new solution is feasible
• The STD vector of the bins in the solution is
  lexicographically greater than the one in the
  original solution
• Contradiction

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Online Algorithm

• VMR
• Let
• Class 0
• Class 1kC
• Class C1

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Online Algorithm

• Algorithm 3
• Classify next item according to the VMR classes
• Place the next item in the first bin of its class
  into which it can be feasibly packed
• If no such bin exists, open a new bin to pack
  this item
• Theorem 2 Algorithm 3 is a (2O(e))-approximation
  algorithm for SBP with normal variables.

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Simulation Study

• Compare our proposed algorithms to previous
  reported ones
• Data set
• Real trace from production data center used to
  compute mean and standard deviation of bandwidth
  consumption of 6000 VMs over a few hours period.
• Synthetic traces with statistical properties
  similar to those of the real traces

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Algorithms

• Algorithms 1-3
• First Fit (FF) with deterministic item sizes
  µißsi
• First Fit Decreasing (FFD) with deterministic
  item sizes µißsi
• Group Packing (GP) Wang et. al 2011
• For the online algorithms (Algorithm 3 and Group
  Packing), we set e0.1.

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Real Instance
(Online)
(Approx.)
(L.B)
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Real Instance
(Approx.)
(Online)
(L.B)
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Real Instance
(Approx.)
(Online)
(L.B)
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Online Algorithms

• Large synthetic instances

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Summary

• We studied SBP under the assumption that virtual
  machines bandwidth demand obeys normal
  distribution
• We showed a 2-approximation algorithm
• We showed (2e)-competitive algorithm
• We observed the existence of a dual PTAS for SBP
• We studied the performance and applicability of
  our algorithms using synthetic and real data
• The performance evaluation showed that our
  proposed algorithms considerably reduce the
  number of bins compared to the best known
  algorithms for the problem