Application of Methods of Queuing Theory to Scheduling in GRID

1
Application of Methods of Queuing Theory to
Scheduling in GRID

• A Queuing Theory-based mathematical model is
  presented, and an explicit form of the optimal
  control procedure obtained as the solution to
  the problem of maximizing the system throughput.

2
Why Queuing Theory?

• Indeed, there are queues in real GRIDs
• The services GRIDs offer to end users much
  resemble the services offered by telephone
  networks, the typical subject of study in Queuing
  Theory
• The complexity of the associated processes leaves
  little options but to use the probabilistic
  techniques

3
Complexity The Principal Limiting Factor to
Modeling

• GRIDs are very complicated systems themselves
• GRIDs are composed of smaller complicated systems
• Computer hardware
• Networks
• Software
• GRIDs are embedded into the larger complicated
  systems
• Scientific organizations
• RD activities
• Globalization processes

4
Stopping Decomposition as Soon as Possible to
Avoid Unnecessary Complexity

• Demarcate the phenomena specific to scheduling in
  GRID, and the generic phenomena
• Model complicated behavior of the components with
  probabilistic techniques
• Find the most general expression of the effects

5
Ultimate Stopper of Decomposition

• No entity in the modeled system should be
  decomposed, if the system persists when that
  entity is replaced with another similar one.

6
Implications

• There is no need to develop detailed models of
  computers, networks, software or interaction
  external to GRID
• There is no need to model the intra-GRID
  interaction, which does not directly affect
  scheduling
• Information about how long it will take to
  process a demand on each node is all we need to
  know about the demand.

7
Mathematical Concepts Involved

• Probability
• Poisson Process
• Multivariate Distribution
• Linear Programming
• Convergence By Law

8
Simplified Model

• There is a finite number of classes of demands
  (all demands from the same class have equal
  complexity)
• Sub-Model of Structure
• Set of N nodes with queues
• Sub-Model of Flow of Demands
• Poisson process of arrivals with intensity ?
• M classes of demands
• Sub-Model of Scheduling Procedure
• Recognizes distinct classes of demands and routes
  the demands to the nodes it chooses

9
Sub-Model Structure
10
Sub-Model Flow of Demands

• Demands from class j arrive with intensity ?j
  ?pj (?1 ?m ?)
• Upon arrival, a demand from class j is routed to
  node i with probability si,j
• A demand from class j requires ?i,j units of
  processing time, if routed to node i
• The computing time is incompressible
  processing two demands with complexities T1 and
  T2 at a particular node requires T1T2 time units
  independently of the order (or level of
  parallelism) in which they are processed

11
Two Important Facts About Poisson Processes

• Let X1 and X2 be independent Poisson processes
  with intensity ?1 and ?2.Then X1 X2 is a Poisson
  process with intensity ?1 ?2.
• Suppose a Poisson process X with intensity ? is
  split into X1 and X2. With probability p events
  are passed to X1 and otherwise to X2. Then X1 and
  X2 are Poisson processes with intensities p? and
  (1-p)?.

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Flow of Demands Scheduling Procedure
13
Sub-Model Scheduling Procedure

• The GRID operates in a stable environment
• Routing of any demand in each moment depends on
  the current state of the system only
• For all nodes load ?ilt1
• ?
• The system can operate in the stationary mode
• The stationary mode is stable

14
Stationary Mode
15
Implications of Stationary Operation

• Incoming demands of class j are routed to node i
  with stationary probability si,j
• Load of node i has the form
• ?i ? ? si,j ?i,j pj lt 1

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Optimization Problem
17
Linear Programming

• It is possible to rewrite the constraints in the
  folowing form
• ?i ? si,j ?i,j pj
• ?i ? ?
• ??min
• Now it is an LP problem

18
From Simplified to Real-World Model

• How to handle non-discrete distributions of
  demands?
• How to handle errors in classification (imperfect
  information)?
• What about non-stationary modes?
• Short-term excesses are not fatal because of
  stability
• Long-term changes in distribution of demands can
  render the S.P. non-optimal

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Approximating Actual Distribution of Demands with
A Discrete Distribution
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A Better Approximation
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What Happens When M???

• Simplified
• s is a matrix
• s NxM?0,1
• ? NxM?0,?)
• ?i ? ? si,j ?i,j pj

• Marginal
• s is a function
• si RM?0,1
• ? multivariate random value (in RM )
• ?i ?E ?isi(?)

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Handling Imperfect Information

• Average values of ?i,j can be used
• The scheduling procedure should be iteratively
  re-evaluated when more information becomes
  available
• In the real world applications, the exact
  distribution of demands is unknown, but can be
  approximated from the history of the system
  operation

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A Comparison

• Let ? be an exponentially distributed random
  value with average 1
• ?i,j 1?
• Trivial procedure distributes demands with equal
  probability to any node
• An optimized procedure is obtained as shown

24
Scheduling Trivial vs. Optimized
Maximum Throughput
Optimized
Trivial
Num. of Nodes
25
Conclusions

• The exact upper bound of throughput for a given
  GRID can be estimated
• A scheduling procedure which achieves this limit
  can be constructed from a solution of an LP
  problem
• The optimal scheduling procedure should be
  non-deterministic
• Trivial and deterministic schedulers are
  generally unlikely to achieve the theoretical
  limit

26
References

• L. Kleinrock, Queueing Systems, 1976
• Andrei Dorokhov, Simulation simple models and
  comparison with queueing theory
  http//csdl.computer.org/comp/proceedings/hpdc/200
  3/1965/00/19650034abs.htm
• Atsuko Takefusa, Osamu Tatebe, Satoshi Matsuoka,
  Youhei Morita, Performance Analysis of
  Scheduling and Replication Algorithms on Grid
  Datafarm Architecture for High-Energy Physics
  Applications
• GNU Linear Programming Kit, http//www.fsf.org

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My Special Thanks To

• Dr. V.A. Ilyin for directing my work in the field
  of GRID systems
• Prof. A.N. Shiryaev for directing my work in the
  Theory of Probability