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## Application of Methods of Queuing Theory to Scheduling in GRID

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Title: 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

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)?.

12
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

16
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)?
• Short-term excesses are not fatal because of
stability
• Long-term changes in distribution of demands can
render the S.P. non-optimal

19
Approximating Actual Distribution of Demands with
A Discrete Distribution
20
A Better Approximation
21
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(?)

22
Handling Imperfect Information
• Average values of ?i,j can be used
• The scheduling procedure should be iteratively
available
• In the real world applications, the exact
distribution of demands is unknown, but can be
approximated from the history of the system
operation

23
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

27
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