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.

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

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

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

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.

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.

Mathematical Concepts Involved

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

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

Sub-Model Structure

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

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)?.

Flow of Demands Scheduling Procedure

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

Stationary Mode

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

Optimization Problem

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

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

Approximating Actual Distribution of Demands with

A Discrete Distribution

A Better Approximation

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(?)

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

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

Scheduling Trivial vs. Optimized

Maximum Throughput

Optimized

Trivial

Num. of Nodes

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

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

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