Application of Methods of Queuing Theory to Scheduling

in GRIDA 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 GRIDsThe services GRIDs offer to end users much resemble the services offered by telephone networks, the typical subject of study in Queuing TheoryThe 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 themselvesGRIDs are composed of smaller complicated systems

Computer hardwareNetworksSoftware

GRIDs are embedded into the larger complicated systems:

Scientific organizationsR&D activitiesGlobalization processes

Stopping Decomposition as Soon as Possible to Avoid Unnecessary Complexity

Demarcate the phenomena specific to scheduling in GRID, and the generic phenomenaModel complicated behavior of the components with probabilistic techniquesFind 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.

ImplicationsThere is no need to develop detailed models of computers, networks, software or interaction external to GRIDThere is no need to model the intra-GRID interaction, which does not directly affect schedulingInformation 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

ProbabilityPoisson ProcessMultivariate DistributionLinear ProgrammingConvergence “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 DemandsPoisson process of arrivals with intensity M classes of demands

Sub-Model of Scheduling ProcedureRecognizes distinct classes of demands and routes the demands to the nodes it chooses

SchedulerDemands Queues Nodes

Sub-Model: Structure

Sub-Model: Flow of DemandsDemands 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 iThe computing time is “incompressible”: processing two demands with complexities T1 and T2 at a particular node requires T1+T2 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).

11 p

mm p

1,11sp

mnmsp ,

jijsp ,

Classification Routing

Flow of Demands & Scheduling Procedure

Sub-Model: Scheduling Procedure

The GRID operates in a stable environmentRouting of any demand in each moment depends on the current state of the system only

For all nodes load i<1

The system can operate in the stationary modeThe stationary mode is stable

Lenth of Queueat node 1

Length of queueat node 2

Length of queueat node N

Probability of findingthe system in a

particular point ofphase space

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 < 1

j

jijiji sp ,,

1i

ji , jis ,ˆ

max

jp

OptimizationGoal

ModelParameters Result

Constraints

i

jis 1,

0, jis

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 stabilityLong-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 matrixs: NxM[0,1]: NxM[0,)i = si,j i,j pj

Marginals is a functionsi: RM[0,1]

: multivariate random value (in RM )i = E isi()

Handling Imperfect Information

Average values of i,j can be usedThe scheduling procedure should be iteratively re-evaluated when more information becomes availableIn 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 1i,j =1+

Trivial procedure distributes demands with equal probability to any nodeAn optimized procedure is obtained as shown

Scheduling: Trivial vs. OptimizedMaximum Throughput

Num. of Nodes

Optimized

Trivial

Conclusions

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

ReferencesL. Kleinrock, “Queueing Systems”, 1976Andrei Dorokhov, “Simulation simple models and comparison with queueing theory” http://csdl.computer.org/comp/proceedings/hpdc/2003/1965/00/19650034abs.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 systemsProf. A.N. Shiryaev for directing my work in the Theory of Probability