cloud computing modeling

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MODELING OF CLOUD

M/G/M QUEUES

HAMZEH KHAZAEI

UNIVERSITY OF MANITOBADEPARTMENT

OF COMPUTER SCIENCE

OCT 28, 2010


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2 /30Agenda

Introduction

Modeling of a Cloud Center

Performance Metrics

mu a on Results

Conclusion


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3 /30Introduction to Cloud Computing Cloud Computing, CC, is a computing paradigm, in

which diff. computing resources, such as,infrastructure, platforms and software applicationsare made accessible, over internet to remote user

.

Delivery as service: so QoS is essential.

QoS has multiple dimensions: response time,

throughput, availability, reliability, and security. Service Level Agreement, SLA: negotiated and

agreed btw customers and service providers


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4 /30Introduction

Our Contribution so far:

an ana ytica mo efor

performance evaluation


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5 /30High level schematic of CC

Cloud Centers: could be viewed as a single point of

access


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7 /30A little bit on M/G/m

So far, there is no exact and close form steady-

state solution for M/G/m queues. So approximation methods are sought.

,

useful for our purpose. (why?)

Most of them are accurate for small value of m, let

say less that 20. And almost all of them lead to reasonable results if

coefficient of variation, CV, is less than unity.

So we needed to develop our model


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8 /30A little bit on Stochastic Process


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9 /30Analytical Model

M/G/m queuing system is used for modeling.

Embedded Markov Chain, EMC is employed tomodel the system.

moments.

The points of arriving instants are selected as

Markov points. (they have Markov property)


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10 /30Analytical Model


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11 /30Markov Chain


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Formal Definitions

Due to ergodicity of the Markov chain, an

equilibrium probability distribution exists for thenumber of tasks in system at arrivals:

In other words, we need to solve following

equations:

In which and P is the one-steptransition matrix.

So immediate step would be finding matrix P.


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Formal Definitions

Cumulative Distribution Function, CDF, of Service

time & arrival: B(x) & A(x) Laplace-Stieltjes Transform, LST, of service time:

B* s

Mean service time: b=1/

We indicate remaining service time, as B+ and

elapsed service time as B-

It can be shown that both of them has the same

probability distribution function as well as LST:


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More details on the system behavior


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Formal Definitions

The probability of having first departure in a busy

server:

The probability of having first departure in an idle

server:

The probability of having k>=1 departures in a

busy server:


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Transition Probabilities


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Regarding region labeled (1): pij=0

Region (2):

Region (3):


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And finally region (4):


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Equilibrium Balance Equations

Now we have the balance equations:

or numer ca so u on we runca e a ance

equations:

Normalization equation:


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Distribution of no. of tasks in system

Now, we can establish PGF of number of tasks in

the system at arrival instant:

Based on PASTA, distribution of tasks in system at

arrival is identical with any arbitrary moment of

time:


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Performance metrics

Mean number of tasks in the system:

Mean response time, by Little's law:


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Distribution of Waiting & Response Time

If W denotes the waiting time in queue in

equilibrium, W(x), W*(s) be the CDF and LSTrespectively.

And if Q z indicates the PGF of number of tasks in

queue in steady-state. We have:

The left hand side of above equation is:


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Distribution of Waiting & Response Time

So we have the LST of waiting time in queue as:

n we now:

We also can have higher moments of response time:


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A little bit on Simulation


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Results


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Results


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Results Response tiem


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Results Response time


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Results higher moments


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Results higher moments


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Thank You !!!Any Question?

cloud computing modeling

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