CS433Modeling and Simulation

Lecture 06 – Part 02

Discrete Markov Chains

Dr. Anis Koubâa

http://10.2.230.10:4040/akoubaa/cs433/

11 Nov 2008

Al-Imam Mohammad Ibn Saud UniversityAl-Imam Mohammad Ibn Saud University

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Goals for Today

Practical example for modeling a

system using Markov Chain

State Holding Time

State Probability and Transient

Behavior

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Example• Learn how to find a model of a given system• Learn how to extract the state space

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Example: Two Processors System

Consider a two processor computer system where, time is divided into time slots and that operates as follows: At most one job can arrive during any time slot and this can happen

with probability α. Jobs are served by whichever processor is available, and if both are

available then the job is given to processor 1. If both processors are busy, then the job is lost. When a processor is busy, it can complete the job with probability β

during any one time slot. If a job is submitted during a slot when both processors are busy but

at least one processor completes a job, then the job is accepted (departures occur before arrivals).

Q1. Describe the automaton that models this system (not included).

Q2. Describe the Markov Chain that describes this model.

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Example: Automaton (not included) Let the number of jobs that are currently processed by the

system by the state, then the State Space is given by X= {0, 1, 2}.

Event set: a: job arrival, d: job departure

Feasible event set: If X=0, then Γ(X)= a If X= 1, 2, then Γ(Χ)= a, d.

State Transition Diagram

0 1 2

a

- / a,d

a

-d d / a,d,d

dd

-/a/ad

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Example: Alternative Automaton(not included) Let (X1,X2) indicate whether processor 1 or 2 are busy, Xi= {0, 1}. Event set:

a: job arrival, di: job departure from processor i Feasible event set:

If X=(0,0), then Γ(X)= a If X=(0,1) then Γ(Χ)= a, d2. If X=(1,0) then Γ(Χ)= a, d1. If X=(0,1) then Γ(Χ)= a, d1, d2.

State Transition Diagram

a

- / a,d1

a

-

d2 d1

d1,d2

-/a/ad1/ad2

-

d1

a,d2

a,d1,d2

00

10

11

01

7

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Example: Markov Chain

For the State Transition Diagram of the Markov Chain, each transition is simply marked with the transition probability

0 1 2

p01

p11

p12

p00p10

p21

p20

p22

00 1p 01p 02 0p

10 1p 11 1 1p 12 1p

220 1p 2

21 2 1 1p 2

22 21 1p

8

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Example: Markov Chain

Suppose that α = 0.5 and β = 0.7, then,

0 1 2

p01

p11

p12

p00p10

p21

p20

p22

0.5 0.5 0

0.35 0.5 0.15

0.245 0.455 0.3ijp

P

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How much time does it take for going from one state to another?

State Holding Time

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State Holding Times

Suppose that at point k, the Markov Chain has transitioned into state Xk=i. An interesting question is how long it will stay at

state i. Let V(i) be the random variable that represents the number of

time slots that Xk=i.

We are interested on the quantity Pr{V(i) = n}

1 1

1

1 1

Pr Pr , ,..., |

Pr | ,...,

Pr ,..., |

k n k n k k

k n k n k

k n k k

V n X i X i X i X ii

X i X i X i

X i X i X i

1 1 2

2 1

Pr | Pr | ...,

Pr ,..., |

k n k n k n k n k

k n k k

X i X i X i X X i

X i X i X i

| | |P A B C P A B C P B C

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State Holding Times

This is the Geometric Distribution with parameter Clearly, V(i) has the memoryless property

1

1 2

2 1

Pr Pr |

Pr | ...,

Pr ,..., |

k n k n

k n k n k

k n k k

V n X i X ii

X i X X i

X i X i X i

1Pr 1 nii iiV n p pi

1 2

2 3

3 1

1 Pr |

Pr | ,...,

Pr ,..., |

ii k n k n

k n k n k

k n k k

p X i X i

X i X i X i

X i X i X i

iip

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State Probabilities

An interesting quantity we are usually interested in is the probability of finding the chain at various states, i.e., we define

Pri kX ik For all possible states, we define the vector

0 1, ...k k k π Using total probability we can write

1 1Pr | Pr

1

i k k kj

ji jj

X i X j X jk

p k k

In vector form, one can write

1k k k π π P 1k k π π POr, if homogeneous Markov Chain

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State Probabilities Example

Suppose that

1 0 00 π

Find π(k) for k=1,2,…

with0.5 0.5 0

0.35 0.5 0.15

0.245 0.455 0.3

P

0.5 0.5 0

1 0 0 0.35 0.5 0.15 0.5 0.5 010.245 0.455 0.3

π

Transient behavior of the system In general, the transient behavior is obtained by

solving the difference equation 1k k π π P