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CS433 : Modeling and Simulation. Lecture 10: Discrete Time Markov Chains Dr. Shafique Ahmad Chaudhry Department of Computer Science E-mail: [email protected] Office # GR-02-C Tel # 2581328. Markov Processes. Stochastic Process X ( t ) is a random variable that varies with time.

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CS433 : Modeling and Simulation

Lecture 10:

Discrete Time Markov Chains

Dr. Shafique Ahmad Chaudhry

Department of Computer Science

E-mail: [email protected]

Office # GR-02-C

Tel # 2581328


Markov processes
Markov Processes

  • Stochastic Process X(t)is a random variable that varies with time.

  • A state of the process is a possible value of X(t)

  • Markov Process

    • The future of a process does not depend on its past, only on its present

    • a Markov process is a stochastic (random) process in which the probability distribution of the current value is conditionally independent of the series of past value, a characteristic called the Markov property.

    • Markov property: the conditional probability distribution of future states of the process, given the present state and all past states, depends only upon the present state and not on any past states

  • Marko Chain: is a discrete-time stochastic process with the Markov property


Classification of states 1
Classification of States: 1

Apathis a sequence of states, where each transition has a positive probability of occurring.

State jis reachable (or accessible)(يمكن الوصول إليه) from state i(ij) if there is a path from i to j –equivalently Pij(n)> 0 for some n≥0, i.e.the probability to go from ito j in nsteps is greater than zero.

States i and j communicate (ij)(يتصل) ifiis reachable fromjandjis reachable fromi.

(Note: a state i always communicates with itself)

A set of states C is a communicating classif every pair of states in C communicates with each other, and no state in C communicates with any state not in C.


Classification of states 11
Classification of States: 1

A state i is said to be an absorbing state if pii= 1.

A subset S of the state space Xis a closed set if no state outside of S is reachable from any state in S (like an absorbing state, but with multiple states), this means pij= 0for every iS and j S

A closed set S of states is irreducible(غير قابل للتخفيض) if any state j Sis reachable from every state iS.

A Markov chain is said to be irreducible if the state space X is irreducible.


Example
Example

Irreducible Markov Chain

p01

p12

p22

p00

p21

p10

p01

p12

p23

0

1

2

3

p32

p10

p00

p14

0

1

2

p22

p33

4

  • Reducible Markov Chain

Absorbing State

Closed irreducible set


Classification of states 2
Classification of States: 2

State iis atransient state(حالة عابرة)if there exists a state j such that j is reachable from ibut i is not reachable from j.

A state that is not transient is recurrent(حالة متكررة) . There are two types of recurrent states:

Positive recurrent: if the expected time to return to the state is finite.

Null recurrent (less common): if the expected time to return to the state is infinite(this requires an infinite number of states).

A state iis periodic with periodk>1, ifkis the smallest number such that all paths leading from state iback to state i have a multiple of k transitions.

A state is aperiodic if it has period k =1.

A state is ergodic if it is positive recurrent and aperiodic.


Classification of states 21
Classification of States: 2

Example from Book

Introduction to Probability: Lecture Notes

D. Bertsekas and J. Tistsiklis – Fall 2000


Transient and recurrent states
Transient and Recurrent States

We define the hittingtime Tijas the random variable that represents the time to go from state j to stat i, and is expressed as:

k is the number of transition in a path from i to j.

Tijis the minimum number of transitions in a path from i to j.

We define the recurrence timeTii as the first time that the Markov Chain returns to state i.

The probability that the first recurrence to state ioccurs at the nth-step is

TiTime for first visit to i given X0 = i.

The probability of recurrence to state iis


Transient and recurrent states1
Transient and Recurrent States

  • The mean recurrence time is

  • A state is recurrent if fi=1

    • If Mi < then it is said Positive Recurrent

    • If Mi =  then it is said Null Recurrent

  • A state is transient if fi<1

  • If , then is the probability of never returning to state i.


Transient and recurrent states2
Transient and Recurrent States

We define Niasthe number of visits to stateigiven X0=i,

Theorem: If Ni is the number of visits to state igiven X0=i,then

Proof

Transition Probability from

state i to state i after n steps


Transient and recurrent states3
Transient and Recurrent States

The probability of reaching state j for first time in n-steps starting from X0 = i.

The probability of ever reaching j starting from state i is


Three theorems
Three Theorems

If a Markov Chain has finite state space, then: at least one of the states is recurrent.

If state i is recurrent and state j is reachable from state i

then: state j is also recurrent.

IfS is a finite closed irreducible set of states, then: every state in S is recurrent.


Positive and null recurrent states
Positive and Null Recurrent States

Let Mi be the mean recurrence time of state i

A state is said to be positive recurrent if Mi<∞.

If Mi=∞ then the state is said to be null-recurrent.

Three Theorems

If state i is positive recurrent and statej is reachable from state i then, state j is also positive recurrent.

If S is a closed irreducible set of states, then every state in S is positive recurrent or, every state in S is null recurrent, or, every state in S is transient.

If S is a finite closed irreducible set of states, then every state in S is positive recurrent.


Example1
Example

p01

p12

p23

0

1

2

3

Positive Recurrent States

Transient States

p32

p10

p00

p14

p22

p33

4

Recurrent State


Periodic and aperiodic states
Periodic and Aperiodic States

Suppose that the structure of the Markov Chain is such that state i is visited after a number of steps that is an integer multiple of an integer d >1. Then the state is called periodic with period d.

If no such integer exists (i.e., d =1) then the state is called aperiodic.

Example

1

0.5

0

1

2

1

0.5

Periodic State d = 2


Steady state analysis
Steady State Analysis

Recall that the state probability, which is the probability of finding the MC at state i after the kth step is given by:

  • An interesting question is what happens in the “long run”, i.e.,

  • This is referred to as steady stateor equilibrium or stationary state probability

  • Questions:

    • Do these limits exists?

    • If they exist, do they converge to a legitimate probability distribution, i.e.,

    • How do we evaluate πj, for all j.


Multi step t step transitions
Multi-step (t-step) Transitions

Example: TAX auditing problem:

Assume that whether a tax payer is audited by Tax department or not in the n + 1 is dependent only on whether he was audit in the previous year or not.

  • If he is not audited in year n, he will not be audited with prob 0.6, and will be audited with prob 0.4

  • If he is audited in year n, he will be audited with prob 0.5, and will not be audited with prob 0.5

How to model this problem as a stochastic process ?

17


The tax auditing example
The Tax Auditing Example

State Space: Two states: s0 = 0 (no audit), s1 = 1 (audit)

Transition matrix

Transition Matrix P is the prob. of transition in one step

How do we calculate the probabilities for transitions involving more than one step?

Notice: p01 = 0.4, is conditional probability of audit next year given no audit this year.

p01 = p (x1 = 1 | x0 = 0)

OR

18


n-Step Transition Probabilities

This idea generalizes to an arbitrary number of steps:

In matrix form, P(2) = P  P,

P(3) = P(2) P = P2 P = P3

or more generally

P(n) = P(m) P(n-m)

The ij'th entry of this reduces to

Pij(n) = Pik(m) Pkj(n-m) 1  m  n1

m

k=0

Chapman - Kolmogorov Equations

“The probability of going from i to k in m steps & then going from k to j in the remaining nm steps, summed over all possible intermediate states k”


Steady state solutions n steps
Steady-State Solutions – n Steps

What happens with t get large?

20



Steady state transition probability
Steady State Transition Probability

Observations: as n gets large, the values in row of the matrix becomes identical OR they asymptotically approach a steady state value

What does it mean?

The probability of being in any future state becomes independent of the initial state as time process

 j = limn Pr (Xn=j |X0=i } = limnpij (n) for all i and j

These asymptoticalvalues are calledSteady-State Probabilities

22


Steady state analysis1
Steady State Analysis

Recall the recursive probability

  • If steady state exists, then π(k+1)π(k), and therefore the steady state probabilities are given by the solution to the equations

and

  • If an Irreducible Markov Chain, then the presence of periodic states prevents the existence of a steady state probability


Steady state analysis2
Steady State Analysis

  • THEOREM: In an irreducible aperiodic Markov chain consisting of positive recurrent states a unique stationary state probabilityvector π exists such that πj > 0 and

where Mj is the mean recurrence time of state j

  • The steady state vector πis determined by solving

and

  • Ergodic Markov chain.


Comments on steady state results
Comments on Steady-State Results

1. Steady-state probabilities might not exist unless the Markov chain is ergodic.

2.Steady-state predictions are never achieved in actuality due to a combination of

(i) errors in estimating P,

(ii) changes in P over time, and

(iii) changes in the nature of dependence relationships among the states.

Nevertheless, the use of steady-state values is an important diagnostic tool for the decision maker.

25


Interpretation of steady state conditions
Interpretation of Steady-State Conditions

Just because an ergodic system has steady-state probabilities does not mean that the system “settles down” into any one state.

j is simply the likelihood of finding the system in state j after a large number of steps.

The limiting probability πj that the process is in state j after a large number of steps is also equals the long-run proportion of time that the process will be in state j.

When the Markov chain is finite, irreducible and periodic, we still have the result that the πj, j Î S, uniquely solves the steady-state equations, but now πj must be interpreted as the long-run proportion of time that the chain is in state j.

26


Discrete birth death example
Discrete Birth-Death Example

1-p

1-p

1-p

0

1

i

p

p

p

p

  • Thus, to find the steady state vector πwe need to solve

and


Discrete birth death example1
Discrete Birth-Death Example

  • In other words

  • Solving these equations we get

  • In general

  • Summing all terms we get


Discrete birth death example2
Discrete Birth-Death Example

  • Therefore, for all states j we get

  • If p<1/2, then

All states are transient

  • If p>1/2, then

All states are positive recurrent


Discrete birth death example3
Discrete Birth-Death Example

  • If p=1/2, then

All states are null recurrent


Reducible markov chains
Reducible Markov Chains

Transient Set T

Irreducible Set S1

Irreducible Set S2

  • In steady state, we know that the Markov chain will eventually end in an irreducible set and the previous analysis still holds, or an absorbing state.

  • The only question that arises, in case there are two or more irreducible sets, is the probability it will end in each set


Reducible markov chains1
Reducible Markov Chains

Transient Set T

Irreducible Set S

s1

r

sn

i

  • Suppose we start from state i. Then, there are two ways to go to S.

    • In one step or

    • Go to r T after k steps, and then to S.

  • Define


Reducible markov chains2
Reducible Markov Chains

  • First consider the one-step transition

  • Next consider the general case for k=2,3,…


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