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CS433 : Modeling and Simulation

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CS433 : Modeling and Simulation

Lecture 10:

Discrete Time Markov Chains

Dr. Shafique Ahmad Chaudhry

Department of Computer Science

E-mail: hazrat.shafique@gmail.com

Office # GR-02-C

Tel # 2581328

- 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

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(ij) 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 (ij)(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.

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 iS and j S

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

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

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

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.

Example from Book

Introduction to Probability: Lecture Notes

D. Bertsekas and J. Tistsiklis – Fall 2000

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

- 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.

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

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

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.

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.

p01

p12

p23

0

1

2

3

Positive Recurrent States

Transient States

p32

p10

p00

p14

p22

p33

4

Recurrent State

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

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.

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

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 n1

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 nm steps, summed over all possible intermediate states k”

What happens with t get large?

20

21

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 } = limnpij (n) for all i and j

These asymptoticalvalues are calledSteady-State Probabilities

22

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

- 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.

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

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

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

- In other words

- Solving these equations we get

- In general

- Summing all terms we get

- 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

- If p=1/2, then

All states are null recurrent

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

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

- First consider the one-step transition

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