1. Markov Chains

2. Summary Markov Chains Discrete Time Markov Chains
Homogeneous and non-homogeneous Markov chains
Transient and steady state Markov chains
Continuous Time Markov Chains

3. Markov Processes Recall the definition of a Markov Process
The future a a process does not depend on its past, only on its present
Since we are dealing with “chains”, X(t) can take discrete values from a finite or a countable infinite set.
For a discrete-time Markov chain, the notation is also simplified to
Where Xk is the value of the state at the kth step

4. Chapman-Kolmogorov Equations
Define the one-step transition probabilities
Clearly, for all i, k, and all feasible transitions from state i
Define the n-step transition probabilities xi x1 xR … xj k u
k+n

5. Chapman-Kolmogorov Equationsxi x1 xR … xj k u
k+n
Using total probability
Using the memoryless property of Marckov chains
Therefore, we obtain the Chapman-Kolmogorov Equation

6. Matrix Form Define the matrix
We can re-write the Chapman-Kolmogorov Equation
Choose, u = k+n-1, then
Forward Chapman-Kolmogorov
One step transition probability

7. Matrix Form Choose, u = k+1, then Backward Chapman-Kolmogorov
One step transition probability

8. Homogeneous Markov Chains
The one-step transition probabilities are independent of time k.
Even though the one step transition is independent of k, this does not mean that the joint probability of Xk+1 and Xk is also independent of k
Note that

9. Example
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).
Describe the automaton that models this system.
Describe the Markov Chain that describe this model.

10. Example: Automaton 1 2 - / a,d a a -/a/ad - d / a,d,d d dd
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
- / a,d a a
-/a/ad 1 2 -
d / a,d,d d dd

11. Example: Alternative Automaton
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, d If X=(0,1) then Γ(Χ)= a, d1, d2.
State Transition Diagram
- / a,d1 a a 00 10 11 01
-/a/ad1/ad2 d1
a,d1,d2 -
d1,d2
a,d2 d2 d1 -

12. Example: Markov Chain 1 2 p11 p01 p12 p22 p00 p21 p10 p20
For the State Transition Diagram of the Markov Chain, each transition is simply marked with the transition probability
p11
p01
p12
p22 1 2
p00
p21
p10
p20

13. Example: Markov Chain 1 2 p01 p11 p12 p00 p10 p21 p20 p221 2
p01
p11
p12
p00
p10
p21
p20
p22
Suppose that α = 0.5 and β = 0.7, then,

14. 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}

15. State Holding Times
This is the Geometric Distribution with parameter pii.
Clearly, V(i) has the memoryless property

16. State Probabilities
An interesting quantity we are usually interested in is the probability of finding the chain at various states, i.e., we define
For all possible states, we define the vector
Using total probability we can write
In vector form, one can write
Or, if homogeneous Markov Chain

17. State Probabilities Example
Suppose that
with
Find π(k) for k=1,2,…
Transient behavior of the system: MCTransient.m
In general, the transient behavior is obtained by solving the difference equation

18. Classification of States
Definitions
State j is reachable from state i if the probability to go from i to j in n >0 steps is greater than zero (State j is reachable from state i if in the state transition diagram there is a path from i to j).
A subset S of the state space X is closed if pij=0 for every iS and j S
A state i is said to be absorbing if it is a single element closed set.
A closed set S of states is irreducible if any state jS is reachable from every state iS.
A Markov chain is said to be irreducible if the state space X is irreducible.

19. Example 1 2 4 1 2 3 Irreducible Markov Chain p01 p12 p00 p10 p21 p221 2
p01
p12
p00
p10
p21
p22
Reducible Markov Chain
p01
p12
p00
p10
p14
p22 4
p23
p32
p33 1 2 3
Absorbing State
Closed irreducible set

20. Transient and Recurrent States
Hitting Time
Recurrence Time Tii is the first time that the MC returns to state i.
Let ρi be the probability that the state will return back to i given it starts from i. Then,
The event that the MC will return to state i given it started from i is equivalent to Tii < ∞, therefore we can write
A state is recurrent if ρi=1 and transient if ρi<1

21. 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.
If S is a finite closed irreducible set of states, then every state in S is recurrent.

22. 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.
Theorems
If state i is positive recurrent and state j 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.

23. Example 4 1 2 3 p01 p12 p00 p10 p14 p22 p23 p32 p33 Transient States1 2 3
Transient States
Positive Recurrent States
Recurrent State

24. 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 2
Periodic State d = 2

25. Steady State Analysis
Recall that 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 state or 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.

26. 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 the presence of periodic states prevents the existence of a steady state probability
Example: periodic.m

27. Steady State Analysis
THEOREM: In an irreducible aperiodic Markov chain consisting of positive recurrent states a unique stationary state probability vector π exists such that πj > 0 and
where Mj is the mean recurrence time of state j
The steady state vector π is determined by solving
and
Ergodic Markov chain.

28. Birth-Death Example 1 i 1-p p1 i
Thus, to find the steady state vector π we need to solve
and

29. Birth-Death Example In other words Solving these equations we get
In general
Summing all terms we get

30. 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

31. Birth-Death Example
If p=1/2, then
All states are null recurrent

32. Reducible Markov Chains
Irreducible Set S1
Transient Set T
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

33. 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

34. Reducible Markov Chains
First consider the one-step transition
Next consider the general case for k=2,3,…

35. Continuous-Time Markov Chains
In this case, transitions can occur at any time
Recall the Markov (memoryless) property
where t1 < t2 < … < tk
Recall that the Markov property implies that
X(tk+1) depends only on X(tk) (state memory)
It does not matter how long the state at X(tk) (age memory).
The transition probabilities now need to be defined for every time instant as pij(t), i.e., the probability that the MC transitions from state i to j at time t.

36. Transition Function Define the transition function
The continuous-time analogue of the Chapman-Kolmokorov equation is
Using the memoryless property
Define H(s,t)=[pij(s,t)], i,j=1,2,… then
Note that H(s, s)= I

37. Transition Rate Matrix
Consider the Chapman-Kolmogorov for s ≤ t ≤ t+Δt
Subtracting H(s,t) from both sides and dividing by Δt
Taking the limit as Δt0
where the transition rate matrix Q(t) is given by

38. Homogeneous Case
In the homogeneous case, the transition functions do not depend on s and t, but only on the difference t-s thus
It follows that
and the transition rate matrix
Thus

39. State Holding Time
The time the MC will spend at each state is a random variable with distribution
where
Explain why…

40. Transition Rate Matrix Q.
Recall that
First consider the qij, i ≠ j, thus the above equation can be written as
Evaluating this at t = 0, we get that
pij(0)= 0 for all i ≠ j
The event that will take the state from i to j has exponential residual lifetime with rate λij, therefore, given that in the interval (t,t+τ) one event has occurred, the probability that this transition will occur is given by Gij(τ)=1-exp{-λijτ}.

41. Transition Rate Matrix Q.
Since Gij(τ)=1-exp{-λijτ}.
In other words qij is the rate of the Poisson process that activates the event that makes the transition from i to j.
Next, consider the qjj, thus
Evaluating this at t = 0, we get that
Probability that chain leaves state j

42. Transition Rate Matrix Q.
The event that the MC will transition out of state i has exponential residual lifetime with rate Λ(i), therefore, the probability that an event will occur in the interval (t,t+τ) given by Gi(τ)=1-exp{- Λ(i)τ}.
Note that for each row i, the sum

43. Transition Probabilities P.
Suppose that state transitions occur at random points in time T1 < T2 <…< Tk <…
Let Xk be the state after the transition at Tk
Define
Recall that in the case of the superposition of two or more Poisson processes, the probability that the next event is from process j is given by λj/Λ.
In this case, we have
and

44. Example
Assume a computer system where jobs arrive according to a Poisson process with rate λ.
Each job is processed using a First In First Out (FIFO) policy.
The processing time of each job is exponential with rate μ.
The computer has buffer to store up to two jobs that wait for processing.
Jobs that find the buffer full are lost.
Draw the state transition diagram.
Find the Rate Transition Matrix Q.
Find the State Transition Matrix P

45. Example 1 2 3 a a a a d d d The rate transition matrix is given by1 2 3 a d d d
The rate transition matrix is given by
The state transition matrix is given by

46. State Probabilities and Transient Analysis
Similar to the discrete-time case, we define
In vector form
With initial probabilities
Using our previous notation (for homogeneous MC)
Obtaining a general solution is not easy!
Differentiating with respect to t gives us more “inside”

47. “Probability Fluid” view
We view πj(t) as the level of a “probability fluid” that is stored at each node j (0=empty, 1=full).
Change in the probability fluid
inflow
outflow i r
qij
qjr … j …
Inflow
Outflow

48. Steady State Analysis
Often we are interested in the “long-run” probabilistic behavior of the Markov chain, i.e.,
These are referred to as steady state probabilities or equilibrium state probabilities or stationary state probabilities
As with the discrete-time case, we need to address the following questions
Under what conditions do the limits exist?
If they exist, do they form legitimate probabilities?
How can we evaluate these limits?

49. … … Steady State Analysis i r j
Theorem: In an irreducible continuous-time Markov Chain consisting of positive recurrent states, a unique stationary state probability vector π with
These vectors are independent of the initial state probability and can be obtained by solving
Using the “probability fluid” view i r
outflow
qij
qjr … j …
0 Change
inflow
Inflow
Outflow

50. Example a a a 1 2 3 a d d d
For the previous example, with the above transition function, what are the steady state probabilities
Solve

51. Example
The solution is obtained

52. Birth-Death Chain 1 i λ0 λ1 λi-1 λi μ1 μi μi+11 i λ1
λi-1 λi μ1 μi
μi+1
Find the steady state probabilities
Similarly to the previous example,
And we solve
and

53. Example The solution is obtained In general Making the sum equal to 1
Solution exists if

54. Uniformization of Markov Chains
In general, discrete-time models are easier to work with, and computers (that are needed to solve such models) operate in discrete-time
Thus, we need a way to turn continuous-time to discrete-time Markov Chains
Uniformization Procedure
Recall that the total rate out of state i is –qii=Λ(i). Pick a uniform rate γ such that γ ≥ Λ(i) for all states i.
The difference γ - Λ(i) implies a “fictitious” event that returns the MC back to state i (self loop).

55. Uniformization of Markov Chains
Uniformization Procedure
Let PUij be the transition probability from state I to state j for the discrete-time uniformized Markov Chain, then i j k … i j k …
qij
qik
Uniformization