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Markov Chains
Jian He
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Outline
• Markov Decision
Probability Review
Process
Stochastic Processes
• Hidden Markov
Definition
Chains
Classification of States • Continuous-time
Limiting Probabilities
Markov chain
Time-reversible Markov
Chain
• Branching Processes
• Random Walk
•
•
•
•
•
•
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Probability Review
• Conditional probability • Expectation of r.v.
P[ A | B ] 
P[ AB ]
P[ B ]
E[ X ] 
• Bayes' Theorem
P[ Ai | B] 

x : p ( x ) 0
• Moment Generating Function
P[ B | Ai ]P[ Ai ]
n
i 1
 xp( x )
 ( t )  E [e ]
P[ B | Ai ]P[ Ai ]
tX
• CDF vs PDF
F (a )  P{ X  (, a]}  
a

 n ( 0)  E[ X n ]
n1
f ( x )dx
3
Stochastic Processes
• Collection of random variables
{ X ( t ), t  T }
• t is often interpreted as time,while X(t) as the
state of the process
• T is called the index set of the process
FX(t)(x)  P{X(t)  x}
• Example:
job waiting in the queue as a function of time
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Definition
• Stochastic process {X n ,n  0 ,1,2 ,...}
• Finite or countable number of possible values
• Markov property
P{ X n1  j | X n  i , X n1  in1 ,..., X 1  i1 , X 0  i0 }
 P{ X n1  j | X n  i }  Pij
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One-step Transition Probability
• Homogeneous Markov chain
P{X n1  j | X n  i}  P{X n  j | X n1  i}
n
• One-step transition probability
P00
P10
P 
Pi 0
P01
P11

Pi 1


P02 
P12 

Pi 2 

6
Chapman-Kolmogorov Equations
(n )
ij
• n-step transition probabilities P
( n)
ij
P
 P{X nk  j | X k  i}
nm
ij
P
P
( n)

 P P
k 0
P
n m
ik kj
n 1
n  0, i, j  0
for all n, m  0, all i, j
P  P
n
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Classification of States
• Accessibility n  0 P  0
i j
• Communicate if i  j and j  i , then i  j
• Properties of communication
if i  j , then j  i
i i
n
ij
if i  j and j  k , then i  k
• class
states communicate with each other
• irreducible only one class
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Classification of States(cont.)

• recurrent state
• transient state
P
n 1

n
ii

n
P
 ii  
n 1
• state i is recurrent if and only if, starting in
state i, the expected number of time periods
that the process is in state i is infinite
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Recurrent vs Transient
• If state i is recurrent, and state i
communicates with state j, then
state j is recurrent
• At least one of the states must be
recurrent in finite-state Markov
chains
• All states of a finite irreducible
Markov chain are recurrent
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Random Walk
• Markov chain
• State space i  0,1,2,...
• Transition probability
Pi ,i 1  p  1  Pi ,i 1
i  0,1,2,...
• irreducible
• all transient or all recurrent ?
2n
00
P
(4 p(1  p)) n
~
n
1
n1 P   if and only if p  2

n
00
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Random Walk(cont.)
• Symmetric random walk equal probability
• Two-dimensional symmetric random walk
P( i , j ),( i 1, j )  P( i , j ),( i 1, j )  P( i , j ),( i , j 1)  P( i , j ),( i , j 1)
2n
00
P
1
~
n


2n
00
P
n 1
1

4

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Limiting Probabilities
•
•
•
•
•
n
d

gcd{
n
|
P
period
ii  0}
aperiodic d=1
if state i has period d, and states i and j
communicate, then state j also has period d.
positive recurrent
n
nPii  
expected return time is finite
ergodic positive recurrent && aperiodic

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Limiting Probabilities(cont.)
• Irreducible ergodic Markov chain
exists, and is independent of i
n
• Definition  j  lim Pij
n 
• Property

 j    i Pij
i 0
lim n Pijn

j0

j 0
j
1
•  j equals the long-run proportion of time that
the process will be in state j
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Limiting Probabilities(Example)
• Transition probability matrix
0 .5 0 .4 0 .1
P  0 .3 0 .4 0 .3
0 .2 0 .3 0 .5
 0  0.5 0  0.3 1  0.2 2
 1  0.4 0  0.4 1  0.3 2
 2  0.1 0  0.3 1  0.5 2
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23
18
 0  , 1 
, 2 
62
62
62
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Limiting Probability(Property)
• Let { X n , n  1} be an irreducible Markov chain
with stationary probabilities  j , j  0 ,and let r be a
bounded function on the state space. Then, with
probability 1,

lim
N 
N
n 1
r( X n )
N
  j  0 r ( j ) j

• If we suppose that we earn a reward r(j)
whenever the chain is in state j, then our
average reward per unit time is  j r ( j ) j
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Advanced Random Walk
• Markov chain with states 0,1,...,n having
P0,1  1, Pi ,i 1  p, Pi ,i 1  q  1  p,1  i  n
•
N i the number of additional transitions that it
takes the chain when it first enters state i until it
enters state i+1
• i  E[ N i ]
• Goals the number of trasitions that it takes
the chain to go from state 0 to state n
n 1
N 0,n   N i
i 0
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Mean Number of Transitions

 1  E[# additional trasitions to reach i  1 |
i
•
chain to i  1]q
•  i  1  E[ N i*1  N i* ]q
 1  q( i 1  i )
•
  q/ p

1 n 1 i 1 j n 1 i

i 1
1
  

 i    j   i   E[ N 0 , n ]  1  p 
i 1 j  0
i 1

p j 0
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Mean Number of Transitions(cont.)
1
• when p  , E[ N 0,n ]  n2
2
n1
n1
1
2


(
n

1
)

 n1
• when p  , E[ N 0,n ]  1 
2
(1   )2
1
• p  exponentially increasing function
2
1
• p  2 for large n, essentially linear in n
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Branching Processes
• A population consisting of individuals able to
produce offspring of the same kind
• Each individual produces j new offsring with
probability Pj , j  0 independently,by the end of
lifetime.
• X n the size of the nth generation
• { X n , n  0,1,2,...} is a Markov chain
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Extinction
• State 0 is a recurrent state P00  1
• All other states are transient, if P0  0

•    jP j   E[ X ]  E[ X ]   n

n
n 1
j 0
X 0  1 


•
 0  lim P{ X n  0 | X 0  1}   0    Pj
n 
j 0
j
0
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Age-dependent Branching Processes
• Attach another random variable 'age'
 t
• CDF fT ( t )  e , t  0
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Mean Number of Individuals
• Theorem
m ( t )  E[ Z ( t )]
There exist   0 and   0 such that
t
m( t ) ~ e
whenever m( t )  1
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Branching Processes(Example)
• Service Capacity of
P2P system in
trasient regime
• N d (t ) the number of
peers available to
serve document d at
time t
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Service Capacity
• Basic Braching Processes Model
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Time Reversible Markov Chains
• Stationary ergodic Markov chain
Pij stationary probabilities  j
• { X n , X n1 , X n 2 ,..., X 0 } a Markov Chain ?
transition probabilities
• Transition probabilities
 j Pji
Qij  P{ X m  j | X m 1  i } 
i
• Theorem
Time reversible, iff Qij  Pij   i Pij   j Pji
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Reversibility(Example)
• Arbitrary connected graph
 ij
Pij 
1
  ij
2
3
3
1
j
 ij
i
j
 j  ij

i
i
 ij
i  ji
6
2
5
4
4
1
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Reversibility(Property)
• An ergodic Markov chain for which Pij  0
whenever Pji  0 is time reversible if and only if
starting in state i, any path back to i has the
same probability as the reversed path. That is, if
Pi ,i1 Pi1 ,i2 ... Pik ,i  Pi ,ik Pik ,ik 1 ... Pi1 ,i
for all states i , i
1
,..., ik
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Reversibility(Extenstion)
• Irreducible Markov chain Pij
• Theorem
i  0 
  Qij : tran prob of reversed chain
i  i  1    : stationary probabilit ies (both)
i


 i Pij   j Q ji 
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Markov Decision Processes
• After observing the state of the process, an action
must be chosen
P{ X n1  j | X 0 , a0 , X 1 , a1 ,..., X n  i , an  a }  Pij (a )
• Limiting Probabilities
 ia  0 for all i , a
   1
    
i
a
a
ja
ia
i
a
ia
Pij (a ) for all j
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Hidden Markov Chains
• finite set of signals 
p( s | j )  1
• emitting signals
s
• observing the sequence of signals

P{ Sn  s | X 1 , S1 ,..., X n1 , Sn1 , X n  j }  p( s | j )
• conditional probability of state
n
P
{
S
 sn , X n  j }
n
P { X n  j | S  sn } 
P{ S n  sn }
P { S n  sn , X n  j }

n
P
{
S
 n, X n  i }
i
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Continuous-Time Markov Chains
• Definition
P{ X ( t  s )  j | X ( s )  i , X ( u)  x( u),0  u  s}
 P{ X ( t  s )  j | X ( s )  i }
•
•
Ti the amount of time stay in state i
P{Ti  s  t | Ti  s}  P{Ti  t }
• memoryless  exponentially distributed
Continunous-Time Markov Chain(property)
• The amount of time the process spends in a
state before making a transition into another
state is exponentially distributed
• Transition probability
Pii  0, all i
P
ij
j
 1, all i
Limiting Probabilities
(Continuous-Time version)
• The mean time stays in one state 1 / v i
• qij  v i Pij
• Properties of Limiting Probabilities
v j Pj   qkj Pk
k j
P
j
1
j
• v j Pj rate at which the process leaves state j
•  qkj Pk rate at which the process enters j
k j
Birth and Death Process
• state the number of people
• whenever there are n persons
• successive arrival time exponentially
distributed with mean 1 /  n
• successvie departure time exponentially
distributed with mean 1 / n
• Continuous-time Markov chain
Birth and Death Process(cont.)
• Transition probabilities
i
P01  1 Pi ,i 1    
i
i
Pi ,i 1 
i
i   i
• Limiting Probabilities
0 P0  1 P1
(n  n ) Pn  n1 Pn1  n1 Pn1
0 1 ...n1
Pn 

01 ...n1
1  2 ... n (1  
)
n 1 1  2 ... n
P0 
1
0 1 ...n1
1 
n 1 1  2 ... n

M/M/1 System
• Poisson arrival(or the interarrival time is
exponential)and service time is exponentially
 t
 t
distributed. Arrival is e and service is  e
M/M/1 System(cont.)
• Limiting Probabilities

P0  1 

  n
Pn  (1  )( )
 
• E[number of customers in the system]

  iPi 
i 0

1 

   system utilization

References
• Sheldon M. Ross, Introduction to Probability
Models, 9th Edition. ISBN 978-7-115-160232/O1. 2007
• G.R. Grimmett and D. R. Stirzaker, Probability
and Random Processes,2nd Edition, Clarendon
Press, Oxford,1992.
• X. Yang and G. de Veciana. Service Capacity
of Peer to Peer Networks. INFOCOM,2004
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