generalizations of poisson process
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Generalizations of Poisson Process. (1). i.e., P k (h ) is independent of n as well as t. This process can be generalized by considering λ no more a constant but a function of n or t or both. The generalized process is again Markovian in nature. Generalizations of Poisson Process. - PowerPoint PPT PresentationTRANSCRIPT
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Generalizations of Poisson Process
i.e., Pk(h) is independent of n as well as t. This process can be generalized by considering λ no more a constant but a function of n or t or both. The generalized process is again Markovian in nature.
(1)
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Generalizations of Poisson Process
This generalized process has excellent interpretations in terms of birth-death processes. Consider a population of organisms, which reproduce to create similar organisms. The population is dynamic as there are additions in terms of births and deletions in terms of deaths. Let n be the size of the population at instant t. Depending upon the nature of additions and deletions in the population, various types of processes can be defined.
Pure Birth Process Let λ is a function of n, the size of the population at instant t. Then
n ≥ 0 and λ0 may or may not equal to zero
(2)
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Birth and Death Process Now, along with additions in the population, we consider deletions
also, i.e., along with births, deaths are also possible. Define
(3)
(2) and (3) together constitute a birth and death process. Through a birth there is an increase by one and through a death, there is a decrease by one in the number of “Individuals”. The probability of more than one birth or more than one death is O(h). We wish to obtain
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Birth and Death Process
To obtain the differential-difference equation for Pn(i), we consider the time interval (0, t+h) = (0, t) + [t, t+h)
Since, births and deaths, both are possible in the population, so the event {N(t+h) = n , n ≥ 1} can occur in the following mutually exclusive ways:
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Birth and Death Process
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Birth and Death Process
(4)
(6)
(5) and (7) represent the differential-difference equations of a birth and death process which play an important role in queuing theory.
(5)As h → o, we have
(7)
(8)
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Birth and Death Process We make the following assertion:
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Births and Death Rates Depending upon the values of λ n and μn , various types of birth and death processes can be defined.
State (0) is absorbing state.
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Birth and Death Process When the specific values of both λ n and μn are considered simultaneously, we get
the following processes:
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Birth and Death Process
If the initial population size is i, i.e, X(0) = i, then we have the initial condition Pi(0) = 1 and Pn(0) = 0, n ≠ i.
(9)
(10)
From Equ.
(5) and (7)
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(9) (10)
(11)
(12)
n =0
n =1
n
Sn
Some Notifications they may help
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Birth and Death Process
constant
9 10
(13)
9
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Birth and Death Process
The second moment M2(t) of X(t) can also be calculated in the same way.
(14)
(13)
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Birth and Death Process
(12)
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Birth and Death Process
(15)
(16)<
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Birth and Death Process
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Birth and Death Process
Finally, the birth and death process is a special type of continuous time Markov process with discrete state space 0, 1, 2, … such that the probability of transition from state i to state j in (∆t) is O(∆t) whenever │i - j│≥ 2. In other words, changes take place through transitions only from a state to its immediate neighbouring state.
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Thanks for your attention
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Some Notifications they may help
In case we have:
1
2
a
b
c If we adding the part P1(t) for both sides as we have in our equation we will get:
BACK
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Birth and Death Process
0 tt+ h
P{N(t+h)= n} = P{N(t)= n-i+j}* P{N(h)= i+j} =Pn-
i+j(t)*Pi(h)*Pj(h)
E00
E10
E01
E11
n
n
n-1
n+1
n
i
0
1
0
1
j
0
0
1
1
P{N(t)= n-i+j} = Pn-
i+j(t)
t
h
P{Eij(h)} = Pi(h)*Pj(h)
Eij
t h
P{N(t+h)= n} = P{N(t)= n-i+j} * P{Eij(h)} = Pn-i+j(t) * Pi(h)*Pj(h) = Pn(t+h)
P{N(t+h)= n} = Pn(t) {1-λn h + O(h)} {1- μn h + O(h)}
P{N(t+h)= n} = Pn-1(t) {λn-1 h + O(h)} {1- μn-1 h + O(h)}
P{N(t+h)= n} = Pn+1(t) {1- λn+1 h + O(h)} {μn+1 h + O(h)}
P{N(t+h)= n} = Pn(t) { λn h + O(h)} {μn h + O(h)}