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Generalizations of Poisson Process
(1)
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.
Generalizations of Poisson Process
This generalized process has excellent interpretations in terms of birthdeath 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
(2)
n ≥ 0 and λ0 may or may not equal to zero
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
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:
Birth and Death Process
Birth and Death Process
(4)
As h → o, we have
(5)
(6)
(7)
(8)
(5) and (7) represent the differential-difference equations of a birth and
death process which play an important role in queuing theory.
Birth and Death Process
We make the following assertion:
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.
Birth and Death Process
When the specific values of both λ n and μn are considered simultaneously, we get
the following processes:
Birth and Death Process
From Equ.
(5)
and
(7)
(9)
(10)
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.
n =1
n =0
(9)
(10)
Sn
n
Some Notifications
they may help
(11)
(12)
Birth and Death Process
constant
9
10
9
(13)
Birth and Death Process
(13)
(14)
The second moment M2(t) of X(t) can also be calculated in the same way.
Birth and Death Process
(12)
Birth and Death Process
(15)
(16)
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Birth and Death Process
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.
Thanks for
your
attention
Some Notifications they may help
1
2
In case we have:
BACK
a
b
c
If we adding the part P1(t) for both sides as we have
in our equation we will get:
Birth and Death Process
0
t
t+ h
t
P{N(t)= n-i+j} = Pn-i+j(t)
t
h
P{Eij(h)} = Pi(h)*Pj(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)
h
n
P{N(t+h)= n} = P{N(t)= n-i+j}* P{N(h)= i+j} =Pn-i+j(t)*Pi(h)*Pj(h)
i
j
Eij
n
P{N(t+h)= n} = Pn(t) {1-λn h + O(h)} {1- μn h + O(h)}
0
0
E00
n-1
P{N(t+h)= n} = Pn-1(t) {λn-1 h + O(h)} {1- μn-1 h + O(h)}
1
0
E10
n+1
P{N(t+h)= n} = Pn+1(t) {1- λn+1 h + O(h)} {μn+1 h + O(h)}
0
1
E01
n
P{N(t+h)= n} = Pn(t) { λn h + O(h)} {μn h + O(h)}
1
1
E11