1. Generalizations of Poisson Process 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. (1)

2. 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 λ 0 may or may not equal to zero (2)

3. 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

4. Birth and Death Process To obtain the differential-difference equation for P n (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:

5. Birth and Death Process

6. (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)

7. Birth and Death Process We make the following assertion:

8. Births and Death Rates Depending upon the values of λ n and μ n, various types of birth and death processes can be defined. State (0) is absorbing state.

9. Birth and Death Process When the specific values of both λ n and μ n are considered simultaneously, we get the following processes:

10. Birth and Death Process If the initial population size is i, i.e, X(0) = i, then we have the initial condition P i (0) = 1 and P n (0) = 0, n ≠ i. (9) (10) From Equ. (5) and (7)

11. (9)(10) (11) (12) n =0 n =1 n SnSn Some Notifications they may help

12. Birth and Death Process constant 9 10 (13) 9

13. Birth and Death Process The second moment M2(t) of X(t) can also be calculated in the same way. (14) (13)

14. Birth and Death Process (12)

15. Birth and Death Process (15) (16) <

16. Birth and Death Process

17. 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.

18. Thanks for your attention

19. Some Notifications they may help In case we have: 1 2 a b c If we adding the part P 1 (t) for both sides as we have in our equation we will get: BACK

20. Birth and Death Process 0tt+ h P{N(t+h)= n} = P{N(t)= n-i+j}* P{N(h)= i+j} =P n-i+j (t)*P i (h)*P j (h) E 00 E 10 E 01 E 11 n n-1 n+1 n i0101i0101 j0011j0011 P{N(t)= n-i+j} = P n-i+j (t) t h P{E ij (h)} = P i (h)*P j (h) E ij t h P{N(t+h)= n} = P{N(t)= n-i+j} * P{E ij (h)} = P n-i+j (t) * P i (h)*P j (h) = P n (t+h) P{N(t+h)= n} = P n (t) {1-λ n h + O(h)} {1- μ n h + O(h)} P{N(t+h)= n} = P n-1 (t) {λ n-1 h + O(h)} {1- μ n-1 h + O(h)} P{N(t+h)= n} = P n+1 (t) {1- λ n+1 h + O(h)} {μ n+1 h + O(h)} P{N(t+h)= n} = P n (t) { λ n h + O(h)} {μ n h + O(h)}