Birth and death processes

Particle conservation in a vessel is governed by the particle-number continuity equation, essentially a population balance to identify particle numbers in each and every size range and account for any changes due to particle formation, growth and destruction, termed particle birth and death processes reflecting formation and loss of particulate entities, respectively. 

In this section we shall present a few of the elementary type reactions that have been solved exactly. By elementary we mean unimolecular and bimolecular reactions, and simple extensions of them. In a more classical stochastic context, these reactions may be thought of as birth and death processes, unimolecular reactions being linear birth and death processes and bimolecular being quadratic. These reactions may be described by a finite or infinite set of states, (x), each member of which corresponds to a specified number of some given type of molecule in the system. One then describes a set of transition probabilities of going from state x to x — i, which in unimolecular reactions depend linearly upon x and in bimolecular reactions depend quadratically upon x. The simplest example is that of the unimolecular irreversible decay of A into B, which occurs particularly in radioactive decay processes. This process seems to have been first studied in a chemical context by Bartholomay.6... 

Exercise. In a population of n bacteria, each individual has a probability a per unit time to die and / to give birth to a new individual. Construct the M-equation ( birth and death process , compare chapter VI). 

The one-step or birth-and-death processes are a special class of Markov processes, which occur in many applications and can be analyzed in some detail. 

Many stochastic processes are of a special type called birth-and-death processes or generation-recombination processes . We employ the less loaded name one-step processes . This type is defined as a continuous time Markov process whose range consists of integers n and whose transition matrix W permits only jumps between adjacent sites,... 

The cascade theory is probably the oldest branching theory. It was developed by the English chaplain, the Reverend Watson16,181 and the biometrician Galton17,181 in 1873 who were evidently stimulated by Darwin s famous book on The Origin of Species . Nowadays cascade theory is widely used in evolution theory19,201, in actuarial mathematics (birth and death processes), in the physics of cosmic ray showers and in the chemistry of combustion due to branched chain reactions21-241. 

Nei M, Gu X, Sitnikova T (1997) Evolution by the birth-and-death process in multigene families of the vertebrate immune system. Proc Natl Acad Sci USA 94 7799-7806... 

In the following, we derive the Kolmogorov differential equation on the basis of a simple model and report its various versions. In principle, this equation gives the rate at which a certain state is occupied by the system at a certain time. This equation is of a fundamental importance to obtain models discrete in space and continuous in time. The models, later discussed, are the Poisson Process, the Pure Birth Process, the Polya Process, the Simple Death Process and the Birth-and-Death Process. In section 2.1-3 this equation, i.e. Eq.2-30, has been derived for Markov chains discrete in space and time. 

For an arbitrary combined material volume element constituting a combined sub-volume Vsv[t) of the particle phase space the integral formulation of the population balance states that the only way in which the number of particles can change is by birth and death processes [95, 96, 35, 93, 94[. The system balance is thus written on the form ... 

Unlike in the random walk problem, the transition rate out of a given state n depends on n The probability per unit time to go from n+1 to n is A (/j+1), and the probability per unit time to go from n to n — 1 is kn. The process described by Eq. (8.83) is an example of a birth-and-death process. In this particular example there is no source feeding molecules into the system, so only death steps take place. 

Kendal, D. G. 1950 An artificial realization of a simple birth-and-death process. Journal of the Royal Statistical Society Series B 12, 116-119. 

Suppose now that X has state space X = 0, 1, 2,. . . and each transition is to a neighboring state the chain goes either up by one or down by one on any transition. Such a process is known as a birth-and-death process and is characterized by the birth rates A, = and death rates p,j = qii-i- The generator of a birth-and-death process takes the form... 

In this subsection we treat several queues of the Ml Ml si K type. These queues have Poisson arrivals, exponential service times, s servers, and capacity K. For these queues, the number-in-system L(f), t 0, is a continuous-time Markov chain, in fact, a birth-and-death process (Subsection 3.6). The Markov property arises from the exponentitility of service and interarrival times—see the discussion following (33). The queueing discipline is taken to be FIFO in every case. The results presented here follow fairly directly from (38). 

Here we increase the number of servers to two. Assume now that p < 2, so that the two servers are sufficient to handle the arriving work. The Ml MU queue is the birth-and-death process with A =... 

This is the birth-and-death process with A = A and p, = np, for all n a 0. The service rate is proportional to the number of jobs in the system, and this captures the idea that each job is being served simultaneously at the seune rate, p,. The steady-state distribution of the number in the system is Poisson (p), again by (38), where p = A/p. is the traffic intensity as usual. Thus, the expected number in the system is simply (L) = p. Since there is no waiting in this system, other quantities are easy to derive but perhaps not very informative. For example, the distribution of the ste y-state time-in-system IP is the seune as the service distribution, namely, exponential with rate p. 

David G. Kendall, The generalized birth and death process Ann. Math. Statist, vol. 19 (1948) pp. 1-15. 

It is often remarked that stochastic models of chemical reactions can be easily extended to birth and death type phenomena that take place in other populations of entities. Although we agree with this approach in principle, we have to remark that from the mathematical point of view the relationship between the three categories, namely stochastic models of reactions, simple birth and death processes (Karlin MacGregor, 1957) and Markov population processes (Kingman, 1969) is not simple and is illustrated in Figure 5.1. 

Fig. 5.1 Logical relationships among population Markov processes, stochastic models of chemical reactions and simple birth and death processes.

The continued fraction representation of the transition factor has been applied for solving one-variable chemical master equations (Haag Hanggi, 1979, 1980). For simple birth and death processes with birth and death rate functions / and x the nearest neighbour transition gj is ... 

A sufficient condition for having Poissonian stationary distribution in a certain class of birth and death processes was given by Whittle (1968). However, his assumptions are slightly different from those of chemical reactions, therefore the search for precisely defined assumptions or for certain classes of reactions is necessary. 

If the stochastic model of a chemical reaction can be identified with a simple birth and death process, and the process has stationary distribution, then it is precisely Poissonian, if the reaction is an open compartmental system. 

Fig. 5.11 Interrelations between different types of processes (MPP, Markov population process CCR, complex chemical reaction SBD, simple birth and death process S, V, simple Markovian jump processes and models of reactions in the scalar and vector case respectively).

Karlin, S. McGregor, J. (1957). The classification of birth and death processes. Trans. Amer. Math. Soc., 86, 366-400. 

Roehner, B. Valent, G. (1982). Solving the birth and death process with quadratic asymptotically symmetric transition rates. SIAM J. Appl. Math., 42, 1020-46. Rogers, T. D. (1977). 23-41. Local models of cell aggregation kinetics. Bull. Math. Biol., 39, 23-41. 

Schuster, P. Sigmund, K. (1984). Random selection—a simple model based on linear birth and death processes. Bull. Math. Biol., 46, 11-17. 

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