Birth-death process example

Examples include birth—death processes, die Poisson process, and die random telegraph process. 

Examples include birth-death processes, the Poisson process, and the random telegraph process. 

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... 

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. 

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. 

In general we may conclude that the choice of the particle state is determined by the variables needed to specify (i) the rate of change of those of direct interest to the application, and (ii) the birth and death processes. The particle state may generally be characterized by a finite dimensional vector, although in some cases it may not be sufficient. For example, in a diffusive mass transfer process of a solute from a population of liquid droplets to a surrounding continuous phase (e.g., liquid-liquid extraction) one would require a concentration profile in the droplet to calculate the transport rate. In this case, the concentration profile would be an infinite dimensional vector. Although mathematical machinery is conceivable for dealing with infinite dimensional state vectors, it is often possible to use finite dimensional approximations such as a truncated Fourier series expansion. Thus it is adequate for most practical applications to assume that the particle state can be described by a finite dimensional vector. ... 

A nuclear reaction is a spontaneous process. In nature, you can find naturally occurring radionuclides that were created in the birth or death of a star light-years away from planet Earth. Some radioisotopes occur so rarely that huge effort has to go into their purification. Separating a stable isotope from chemically identical radioisotopes for use in a nuclear reactor is an energy-intensive process. Lots of material, such as pitchblende for example, has to be mined in order to purify radioisotopes of uranium for use in a nuclear power plant. Once again, this is because they are naturally occurring and occur so rarely. 

We have in this chapter developed the various features of formulation of population balance. Section 2.11 discussed several examples in which the different features were demonstrated. However, in most of the examples, the net birth term could be dealt with through the boundary conditions. In the next chapter it will be our concern to investigate closely the nature of the birth and death terms in population balance due to breakage and aggregation processes... 

The reader is invited to revisit the examples in Section 2.11 to develop a proper appreciation for the birth and death rates in population balance equations that appear through the boundary conditions. In this regard, the example in Section 2.11.5 presents the boundary condition (2.11.23), which is a particularly interesting example of a birth process occurring at a boundary. 

Kinetics is not only important for chemical or biological processes, but also for everyday life, as inspected in Example 4.3.1 for the social process of the birth and death of a rumor with interacting subpopulations of ignorants, spreaders, and stiflers. 

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