Birth-death processes

Then, since in any system accumulation is the net result of both evolution and birth/death processes, and since any latex particle can be characterized by a set of physical quantities which will fully specify a given particle or class of particles, one can obtain the following population balance equation (33) ... 

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

Since the formal chemical kinetics operates with large numbers of particles participating in reaction, they could be considered as continuous variables. However, taking into account the atomistic nature of defects, consider hereafter these numbers N as random integer variables. The chemical reaction can be treated now as the birth-death process with individual reaction events accompanied by creation and disappearance of several particles, in a line with the actual reaction scheme [16, 21, 27, 64, 65], Describing the state of a system by a vector N = TV),..., Ns, we can use the Chapmen-Kolmogorov master equation [27] for the distribution function P(N, t)... 

N(t) can be interpreted as the size of the population at time t. The most simple kind of birth-death process is that for which the rates of birth and death are time independent. A birth-death process N(t) is called a pure birth process (respectively pure death process) if Dm = 0 (Bm = 0) for any m. 

From the Kolmogorov equations (4.41) and (4.42), one obtains the difference-differential equations for the birth-death process. The backward equation is given by... 

It is of importance to point out that if the right-hand side is truncated after two terms (diffusion approximation), the last relation leads to an expression similar to the familiar Fokker-Planck equation (4.116). The approximation of a master equation of a birth-death process by a diffusion equation can lead to false results. Van Kampen has critically examined the Kramers-Moyal expansion and proposed a procedure based on the concept of system size expansion.135 It can be stated that any diffusion equation can be approximated by a one-step process, but the converse is not true. 

Depending on the Xx - x and lx relationships, two cases will be considered, viz., the linear birth-death process and the z-stage reversible-consecutive process. 

The linear birth-death process. It is assumed here that p-x =... 

Kopp-Schneider, A. (1992). Birth-death processes with piecewise constant rates. Stat Probabil Lett 13, 121-127. 

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

Under the prudent criterion (PC) trams that enter a degraded state drop-off passengers and reach the workshop (this strategy allows to minimize the risk of stop on line). In this case, the resulting model becomes a birth-death process on the c axis of Figure 6. 

In detail, in (Koutras VR Platis A. N., 2006), a birth-death process was proposed to model wehsite visitors arrival and service. An optimization problem was solved to determine the optimal trade off between resource availahihty for high priority visitors and free resource access to lower priority visitors. The major contribution of this paper consisted in deriving formulas for the probabihty that a website visitor has no further access to resources and in determining the optimal reserved resources assuring the above trade off. 

As a simple illustration of how reactions may be treated stochastically on mesoscopic scales, we show how one may construct a mesoscopic Markov process model for the Schlogl reaction (Eq. [3]). As before, we assume that species A and B in the Schlogl mechanism are pool species and their concentrations do not change in time Their effect is incorporated into the rate constants of the reaction. A Markov model of the reactions is a birth-death process, where chemical species are born in some reactions (X is born in the reactions (1) A -> X and (3) 2X - - B 3X) and dies in others (X dies in the reactions (2) X A and (4) 3X 2X - - B). Suppose there are ttx molecules of species X in a cell. Then the probability that a reaction X — A occurs and changes the number of molecules in the cell from x to — 1 is just... 

We let X(t) denote the size of a population at time t with the initial condition X(0) = i. Given X(t) = k, the following are for the birth-death process ... 

Deep-bed filtration involves the flow of particles through randomly distributed passages thus, it tends to be stochastic in nature. The filtration process has been modeled as a pure birth process [12,13], a birth-death process [14-17], a random-walk process [18], and a stochastic diffusion process [19]. 

In Litwiniszyn s pure birth model, the entire bed is considered as on state, and the number of blocked pores in a unit volume of the bed is considered as a random variable. In his birth-death model, the number of trapped particles over the entire bed is a random variable. Fan and his co-workers have extended the pure birth and birth-death process models by incorporating... 

To conclude, we can assert that despite having a globally stable steady state, the birth-death process is a non-equilibrium phenomenon (thermodynamically speaking) because, in the steady state ... 

Abstract In this chapter we generalize the birth-death process analyzed in the previous chapter to account for enzymatic molecule synthesis, rather than simple Poissonian production. To facilitate the analysis we assume a time-scale separation in the enzymatic reactions, and use it to reduce the complexity of the complete system. With this simplification the generalized birth-death process can be separated into two different subsystems that can be studied separately, and correspond to the systems studied in Chaps. 3 and 4. The simplification procedure, introduced in Sect. 5.1, is a very useful mathematical tool way beyond the scope of the present chapter. 

Most biochemical reactions are catalyzed by enzymes. Therefore, although quite instructive, the model for the birth-death process studied in the previous chapter is not good enough an approximation in many instances. Typically, an enzymatic process consists of a series of chemical reactions that occur at different rates, and in some occasions it is possible to identify two well-separated time scales. When this occurs, the time-scale separation can be exploited to simplify the analysis of the whole system. Below we introduce a methodology to perform such simplification. 

Notice that Eq. (5.17) is the same as Eq. (4.10). Hence, the quasi-stationary approximation, allowed us to reduce the system of chemical reactions in (5.1) to a birth-death process in which the effective production and degradation rates are ... 

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