Exponential distribution, continuous

In studies of low-mobility insulators, two types of continuous trap distributions are commonly used the exponential distribution of traps [364] and the Gaussian distribution of traps [365]. 

Due to the functional form of Eq. (185), which, except for the constant coefficient, is identical to the SCL j-U characteristics for a continuous exponential distribution of the form (175) or (176) derived in the case of the infinitely sharp point traps, the experimental data arising from the discrete macrotrap background can, without scrutiny, be mistakenly attributed to the continuous exponential distribution of point traps. The transition from the low-to high-field regions of the current occurs at a voltage (see Fig. 71)... 

The age of a fluid element is defined as the time it has resided within the reactor. The concept of a fluid element being a small volume relative to the size of the reactor yet sufficiently large to exhibit continuous properties such as density and concentration was first put forth by Danckwerts in 1953. Consider the following experiment a tracer (could be a particular chemical or radioactive species) is injected into a reactor, and the outlet stream is monitored as a function of time. The results of these experiments for an ideal PFR and CSTR are illustrated in Figure 8.2.1. If an impulse is injected into a PFR, an impulse will appear in the outlet because there is no fluid mixing. The pulse will appear at a time ti = to + t, where t is the space time (r = V/v). However, with the CSTR, the pulse emerges as an exponential decay in tracer concentration, since there is an exponential distribution in residence times [see Equation (3.3.11)]. For all nonideal reactors, the results must lie between these two limiting cases. 

Hence, we can generate an exponential random variable by generating a random variable U and then use Eq. (33.6) to draw a sample from an exponential distribution with mean X. This method of simulating continuous variables is called the inverse transformation method. Although the method can be applied to any distribution, either continuous or discrete, the problem with the inverse transformation approach is that it is often difficult to invert the CDF, if it even exists, to an analytical solution. 

The continuous phase is well mixed but there is a dispersed phase. The particles in the dispersed phase behave as PFRs. They are in contact with the continuous phase but are isolated from each other and have an exponential distribution of residence times. This case is treated in Examples 11.17 and 15.12. 

A perfect mixer has an exponential distribution of residence times W t) = exp(—r/7). Can any other continuous flow system have this distribution Perhaps, surprisingly, the answer to this question is a definite yes. To construct an example, suppose the feed to a reactor is encapsulated. The size of the capsules is not critical. They must be large enough to contain many molecules but must remain small compared to the dimensions of the reactor. Imagine them as small ping-pong balls as in Figure 15.1 la. 

In those cases where it is not practical to mix discrete classes of narrowly sized particles, an approximation yields a continuous particle size distribution. Mathematically, it is described by the Fuller distribution [B.42], an exponential distribution in which the exponent must be between 1/3 and 2/3. 

Exponential Distribution A third probability distribution arising often in computational biology is the exponential distribution. This distribution can be used to model lifetimes, analogous to the use of the geometric distribution in the discrete case as such it is an example of a continuous waiting time distribution. A r.v. X with this distribution [denoted by X exp(A)] has range [0, - - >] and density function... 

Memoryless Property The exponential distribution shares the memoryless property with the geometric distribution, and it is the only continuous distribution having this property. The proof of the property is simple ... 

The generator Q determines how a continuous-time Markov chain evolves via (26). There is another, more direct prescription for the evolution of the chain in terms of Q. If the chain is now in state i, then the time T until the next change of state has the exponential distribution with rate qj. 

Thus, the transitions are always from a state n to the state n + 1. The transitions are, of course, arrivals because they cause the count N to increase by 1. The probability of a transition in a short interval of time h is approximately Xh for any n by (26). This observation corresponds precisely with the description of the Poisson process in terms of coin tossing in Section 2. Moreover, the fact that the time tetween arrivals in a Poisson process is exponential may be seen now as a consequence of the fact, expressed in (33), that the holding times in any continuous-time Markov chain are exponentially distributed. 

As mentioned on page 61, CTRWs are known as semi-Markov processes in the mathematical literature. In this section we provide a brief account of semi-Markov processes. They were introduced by P. Levy and W. L. Smith [253,415]. Recall that for a continuous-time Markov chain, the transitions between states at random times T are determined by the discrete chain X with the transition matrix H = (hij). The waiting time = T - for a given state i is exponentially distributed with the transition rate k , which depends only on the current state i. The natural generalization is to allow arbitrary distributions for the waiting times. This leads to a semi-Markov process. The reason for such a name is that the underlying process is a two-component Markov chain (X , T ). Here the random sequence X represents the state at the th transition, and T is the time of the nth transition. Obviously,... 

For now, assume in addition that the total repair time is exponentially distributed. Under this assumption, each functional group can be modelled as a continuous time Markov chain with -f 1 states. We let state / 0,..., ff correspond to the situation in which there are i pieces of defect equipment in the group. When in state i < R + 1, we move to state i + 1 with rate This transition corresponds to a failure in a piece of equipment within the group. When in state / > 0, we move to state / — 1 with rate t . This transition corresponds to a fiiushed repair. 

For the single tram all the failure (repair) transitions shown in Figure 1 by continuous (dotted) arrows are possible. Times to failures are assumed to be exponentially distributed. 

Secondly, how do we understand the term, present MTTF/meian life time One of the reasons why the exponential distribution is over-used may be because it does not have to deal with this tricky question. In this case, life time estimations are based on experiences from the past, but the failure rate is changing continually and is not the same today as it was yesterday. In some cases we have several of the same type of component and we may base the estimation on the most recent experience from all of them. If there is only one or few similar components or there have been very few or no failures, we can either obtain expert judgment and have a much higher rmcertainty or we can spend some time and resources investigating the components more closely and estimate a remaining life time based on expert observations and measurements. 

Dellaert studies two lead time policies, CON and DEL, where DEL considers the probability distribution of the flow time in steady-state while quoting lead times. He models the problem as a continuous-time Markov chain, where the states are denoted by (n, 5)=(number of jobs, state of the machine). Interarrival, service and setup times are assumed to follow the exponential distribution, although the results can also be generalized to other distributions, such as Erlang. For both policies, he derives the pdf of the flow time, and relying on the results in [88] (the optimal lead time is a unique minimum of strictly convex functions), he claims that the optimal solution can be found by binary search. 

Both the RBC distribution (8) and the geometric distribution (11) are defined only for specific integer bubble sizes, and derivatives of their distribution functions do not exist. For subsequent developments we need an equivalent continuous distribution. Fortunately, for N and k large with respect to m, both discrete distributions can be closely approximated by the exponential distribution if its mean is set to the RBC mean volume given by (10). The exponential probability density is... 

Continuing from remarks ( 62)-( 63), the following conclusion can be drawn The formula t F = h is often used for the estimation of the natural linewidth. This formula is sometimes interpreted as the time-energy equivalent of the Heisenberg relation, where r is the uncertainty (standard deviation) of the lifetime and F (FWHM) is that of the energy state. It should be stressed, however, that while r can play the assigned role (because the standard deviation is equal to the expected value in the case of the exponential distribution), the quantity F cannot be interpreted as standard deviation, since the Cauchy distribution does not have any. 

Table I-1 provides the hazard rate (HR) calculated from the simplified equation, which is typically the basis of Layers of Protection Analysis (LOPA), and a more rigorous equation based on the exponential distribution. The shaded area shows that for high-demand mode SIFs, the simplified equation yields a hazard rate that exceeds the failure rate k of the SIF, which is 0.1/year. In the case of either high-demand or continuous mode, the simplified mathematics developed for low-demand mode are not adequate, and more advanced assistance should be obtained from someone knowledgeable in the applicable mathematics of modeling such cases.

For functions which have a high demand rate or operate continuously, the accident rate is the iliue rate, X, which is the appropriate measure of performance. An alternative measure is mean time to failure (MTTF) of the hmction. Provided failures are exponentially distributed, MTTF is the reciprocal of X. 

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