Poisson process exponential waiting times

The outflow of a CSTR is a Poisson process, i.e., fluid elements are randomly selected regardless of theirposition in the reactor. The waiting time before selection for a Poisson process has an exponential probability distribution. 

The exponential distribution with parameter X is the distribution of waiting times ( distance in time) between events which take place at a mean rate of X. It is also the distribution of distances between features which have a uniform probability of occurrence (Poisson process), such as the simplest model of faults on a map. The gamma distribution with parameter n and X l, where n is an integer is the distribution of the waiting time between the first and the nth successive events in a Poisson process. Alternatively, the distribution /(t), such as... 

As can be seen from the figure, both distributions are purely exponential. The exponential shape of the distributions shows that the waiting time of an embedded atom is governed by a Poisson process with rate t 1. This implies that subsequent long jumps are independent, which we take as proof... 

This is the exponential ) density. So the waiting time for the first occurrence of a Poisson process with intensity rate A has the exponential(X) distribution. Let Ti,T2,. .. be the times of the first, second,. .. occurrence of the Poisson process respectively. The interarrival times T2 — Ti, T3 - T2,... between occurrences of the Poisson process with intensity rate A will all have the exponential(X) distribution. The "arrivals occurring according to a Poisson process with rate A" is equivalent to "the interarrival times have the exponential(X) distribution."... 

Many commonly used distributions are members of the one-dimensional exponential family. These include the binomial n, n) and Poisson fi) distributions for count data, the geometric ir) and negative binomial r, tt) distributions for waiting time in a Bernoulli process, the exponential ) and gamma n, A) distributions for waiting times in a Poisson process, and the normal p, o ) where [Pg.89]

Piecewise constant hazard rates. Sometimes we know the hazard rate is not constant. We can model this by partitioning the time axis and having a constant hazard rate over each section of the axis. Since the waiting times for the first arrival for a Poisson Process have the exponential distribution... 

Having a pieture always helps to better understand a concept. With this purpose in mind, let us device an algorithm to simulate individual realizations of a Poisson process. The key for this algorithm is the waiting times between consecutive events recall that the set of waiting times in a given realization of a Poisson process can be viewed as realizations of a random variable that obeys an exponential distribution with mean A. Once we have understood this, it is straightforward to device the... 

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. 

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