By Arun C. Murthy, Vinod Kumar Vavilapalli, Doug Eadline, Joseph Niemiec, Jeff Markham

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**Extra info for Apache Hadoop YARN: Moving beyond MapReduce and Batch Processing with Apache Hadoop 2 (Addison-Wesley Data & Analytics Series)**

**Example text**

7 Numerical Solution Accuracy Since we have a steady state analytical solution, we can evaluate the accuracy of the numerical method under consideration. However, to do so, we will first clarify how we compute stationary solutions using the numerical scheme. 11) holds at time t D 0, it will hold for all subsequent time steps. More precisely, if we define rn D x M X . 39) iD1 and r0 D 1, then, by the construction of the scheme, we have rn D 1 for all n 1. Since the solution we are considering converges rapidly to a stationary solution, it is useful to be able to compute the stationary solution directly.

We will immediately see that in mathematical terms the problem is very similar to the calcium release problem. For the ion channel case, however, the stochastic equation is one-dimensional and so is the associated deterministic partial differential equation. The basic Markov model is still based on the open and closed states, but we will also see that an inactivated state plays a central role. Optimal theoretical drugs will be derived and we will observe that they work nicely. 7 Mutations Described by Markov Models A trademark of mutations affecting ion channels and calcium release mechanisms is that they change the open probability and possibly also the mean open time and other characteristics of the channels involved.

9 the counting is based on the time interval running from t D t =2 to t D t , with t D 1 s. By considering the simulations shown in Fig. 7, we know that in this interval the probability density functions have reached their steady state solutions. In the figure, the histogram is computed running 500 Monte Carlo simulations. The figure clearly shows that the probability density approach gives the average of a large number of Monte Carlo simulations. We will see this repeated over and over in this text.