A system experiences shocks that occur in accordance with a Poisson process havi

Question

A system experiences shocks that occur in accordance with a Poisson process having a rate of 1/hour. Each shock has a certain amount of damage associated with it. These damages are assumed to be independent random variables (which are also independent of the times which occur), having the common density function f(x) = x e^-x 0, x > 0 Damages dissipate in time at an exponential rate alpha-that is, a shock whose initial damage is x will have remaining damage value xe^as at time s after it occurs. In addition, the damage values are cumulative. Thus, for example, if by time t there have been a total of two shocks, which originated at times t_1 and t_2 and had initial damages x_1 and x_2, then the total damage at time t is sigma^2_i = 1 x_i e^-a(t - t_i). The system fails when the total damage exceeds some fixed constant C. (a) Suppose we are interested in utilizing a simulation study to estimate the mean time at which the system fails. Define the "events" and "variables" of this model and draw a flow diagram indicating how the simulation is to be run. (b) Write a program that would generate k runs. c) Verify your program by comparing output with a by-hand calculation. (d) With alpha = 0.5, C = 5, and k = 1000, run your program and use the output to estimate the expected time until the system fails.

Explanation / Answer

You will want to have a race course that gives runners a unique experience. This could mean a nice gentle downhill course, or perhaps conversely, a challenging uphill one, depending upon the demographic you want to cater to. Often runners want a route that is flat or gentle downhill, so if you want to draw more participants, it is ideal to avoid hills. If it is necessary for you to minimize expenses, you may want to have an out-and-back or a loop course to avoid the expense of transporting runners. You can use our FREE race map creator to figure out different course options, and look at the elevation chart change as you alter the route.

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