The number of tasks N that a server in a Facebook data center receives in an...

 

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The number of tasks N that a server in a Facebook data center receives in an hour is a geometric random variable with parameter p. The execution time of task i, denoted by X;, i = 1,2,..., N, is exponentially distributed r.v. with mean 1. We assume that the X;'s are i.i.d. We are interested in S = E, ri, the total time that the server will take to finish all the tasks submitted in an hour. 1. Find E[S]. 2. Let s(w) be the characteristic function of S. Prove that Ps(w) = MN(PX1 (w))| %3D The number of tasks N that a server in a Facebook data center receives in an hour is a geometric random variable with parameter p. The execution time of task i, denoted by X;, i = 1,2,..., N, is exponentially distributed r.v. with mean 1. We assume that the X;'s are i.i.d. We are interested in S = E, ri, the total time that the server will take to finish all the tasks submitted in an hour. 1. Find E[S]. 2. Let s(w) be the characteristic function of S. Prove that Ps(w) = MN(PX1 (w))| %3D