Search Process. Items arrive to a system of m cells labeled 1,...,m at times that form a Poisson process with rate λ. Each arrival independently enters cell i with probability αi and remains there...

Search Process. Items arrive to a system of m cells labeled 1,...,m at times that form a Poisson process with rate λ. Each arrival independently enters cell i with probability αi and remains there until it is deleted by a search. Independently of the arrivals, searches occur at times that form a Poisson process with rate μ, and each search is performed at cell i with probability δi. If items are in the search cell, one item is deleted; otherwise no items are deleted and the search is terminated. Let X(t)=(X1(t),...,Xm(t)) denote the numbers of items in the cells at time t. Justify that X(t) is a CTMC and classify its states. Find an invariant measure for it. Prove that X(t) is positive recurrent if and only if αiλ ≤ δiμ, 1 ≤ i ≤ m. Find the mean and variance of the number of items in node i in equilibrium (that is E[Xi(0)] and Var[Xi(0)] when X(t) is stationary).