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    A computer virus is modelled by a branching process which branches at times of a renewal process. It begins only one machine. At renewal times of a discrete-time renewal process, with mean inter-renewal time μ, a command server sends an instruction to all computers with active infections and tells them to spread.
    All actively infected machines get one chance to spread, and upon recieving the 'spread' instruction from the command server they independently attempt to spread the worm to a number of new machines according to some distribution S with mean λ, and then those active worms go dormant forever.
    Newly infected machines start with active infections which spend their time building up a database of machines to infect when they themselves recieve the 'spread' instruction from the command server, and the process repeats.

    Write Z_n for the total number of infected machines by time n. Also write N_n for the number of command server instructions sent by time n.

    Relevant formula could be total progeny, mean of a renewal process E(N_n) = sum (u_m) from m=1 to m=n.

    Show that for the process detailed above

    E(Z_n) =
    1 + E(N_n) for λ=1
    E( (λ^(1 + N_n) - 1)/λ-1 ) for λ≠1
 
 
 
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