Messages arrive to a web page at times that form a Poisson process with rate λ. The messages are independently of high priority with probability p, and each of these high priority messages is...

Messages arrive to a web page at times that form a Poisson process with rate λ. The messages are independently of high priority with probability p, and each of these high priority messages is independently of type i with probability p_i, ≤ i ≤ m. Let N_i(t) denote the number of high priority messages of type i that arrive in [0, t]. Are N1,...,Nm independent Poisson processes? If so, specify their rates.