Question: If there are N robots operating independently according to the above laws and pk (t ) is the probability that exactly k are idle at

If there are N robots operating independently according to the above laws and pk (t ) is the probability that exactly k are idle at time t , show that p′k (t ) = λ(N − k + 1) pk−1(t ) − λNpk (t ) + λ(k + 1) pk+1(t ), for k = 0, 1, . . . , N, subject to the rule that p−1(t ) = pN+1(t) = 0.

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