I am looking at a queuing problem with no aftereffects. We have the following convention $v_{k}(t) = P({k \text{events happen in the time interval} (0,t)})$. We have the following equation we are trying to simplify and solve.
$v^\prime_k (t) = -\lambda v_k(t) + \sum_{1}^{k} p_i v_{k-i}(t)$
We use the characteristic method to solve the above DE. $F(t,x) = \sum_0^\infty v_k(t) x^k$.
Substituting in we get the following;
$\frac{\partial F}{\partial t} = - \lambda F + \lambda \sum_{k=1}^\infty x^k \sum_{l=1}^k p_l v_{k-l}(t)$
Now the book I am reading reduces the above to the following, I am not sure what I am missing here.
$\frac{\partial F}{\partial t} = - \lambda F + \lambda \sum_{l=1}^\infty p_l x^l \sum_{q=0}^\infty v_{q}(t) x^q$
What is the magic that goes to the last step.
Since $v_k(t)$ is nonnegative for all $k,t$ we may use Tonelli's theorem to interchange the order of summation: $$ \sum_{k=1}^\infty x^k \sum_{l=1}^kp_lv_{k-l}(t) = \sum_{l=1}^\infty p_l\sum_{k=l}^\infty x_lv_{k-l}(t). $$ The change of variables $q=k-l$ yields $$ \sum_{l=1}^\infty p_l\sum_{k=l}^\infty x_lv_{k-l}(t) = \sum_{l=1}^\infty p_lx^l\sum_{q=0}^\infty v_q(t)x^q. $$