Given that:
$$W_b = \frac{\sum_{i=1}^{N_b} \frac{\sum_{j=1}^{M_i} W_{i,j}}{M_i}}{N_b}$$
and
$$W = \frac{\sum_{i=1}^{N_b} \left(\sum_{j=1}^{M_i} W_{i,j}\right)}{\sum_{i=1}^{N_b} M_i}$$
and it is known for a fact that $W_b \leq W$ for finite $N_b$, how do you prove, as $N_b \rightarrow \infty$ that the relation
$$W_b \leq W$$
is true?
I'm really struggling to get started with this one. It just seems to me that since this relation is true for every finite number, why would it not be true for $\infty$?
$\bf{Is\ this\ a\ possible\ proof?}$
The statement $W_b \leq W$ has already been proved for all possible values of $N_b$, including $N_b \rightarrow \infty$. The analysis does not depend on the value of $N_b$, but rather on the relationships within each term of the expressions. That is,
The denominator of $W_b$ ($N_b$) is always less than or equal to the denominator of $W$ ($\sum_{i=1}^{N_b} M_i$) because each busy period ($N_b$) has at least one customer ($M_i$).
The numerator of $W_b$ is the sum of the averages of waiting times during each busy period. We calculate the average waiting time for each busy period by dividing the sum of waiting times for that period ($\sum_{j=1}^{M_i} W_{i,j}$) by the number of customers served during that period ($M_i$).
The numerator of $W$ is the sum of all waiting times during each busy period.
Given these facts, each term in the numerator of $W_b$ is less than or equal to the corresponding term in the numerator of $W$, and the denominator of $W_b$ is less than or equal to the denominator of $W$. Therefore, $W_b \leq W$.
The limit of $N_b \rightarrow \infty$ does not change this relationship. As $N_b$ increases, both $W$ and $W_b$ tend to be more accurate representations of the average waiting time in the system and during a busy period, respectively. However, the relationship $W_b \leq W$ still holds. In other words, the average waiting time during a busy period will always be less than or equal to the average waiting time in the system as a whole.