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Let $Z_{t}$ be the workload at time to be the sum of all remaining service times of all customers in a system t.

In short, $Z_{t}$ has the dimension workload per unit customer in the interval between t and $t+dt$. Over a time interval between 0 and t and for a non-uniform $Z_{t}$, the total workload is

$Z=\int_{0}^{t}Z_{s}.ds$

So far, all looks to be reasonable.

Now, the author makes a claim that $y is the rate paid by each customer in the queue when his/her remaining service time is y.

"Remaining service time" comes across as very poorly worded to me.

As a customer is served, their service time decreases and so does the rate paid by that customer. This seems to suggest a cumulative total sum paid by any customer which isn't reasonable. After all, if a customer starts being served by a server and has a service time of x and pays x, then after one minute of being served, his remaining service time is x-1 and pays rate $(x-1)$. This implies that he pays $x+(x-1)$ in total after a minute of service. If this is reasonable, I would appreciate if someone shed light on my doubts.

He goes on to claim that if Y is the expected total payment made by a customer then $Y\lambda$ = Z is the average workload.

I fail to understand how this may be true. The dimensions are different!

Any help is appreciated.

Below is an excerpt:

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