Let the long-run average workload in a system be $Z=\int_{0}^{t}Z_{s}.ds$ where $Z_{t}$ be the workload at time to be the sum of all remaining service times of all customers in a system t.

By the arrival theorem or PASTA:

The probability of steady state seen by an independent observer outside the system is the same as the probability of the steady state as seen by an arriving customer.

Claim: $Z = W_{Q}$ where $W_{Q}$ is the average waiting time in the queue

Why is this claim true?


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