The derivation of the formula for the expected processing times for a sequence of tests that have to be passed before an action is performed is shown below as well as a few simple examples to illustrate the calculations.
The expected processing times appear to favor the more probable cases being tested first even though they take longer but this may not always be true.
A dedicated processor would only have to check for the task it was intended to handle so if we ignore waiting times and there was one queue for each test case the expected processing time would be the the expected time defined by ΣpkTk.
When waiting times become unexpectedly large more processors might be called for to improve efficiency. Letting people decide for themselves what line to get into could help improve processing time. If troublesome cases get into the wrong line a special processing queue might be set up so they can be redirected there. A directory mechanism could be as simple as a sign or maybe an app for unusual situations.
Edit (Mar 2): Added dedicated server processing times for the examples and did a little cleanup.
Supplemental (Mar 2): Testing and tasking both take time and the multiplier formula for sequential actions above applies to general purpose servers dealing with queues or streams requesting actions. Sometimes a referral is the best response. When no action is required it may be best to decline a request.
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