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but I figure if someone knows how to answer it, it may be someone here.

Basically I have this weird distribution where the customer service agent speed (in terms of contacts per hour) is very strongly correlated with the % of priority emails answered.

Mind you, the priority emails are a different topic, and may, or may not, be easier or harder to answer.

I'm thinking I should hold off telling you which direction the correlation is, to not bias your judgement.

But I want to know if this distribution is due to cherry-picking emails ----- or a random phenomenon.

Let's say we have 5 agents who answer 20 emails/ hour, and 5 agents who answer 10 emails/ hour.

If 6% of emails are "high priority" and immediately flow to the top of the pile, and come in at random intervals (poisson distribution) ---- would the "fast" agents statistically answer a higher (% of their total) priority emails, equal %, or lower % than the "slow" agents?

Mind you the next email is only fed once the agent is done on the current one (duration of work twice as long for slow agent of course).

I know this is a complicated question but just throwing it out there. I'm kind of interested in how to tackle this, if not just somehow running a Monte Carlo simulation.

John Babson
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    **It depends.** *E.g.*, if the agents are not overworked and the system is configured to give the next email to the most efficient available agent, then (obviously) the faster agents will tend to process the priority emails. Evidently, a lot will depend on how agents are matched to emails. Another issue concerns the nature of those e-mails. If, for instance, it takes a very long time to process a priority e-mail, then naturally the slower agents will happen to be those who encountered more priority e-mails! (For a simulation solution, see http://stats.stackexchange.com/questions/129322 .) – whuber Jan 07 '15 at 21:30
  • I knew I may not have given enough parameters! Ha -- Your link says 404 not found, though. Actually, the agents aren't overworked, but neither are they underworked. For reasons I won't get into, it's optimal to get down to zero emails at closing time, and not a minute before. So there's always a shrinking backlog -- in this case the email backlog is never empty. Okay, I'll tell you the fast agents have LESS priority emails completed, and the correlation to speed is high (40% r^2 among 30 agents). Cherry pick or not, this would indicate the priority emails are slower to complete. – John Babson Jan 07 '15 at 21:51
  • Your statement that if the email is slower to complete, slower agents would encounter it more, is interesting, though I still don't quite get it. I guess that makes sense if the "priority email" complete times are less variable among agents, thus the fast agents will "race through" lots of easy emails in between them, giving them less %. Thanks for the help. I guess that less variability part is key. – John Babson Jan 07 '15 at 21:54
  • EDIT: Realized your link included the period. That's an interesting link but it seems more like basic Erlang C, which every call center basically uses constantly. But it may help if I decide to do a monte carlo. – John Babson Jan 07 '15 at 22:01
  • (I fixed the link: it's just a tiny SE bug. The value of the simulation is that it enables you to explore actual realizations of the queuing process.) Please note $R^2=40\%$ might not be particularly high; it could easily result from random variation (depending on circumstances). My statement about the relationship is easily understood from a causal viewpoint: *because* it takes longer (hypothetically) to process a priority e-mail, *therefore* the agents who *happen* (at random) to process more priority e-mails would take more time overall, *ceteris paribus*. – whuber Jan 07 '15 at 22:44
  • I know if we're trying to build a model, that correlation is not great. But really, I order agents by CPH (contacts per hour), and then % of priority tickets answered is nearly almost in the exact same order. This reveals that it they are linked in some manner. I feel it's possible it's inherent to the system that faster agents answer less priority mails as a % of their totals. Thinking on it, it's probably more likely it's a natural occurrence. If it wasn't, % priority would be staggered depending on level of cherry-picking, not gradually increased as speed goes down. if that makes sense – John Babson Jan 07 '15 at 22:53
  • The two key questions your description does not answer are (1) how strong is the effect (that is, how much slower do the agents become as a function of proportion of priority tickets processed) and (2) how significant is it (that is, to what extent could it be explained by chance variations). Although you say the orderings are about the same, when you repeat the analysis with new data, is it always the same agents who are the slowest or not? – whuber Jan 07 '15 at 22:58
  • With a range of 5-12% emails completed being priority, each additional 1% of your emails being priority will decrease your Contact Per Hour by nearly 1.0 (average is about 12). Actually, we just changed software systems --- in previous months, with the old software (where you can take multiple emails at once) -- there was no apparent pattern. I guess this is evidence that it's unintentional phenomenon, perhaps. – John Babson Jan 07 '15 at 23:39

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