# Distribution of "priority" emails among agents with different speeds (strange question)

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.

• 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 stats.stackexchange.com/questions/129322 .)
– whuber
Jan 7, 2015 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. Jan 7, 2015 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. Jan 7, 2015 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. Jan 7, 2015 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 7, 2015 at 22:44