## Queuing Theory and iiNet, Part II

It’s an interesting read, but the author makes a lot of basic errors. Unfortunately, customers refuse to line up and call at regular intervals and spend the average amount of time on the call. The reality is obviously more bursty than that and needs non-linear modelling.

The feedback from Michael Malone above was in response to my previous blog post on Applying Queuing Theory to iiNet Call Centre Data. I don’t accept that I made “a lot of basic errors”, but I did make a lot of assumptions. Or perhaps the statistician George E. P. Box said it better, “Essentially, all models are wrong, but some are useful.”

But Michael is correct – customers don’t line up and call at regular intervals, and the reality is more “bursty” (i.e. Poisson). My model is inadequate because it doesn’t take into account all the natural variation in the system.

One way of dealing with, or incorporating, this random variation into the model is by applying Monte Carlo methods.

Take the iiNet Support Phone Call Waiting Statistics for 6 February 2012, specifically for the hour 11am to noon. I chose this time block because the values are relatively easy to read off the graph’s scale – (a bit over) 664 calls and an average time in the queue of 24 minutes.

Now if we assume Average Handling Time (AHT), including time on the call itself followed by off-phone wrap-up  time, was 12 minutes, then my model says there were 664*(12/60) / (24/60 +1) = 95 iiNet Customer Service Officers (CSOs) actually taking support calls between 11am and noon on 6 February 2012. That’s an estimate of average number of CSOs actually on the phones and taking calls during that hour, excluding those on a break, performing other tasks, and so on. Just those handling calls.

But there will be a lot of variation in conditions amongst those 664 calls. I constructed a little Monte Carlo simulation and ran 20,000 iterations of the model with random variation in call arrival rates, AHT, and queue wait times.

Assumptions:

Little’s Law applies
Average time in the queue of 24 minutes
AHT (time on the actual call itself plus off-call wrap-up) of 12 minutes

then the result of the 20,000 monte carlo runs is a new estimate of 135 iiNet CSOs taking support calls between 11am and noon on 6 February 2012.

I ran a few more simulations, plugging in different values for number of CSOs handling calls (all else remaining equal – i.e. 664 calls an hour; AHT=12 minutes) to see what it did for average time in the queue. The results are summarised in the table below:

Modelling suggests that if iiNet wanted to bring the average time in the phone call support queue down to a sub-5 minute level during that particular hour of interest, an additional 85% in active phone support resourcing would need to be applied.

The table of results is graphically presented below (y-axis is time in queue, x-axis is CSOs)

Looks nice and non-linear to me 🙂 You can see a law of diminishing returns thing start to take place around about the point of the graph corresponding to 160 CSOs / 16.5 minute average queue wait time.

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