Dimensioning a call centre to meet service-level targets

31 July 2012

A client recently enquired as to whether STEM could be used to relate customer demand on a call centre to underlying server costs according to varying service-level targets such as, ‘the probability of a wait in excess of 5 seconds should not exceed 10%’.

The objective is to be able to dimension the underlying system of resources to achieve a target maximum probability for the wait in a M/M/k queue to exceed a certain wait threshold. For a given arrival rate, service time and postulated Halfin-Whitt delay function, P(y), how to find the value of y which satisfies prob (wait > threshold) = target, from which the number of servers required can then be inferred?

Note: conceptually the wait threshold described above represents the limit of acceptable performance and such a target probability would be referred to as a blocking probability in the context of the Erlang B formula analysis more familiar to current STEM users. The waiting probability arising from the Halfin-Whitt delay function is actually very similar to that given by the Erlang C formula.

1. A typical goal-seek problem

There are many references to the underlying mathematical analysis in the literature, but the specific query was whether there was any way that a user could embed the required goal-seek algorithm into a STEM model. If you generalise the structure of this problem as follows, with f(y) representing the combination of the delay function and the subsequent probability calculation, then the real question is how would you get STEM to calculate y?

Figure 1: Generic structure of a goal-seek problem

The idea is to seed the y and dy variable with appropriate values, such as 0 and 0.1 respectively, and then calculate the current value of f(y). According to whether this is below or above the target (i.e., a positive or negative difference) then set dy to be the same as before if still below the target, otherwise set dy = –previous dy / 2 in order to home-in on the target. Then set y = previous y + dy and continue … which looks a lot like the prescription for a macro script!

2. Iteration in the Editor

One of the lesser known and more infrequently used capabilities of the STEM Editor is its ability to use controlled iteration to resolve convergent calculations based on circular references between a set of input fields in a model. A simple example would be:

  • User 1 = User 2 / 2
  • User 2 = User 1 + 1

Such a construct is not much use by itself, but becomes interesting if other parameters feed into the loop (e.g., set User 2 = User 1 + User 3 above), and can be the only way to resolve certain arithmetical systems where no closed form exists for the solution.

With a healthy dose of confidence and only a modicum of care, it is not so difficult to capture the parameters and terms of the Halfin-Whitt delay function above within a series of User Data in a STEM model. If you manually seed y = 0 and dy = 0.1, and then replace first dy and then y with the appropriate formulae, STEM will then iterate through the formulae as desired so as to home in on the required value of y, and then calculate the required number of servers.

Figure 2: Using iteration to find a convergent solution for circular references in the STEM Editor

With a degree in ingenuity from a well-known UK university, it is possible to extend the simulation to include a reset variable which will detect a change in the input parameters and force an automatic re-seed of dy and y so that y can be re-calculated automatically after changing any of the input parameters.

Figure 3: Goal-seek problem with automatic reset to seed new calculation

Since the User Data are all time-series variables, it is actually possible to generate a graph of the output by making any of the inputs (including the target probability) vary in time, so you can get a direct understanding of the impact of changing the service level.

Note: the alert reader may have noticed that a standard simulation of the ‘error function’, erf(), was slipped into the final STEM 7.3 release at the last minute. It was required as part of the formulation of the Halfin-Whitt delay function in order to implement f(y). So if you fancy a challenge, there is nothing to stop you implementing your own version of this function once you have downloaded and installed STEM 7.3!

3. A new Goal Seek transformation

This is all very well if all the parameters are known in advance in the STEM Editor, but the calculation will be much more usefully applied to a dynamic quantity such as the output of a transformation calculated at run-time. In fact the support for aggregate measures in STEM 7.3 lends itself very well to this kind of problem where you might want to calculate and work with the actual volume of call-centre events in a given period.

So the obvious next step is to look to implement a new type of Goal Seek transformation in STEM which would encapsulate an equivalent iterative logic as described above and with the general structure outlined in Figure 3. An important detail to understand is that, while this transformation will benefit from multiple dynamic inputs plus static user data just like the current Expression transformation, the planned iteration will be limited to the internal calculation of the output of that transformation. It will continue to be the case that circular references between separate transformations are not allowed.

In fact, the Goal Seek transformation will have a lot in common with the Expression transformation, including the Cost Allocation factors to weight individual inputs, but its output will not be defined directly by the expression. For each period of a model run, the transformation will iterate a search variable, y, to achieve a given target for a user-defined function (the expression) of (i) its inputs, (ii) any static user data and (iii) the search variable, and instead the transformation’s output will be y.

According to our draft specification, and compared to the current interface for an Expression transformation, the Goal Seek transformation will require the following additional parameters:

  • Target Value as time series, default = 0.0
  • Seed Output as time series, default = 0.0
  • Seed Increment as time series, default = 1.0
  • Refine Factor as double, default = 0.5
  • Maximum Iterations as integer, default = 100
  • Tolerance as double, default = 1.0e–09.

Figure 4: Mock-up of the dialog interface for a Goal Seek transformation

It is clear that by no means all expressions will be amenable to a goal-seek strategy. The expression must at the very least reference the output variable, y, and ideally it will be a monotonic function with respect to y; though that may not be essential so long as the seed values are well chosen. A run-time error will be raised if the output fails to converge to within the desired Tolerance before the specified Maximum Iterations.

This is still very much work-in-progress and subject to prioritisation of development tasks beyond STEM version 7.3. Part of the reason for publicising these ideas is to invite comment and feedback before the specification is actually finalised.

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