STEM newsletter

Modelling subscriber churn

30 April 2006

Subscriber churn, i.e. the turnover within a subscriber base of new customers balanced by those leaving the network in a given period, is an inevitable characteristic of competitive markets. STEM includes a Churn Proportion input for services which is propagated through demand for resources and transformations to facilitate a straightforward and intrinsic calculation of simple churn costs, such as a relocation expense for a re-assigned subscriber set-top box.

However, not all churn effects can be driven by the churn proportion within the current period alone. This technical article describes approaches to modelling two different churn situations: provision of new handsets for all new subscribers to a mobile network, and deployment of final drops (from the kerb) to homes in a residential development.

Registered STEM users can download the model files from this article.

Forecasting capex for handset provision

Mobile operators typically offer a handset as part of a new subscription agreement. The so-called ‘handset subsidy’ refers to the fact that some or all of the off-putting cost of a new handset is covered by the operator in return for a minimum term in the contract.

Consider a service with an annual churn rate, c = 10%. In a given period, new handsets are required both for the explicit increase in the number of contracted subscribers, x, but also for the hidden new subscribers replacing those who have left.

So in a period, n, the number of new customers can be calculated as (xnxn–1) + xn · cn · lenn, where len is the length of the period in years. (Only 2.5% will leave in a quarter of a year.)

Over time, the total demand for new handsets will be the reported number of subscribers, plus the accumulation xi · ci · leni for i = 0 to n. This is straightforward to model in STEM:

  1. 10% is entered directly as the Churn Proportion input for a simple Subscribers service.
  2. A multiplier transformation, Churners, takes the service demand in connections as its input, and the multiplier is defined as Subscribers.ChurnProp * periodLen (), using the built-in periodLen() function to capture the length of each period in a time-series calculation.
  3. An expression transformation, Total churners, takes the output from Churners as its input and defines its output as prevThis (0) + Input1, using another built-in function designed specifically to facilitate an accumulation without a circular reference.
  4. Another expression transformation, Total subscribers, takes the service demand in connections as one input and Total churners as the second, and defines its output in turn as Input1 + Input2.
  5. A requirement is defined from Total subscribers for a Handset resource, which captures the initial capex per historical subscriber.

Calculation of cumulative historical subscribers vs. active subscribers.

Note: the Subscribers service and the Total churners transformation could both drive the resource directly, and achieve the sum implicitly, but we will need to reference the combined total again later, so the Total subscribers transformation makes this cleaner.

Replacement

STEM automatically models equipment replacement for a stable assignment of demand to a resource according to the resource’s Physical Lifetime input. This case is more complicated because a proportion of handsets need not be renewed in future by the operator because the relevant customers will have left (churned off) the network.

Suppose the desired replacement cycle is three years. First of all, the Physical Lifetime of the original Handset resource is set to 15 years, long enough to avoid replacement (within the model run) for all the historical Total subscribers.

Looking at the number of new handsets assigned three years ago, yn –3, a proportion, ci of their owners leave the network each year. So if ci is constant, the number needing to be replaced in year n can be calculated as yn–3 · (1 – c)3. (More generally, this could be expressed as yn–3 · (1 – cn–3) · (1 – cn–2) · (1 – cn–1) in an annual model, and you could argue that the (1 – cn–3) term could be disregarded because of the minimum contract term.)

Again this is straightforward to model with a time-lag transformation if churn is constant. If the incremental replacement demand in a period is always the same constant proportion of the incremental total demand three years previously, it follows that the total replacement demand over time is the same constant proportion of the original total demand three years previously:

  1. A time-lag transformation, Total subs lag, takes the output from Total subscribers as its input, and defines its output with a lag of three years.
  2. A multiplier transformation, Replacement subs, takes the output from Total subs lag as its input, and the multiplier is defined as (1 – Subscribers.ChurnProp) ^ 3.
  3. A requirement is defined from Replacement subs for the same Handset resource, as it is likely that the same handsets will be offered to all relevant subscribers who require a new or replacement handset in a given period. (Choice of handset or change of supplier can be modelled separately, using this initial Handset resource as a counter for the total number of handsets to be provided over time.)

Active subscribers, cumulative historical subscribers and handset replacement

If the churn rate is not constant, then the prevThis() function can be used again to have a parallel succession of transformations calculate the gradually diminishing number of subscribers from one year to the next and requiring a replacement handset after three years.

Modelling one-off infrastructure costs in a limited market

Consider a FTTH roll-out, where an initial deployment of FTTC paves the way for final drops to individual homes in a residential development of 1000 homes. Because of competition with satellite and wireless providers, churn is just as much an issue here, with the added twist that the final drop to a home really is a one-off cost: it need not be repeated if a new subscriber moves into a house vacated by a former subscriber.

Each year, there will be a proportion of new subscribers replacing leavers, so the number of final drops will gradually exceed the number of registered subscribers. But this effect is clearly constrained by the size of the development, and so it impacts on the model in two ways. First, the probability of a churned subscriber requiring a new drop is a function of the number of cumulative drops to date, specifically (1000 – xu ) / 1000, where u is the number of currently unused drops (i.e. the effective additional churn), and x measures active subscribers as before.

In a given period, n, we could calculate the increment to u as xn · cn · lenn · (1000 – xn–1un–1) / 1000, looking back one period to ensure that the formula avoids circularity.

Second, the number of unused drops at a given point in time may exceed the difference between the number of active subscribers and total homes later if the penetration is still growing. So we have to allow for the fact that an increase in the active subscriber base may arise from a home which has previously been connected. In other words, the number of unused drops may decrease as penetration increases. More precisely, we can calculate a decrement to u as (xnxn–1) · u–1 / (1000 – x – 1), again looking back one period for the probability to avoid a circularity.

These rules are readily captured in User Data for a service, and then a one-off final drop resource can be driven by x + u:

  1. Market segment, Homes passed, defines the roll-out plan for FTTC.
  2. A service, FTTH, defines the penetration into that addressable market.
  3. Select User Data from the icon menu for the service and, for each of the following five items, select Rename Field from the Edit menu to customise these fields.
  4. Customers = CustomerBase * Penetration.RefValue.
  5. Add drops = Customers * ChurnProp * periodLen () * prev (if (CustomerBase > 0, (CustomerBase – Customers – “Unused drops”) / (CustomerBase – Customers), 0), 1).
  6. Sub drops = (Customers – prev (Customers, 0)) * prev (if (CustomerBase > 0, “Unused drops” / (CustomerBase – Customers), 0), 1).
  7. New unused drops = min (“Add drops” – “Sub drops”, CustomerBase – Customers – prev (“Unused drops”, 0)) .
  8. Unused drops = prev (“Unused drops”, 0) + “New unused drops”.
  9. An expression transformation, Homes with final drop, takes the service demand in connections as its input (for cost allocation purposes), but defines its output directly as FTTH.Customers + FTTH.“Unused drops”.
  10. A requirement is defined from Homes with final drop for a Final drop resource, which captures the one-off capex per home.

Note the if (CustomerBase > 0, …) checks in (5) and (6) and the rounding protection in (7).

The following specimen results assume straight-line roll-out (to Y3) and penetration (to 60% in Y6) in order to highlight the constrained churn effect.

Dynamic model of unused (and total) final drops

STEM Support offers mathematical insight and experience

These two examples are based on real-life STEM Support queries, where the main value-add from Implied Logic has been providing the mathematical insight to rationalise the qualitative effects understood by the respective users and to transform them into quantitative rules. Once a problem can be stated with mathematical equations, it is a matter only of experience to be able to capture the equivalent logic in STEM and to fast-track to the results.


Registered STEM users can download the model files from this article.

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