If a benchmarking study compares tariffs in different countries based on a unit of time, the rounding effects of billing mechanisms have to be taken into account. For example, if calls in one country are billed by the minute, and in another country by the second, charges must be converted to a single unit (either per minute or per second) to allow a meaningful comparison.
Usage pricing of timed interconnection or call charges uses a pulse system, where a pulse could last 1 second, 60 seconds, or any other length of time. The pulse system affects the actual billing of payments. This note describes a methodology to estimate the effective charges that result from rounding up calls to the minute or any other pulse length. For example, if billing of calls is based on measuring the duration of each call in one-minute intervals, then fractions of a minute are rounded upwards, so a call that takes 20 seconds is billed as if it lasted one minute, and a call that takes 1 minute and 1 second is billed as if it lasted two minutes, and so on.
A Methodology for Estimating Rounding Effects
To estimate the effective per minute billing rate, the following information is needed:
- The distribution of call durations. The methodology described here assumes that call duration follows a negative exponential distribution, [1]
- The pulse system for each call (for example, 60 seconds), and
- The average call duration.
At the moment that the called party answers the call, an initial pulse is transmitted to the register of the calling subscriber. As long as the conversation continues, the register periodically accumulates pulses every x seconds (where x is the pulse duration, for example 60 seconds). Under the assumption that the distribution of call durations follows a negative exponential distribution, the total number of pulses Np accumulated in Nc conversations can be estimated as follows:
- Let m be the average duration of all calls. The calls receiving the first periodic pulse are those calls that last at least x seconds, or Nc * exp (-x / m). In other words, x is the time unit used for billing calls (the pulse duration)
- The calls receiving the second periodic pulse are those that last at least 2x seconds, or Nc * exp (-2x / m), and so on. Thus, the total number of pulses is the sum of the following series:
or
The average number of revenue units (pulses) per call, r(x,m), is:
The formula of average revenue per call assumes that the first periodic pulse occurs exactly x seconds after the call begins. [2]
Numerical Example
The numerical example below demonstrates this methodology.
Suppose that:
- The billing rate per pulse is US$ 0.02
- The time unit of billing is one minute (x = 60 seconds). Thus customers pay US$0.02 for every minute or fraction of a minute, and
- The average call length is 2.5 minutes (m = 150 seconds).
Then the revenue units per call, r, is calculated as follows:
r(60,150) = 3.03324478 billable minutes.
Average revenue per call is therefore:
Average revenue = 3.0332447 * US$0.02
= US$ 0.0606649
The effective rate per minute is obtained by dividing average call revenue by the average call length:
US$ 0.0606649 / 2.5 minutes = US$ 0.024 per minute.
The rounding effect in this example is 21 percent:
(US$0.024 – US$0.020) / US$0.020 = 21%
Note that the shorter the call duration, the higher the effective rate per minute. For example, if the call duration in the above example was 1.5 minutes instead of 2.5 minutes, the rounding effect would be 37 percent.
Endnotes:
[1] See Mitchell B. (1979). “ Telephone Call Pricing in Europe: Localizing the Pulse.” Mimeo. The Rand Corporation.
[2] Mitchell (1979) develops other variants of this base line scenario. For instance, one could assume that the length of the first interval between the initial pulse and the first periodic pulse is not x but a random value. Briceno (2000) applies the methodology to estimate effective charges in the case of a two-part fixed-to-mobile tariff (i.e. a set-up charge independent of the duration of a call, and a time sensitive charge). See Briceno, A. (2000). “Fixed-Mobile Interconnection: The Case of Mexico.” Paper presented at the ITU’s Conference on Fixed to Mobile Interconnection. Geneva. June.