# Palm calculus

In the study of stochastic processes, **Palm calculus**, named after Swedish teletrafficist Conny Palm, is the study of the relationship between probabilities conditioned on a specified event and time average probabilities. A Palm probability or Palm expectation, often denoted or , is a probability or expectation conditioned on a specified event occurring at time 0.

## Little's formula

A simple example of a formula from Palm calculus is Little's law , which states that the time-average number of users (*L*) in a system is equal to the product of the rate () at which users arrive and the Palm-average waiting time (*W*) that a user spends in the system. That is, the average *W* gives equal weight to the waiting time of all customers, rather than being the time average of "the waiting times of the customers currently in the system".

## Feller's paradox

An important example of the use of Palm probabilities is Feller's paradox, often associated with the analysis of an M/G/1 queue. This states that the (time-)average time between the previous and next points in a point process is greater than the expected interval between points. The latter is the Palm expectation of the former, conditioning on the event that a point occurs at the time of the observation. This paradox occurs because large intervals are given greater weight in the time average than small intervals.

## References

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- Palm, C. (1943) "Intensitätsschwankungen im Fernsprechverkehr"
*Ericsson Techniks*, No. 44 Template:MR