In the study of stochastic processes, Palm calculus, named after Swedish teletrafficist Conny Palm, is the study of the relationship between

probabilities Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speakin ...

conditioned on a specified event and time-average probabilities. A Palm probability or Palm

expectation Expectation or Expectations may refer to: Science * Expectation (epistemic) * Expected value, in mathematical probability theory * Expectation value (quantum mechanics) * Expectation–maximization algorithm, in statistics Music * ''Expectation' ...

, often denoted

P^0(\cdot)

E^0

cdot CDOT may refer to: *\cdot – the LaTeX input for the dot operator (⋅) *Cdot, a rapper from Sumter, South Carolina * Centre for Development of Telematics, India *Chicago Department of Transportation * Clustered Data ONTAP, an operating system fr ...

/math>, 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

L=\lambda W

, which states that the time-average number of users (''L'') in a system is equal to the product of the rate (

\lambda

) 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 In queueing theory, a discipline within the mathematical theory of probability, an M/G/1 queue is a queue model where arrivals are Markovian (modulated by a Poisson process), service times have a General distribution and there is a single server. ...

. This states that the (time-)average time between the previous and next points in a

point process In statistics and probability theory, a point process or point field is a collection of mathematical points randomly located on a mathematical space such as the real line or Euclidean space. Kallenberg, O. (1986). ''Random Measures'', 4th edition ...

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

* *Palm, C. (1943) "Intensitätsschwankungen im Fernsprechverkehr" ''Ericsson Techniks'', No. 44 Queueing theory Stochastic calculus Telecommunication theory {{probability-stub