Palm calculus

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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[edit]

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[edit]

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[edit]

  • Le Boudec, Jean-Yves (2007). "Understanding the simulation of mobility models with Palm calculus" (PDF). Performance Evaluation. 64 (2): 126&ndash, 147. doi:10.1016/j.peva.2006.03.001.
  • Palm, C. (1943) "Intensitätsschwankungen im Fernsprechverkehr" Ericsson Techniks, No. 44 MR11402