Selection from poisson processes

Summary

There are givenk Poisson processes with mean arrival times 1/λ₁,...1/λ _k . Let λ_[1]≦λ_[2]≦...≦λ_[k] denote the ordered set of values λ₁...,λ_[k]. We consider three procedures for selecting the process corresponding to λ_[k]. The processes are observed until there areN arrivals from any of the given processes, when the processes are observed continuously, or until there are at leastN arrivals, when the processes are observed at successive intervals of time whereN is a pre-determined positive integer. In the continuous case, the process for which theNth arrival time is shortest, is selected. In the discrete case, the selection involves certain randomization.

Given (λ_[k]/λ_[k-1])≧0>1, it is shown that the probability of a correct selection (Pcs) is minimized whenθλ_[1]=θλ_[2]=...=θλ_[k-1]=θλ_[k]=θλ, say. The Pcs for this configuration is independent of λ for two of the given procedures, and monotone increasing in λ for the third. The value ofN is determined by a lower bound placed on the value of the Pcs.

The problem of selecting from given Poisson processes for the discrete case is related to the problem of selecting from given Poisson populations. An application of the given procedures to a problem of selecting the “most probable event” from a multinomial population, is considered.