Inhomogeneous Poisson process

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In probability theory, an inhomogeneous Poisson process (or non-homogeneous Poisson process) is a Poisson process with rate parameter \lambda (t) such that the rate parameter of the process is a function of time.[1] Inhomogeneous Poisson processes have been shown to describe numerous random phenomena[2] including cyclone prediction,[3] arrival times of calls to a call centre in a hospital laboratory[4] and call centre,[5] arrival times of aircraft to airspace around an airport[6] and database transaction times.[7]

The Cox process is an extension of this model where λ(t) itself can be a stochastic or random process.


Write N(t) for the number of events by time t. A stochastic process is an inhomogeneous Poisson process for some small value h if:[1][8]

  1. N(0)=0
  2. Non-overlapping increments are independent
  3. P(N(t+h)-N(t)=1) = \lambda(t) h + o(h)
  4. P(N(t+h)-N(t)>1) = o(h)

for all t and where, in little o notation, \scriptstyle \frac {o(h)}{h} \rightarrow 0\; \mathrm{as}\, h\, \rightarrow 0. In the case of point processes with refractoriness (e.g., neural spike trains) a stronger version of property 4 holds:[9] P(N(t+h)-N(t)>1) = o(h^2).


Write N(t) for the number of events by time t and \scriptstyle m(t) = \int_0^{t} \lambda (u)\text{d}u for the mean. Then N(t) has a Poisson distribution with parameter m(t), that is for k = 0, 1, 2, 3….[10]

\mathbb P(N(t)=k) = \frac{m(t)^k}{k!}e^{-m(t)}.


Traffic on the AT&T long distance network was shown to be described by a inhomogeneous Poisson process with piecewise linear rate function.[11] Ordinary least squares, iterative weighted least squares and maximum likelihood methods were evaluated and maximum likelihood shown to perform best overall for the data.


To simulate a inhomogeneous Poisson process with intensity function λ(t), choose a sufficiently large λ so that λ(t) = λ p(t) and simulate a Poisson process with rate parameter λ. Accept an event from the Poisson simulation at time t with probability p(t).[1][12] For a log-linear rate function a more efficient method was published by Lewis and Shedler in 1975.[13]


  1. ^ a b c Ross, Sheldon M. (2006). Simulation. Academic Press. p. 32. ISBN 0-12-598063-9. 
  2. ^ Leemis, Larry (May 2003). "Estimating and Simulating Nonhomogeneous Poisson Processes". William and Mary Mathematics Department. Retrieved Sep 26, 2011. 
  3. ^ Lee, Sanghoon; Wilson, James R.; Crawford, Melba M. (1991). "Modeling and simulation of a nonhomogeneous poisson process having cyclic behavior". Communications in Statistics - Simulation and Computation 20 (2-3): 777–809. doi:10.1080/03610919108812984. 
  4. ^ Kao, Edward P. C.; Chang, Sheng-Lin (November 1988). "Modeling Time-Dependent Arrivals to Service Systems: A Case in Using a Piecewise-Polynomial Rate Function in a Nonhomogeneous Poisson Process". Management Science (INFORMS) 34 (11): 1367–1379. doi:10.1287/mnsc.34.11.1367. JSTOR 2631999. 
  5. ^ Weinberg, J.; Brown, L. D.; Stroud, J. R. (2007). "Bayesian Forecasting of an Inhomogeneous Poisson Process with Applications to Call Center Data". Journal of the American Statistical Association 102 (480): 1185. doi:10.1198/016214506000001455.  edit
  6. ^ Galliher, Herbert P.; Wheeler, R. Clyde (March–April 1958). "Nonstationary Queuing Probabilities for Landing Congestion of Aircraft". Operations Research 6 (2): 264–275. doi:10.1287/opre.6.2.264. JSTOR 167618. 
  7. ^ Lewis, P. A. W.; Shedler, G. S. (September 1976). "Statistical Analysis of Non-stationary Series of Events in a Data Base System". IBM Journal of Research and Development 20 (5). doi:10.1147/rd.205.0465. CiteSeerX: 
  8. ^ Srinivasan (1974). "Chapter 2". Stochastic point processes and their applications. ISBN 0-85264-223-7. 
  9. ^ L. Citi, D. Ba, E.N. Brown, and R. Barbieri (2014). "Likelihood methods for point processes with refractoriness". Neural Computation. doi:10.1162/NECO_a_00548. 
  10. ^ Pham, Hoang (2006). "Software Reliability Modeling". System Software Reliability. Springer Series in Reliability Engineering. pp. 153–177. doi:10.1007/1-84628-295-0_5. ISBN 978-1-85233-950-0.  edit
  11. ^ Massey, W. A.; Parker, G. A.; Whitt, W. (1996). "Estimating the parameters of a nonhomogeneous Poisson process with linear rate". Telecommunication Systems 5 (2): 361. doi:10.1007/BF02112523.  edit
  12. ^ Lewis, P. A. W.; Shedler, G. S. (1979). "Simulation of nonhomogeneous poisson processes by thinning". Naval Research Logistics Quarterly 26 (3): 403. doi:10.1002/nav.3800260304.  edit
  13. ^ Lewis, P. A. W.; Shedler, G. S. (1976). "Simulation of nonhomogeneous Poisson processes with log linear rate function". Biometrika 63 (3): 501. doi:10.1093/biomet/63.3.501. JSTOR 2335727.  edit