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The Stuttering Poisson Distribution (SPD) is a non-negative discrete compound Poisson distribution that describes two or more events that happen in quickly in bursts. For example, the events might occur in groups or batches [1].

The distribution has the probability generating function (PGF):

Where P_x is the probability a discrete random variable will take the value x.

Calculating Stuttering Poisson Distribution Probability

A general formula for calculating the probability of observing a demand equal to x is given by [2]:

For low demand (x = 1 or x = 2), the formula simplifies to

The following table shows Poisson (λ = 2) and stuttering Poisson distribution (λ = 1 and  ρ = 5) probabilities and cumulative probabilities:

The term “stuttering” Poisson is mostly used in older literature, it does make an appearance in a few modern texts. Many modern authors call the distribution a Poisson-stopped sum or multiple Poisson. The SPD does have a variety of other names in the literature. For example, Cox [3] called the process a “cumulative process associated with a Poisson process.” It’s also referred to as:

• Composed distribution [4]
• Compound Poisson [5],
• Distribution par grappes [6],
• Poison distributions with events in clusters [7]
• Poisson power-series distribution [8]
• Pollaczek-Geiringer distribution.

The name “stuttering” Poisson distribution originated with Galliher et al. [9].  Patel [10] introduced the triple- and quadruple stuttering Poisson distributions.

References

[1] Huiming, Z. et al. (2012). Some Properties of the Generalized Stuttering Poisson Distribution and Its Applications. Studies in Mathematical Sciences. Vol. 5, No. 1, 2012, pp. [11–26] www.cscanada.net DOI: 10.3968/j.sms.1923845220120501.Z0697

[2] Syntetos, A. & Boylan, J. (2021). Intermittent Demand Forecasting. Wiley.

[3] Cox, D. R. (1962). Renewal Theory, Methuen, London

[4] Janossy L. et al. (1950). 0n composed Poisson distributions, I, Acta. Math. Acad. ScL Hung., 1, pp. 209–224.

[5] Feller, W. (1957). An Introduction to Probability Theory and Its Applications (2^nd ed.). Vol 1. New York. Wiley.

[6] Thyrion, P. (1960). Note sur les distribution “par grappes.” Association Royal des Actuaires Belges Bulletin, 60. 49-66.

[7] Castoldi, L. (1963). Poisson processes with events in clusters. Rendiconti del Seminaro della Facolta di Scienze della Universita di Cagliari, 33, 433-437.

[8] KHATRI, C. G. & PATEL, I. R. (1961). Three classes of univariate discrete distributions. Biometrics, 17, 567-75. [9] GALLIHER, H. P., MORSE, P. M. and SIMOND, M. (1959). ‘ Dynamics of Two Classes of Continuous-Review Inventory Systems ‘, Opns. Res. 7, 362-383.

[10] Patel, Y. C. (1976). Estimation of the parameters of the triple and quadruple stuttering Poisson distributions. Technometrics, 18,  67-73.