Counting distributions for an erlang process

1961 ◽  
Vol 13 (1) ◽  
pp. 91-103 ◽  
Author(s):  
John R.B. Whittlesey ◽  
Frank A. Haight
Keyword(s):  
Erlang Process

scholarly journals Generalized fractional Poisson process and related stochastic dynamics

2020 ◽  
Vol 23 (3) ◽  
pp. 656-693 ◽  
Author(s):  
Thomas M. Michelitsch ◽  
Alejandro P. Riascos
Keyword(s):  
Poisson Process ◽  
Continuous Time ◽  
Counting Process ◽  
Stochastic Dynamics ◽  
Waiting Times ◽  
Fractional Diffusion ◽  
Diffusion Limit ◽  
Fractional Operators ◽  
Fractional Poisson Process ◽  
Erlang Process

AbstractWe survey the ‘generalized fractional Poisson process’ (GFPP). The GFPP is a renewal process generalizing Laskin’s fractional Poisson counting process and was first introduced by Cahoy and Polito. The GFPP contains two index parameters with admissible ranges 0 < β ≤ 1, α > 0 and a parameter characterizing the time scale. The GFPP involves Prabhakar generalized Mittag-Leffler functions and contains for special choices of the parameters the Laskin fractional Poisson process, the Erlang process and the standard Poisson process. We demonstrate this by means of explicit formulas. We develop the Montroll-Weiss continuous-time random walk (CTRW) for the GFPP on undirected networks which has Prabhakar distributed waiting times between the jumps of the walker. For this walk, we derive a generalized fractional Kolmogorov-Feller equation which involves Prabhakar generalized fractional operators governing the stochastic motions on the network. We analyze in d dimensions the ‘well-scaled’ diffusion limit and obtain a fractional diffusion equation which is of the same type as for a walk with Mittag-Leffler distributed waiting times. The GFPP has the potential to capture various aspects in the dynamics of certain complex systems.

The variance function of the Erlang process

1970 ◽  
Vol 22 (1) ◽  
pp. 327-337
Author(s):  
R. J. Serfling
Keyword(s):  
Variance Function ◽  
Erlang Process

scholarly journals Incomplete Gamma functions for evaluating Erlang process probabilities

1963 ◽  
Vol 17 (81) ◽  
pp. 11-11 ◽  
Author(s):  
John R. B. Whittlesey
Keyword(s):  
Gamma Functions ◽  
Erlang Process ◽  
Incomplete Gamma Functions
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