Heine process as a q-analog of the Poisson process—waiting and interarrival times

ABSTRACT

In this study, we introduce the Heine process, {X_q(t), t > 0}, 0 < q < 1, where the random variable X_q(t), for every t > 0, represents the number of events (occurrences or arrivals) during a time interval (0, t]. The Heine process is introduced as a q-analog of the basic Poisson process. Also, in this study, we prove that the distribution of the waiting time W_{ν, q},  ν ⩾ 1, up to the νth arrival, is a q-Erlang distribution and the interarrival times T_{k, q} = W_{k, q} − W_{k − 1, q}, k = 1, 2, …, ν with W_{0, q} = 0 are independent and equidistributed with a q-Exponential distribution.

KEYWORDS:

• q-Binomial distribution
• q-Erlang distribution
• q-Exponential distribution
• Heine distribution
• Heine process.

MATHEMATICS SUBJECT CLASSIFICATION:

• 60F99
• 60G99