A New Compound Negative Binomial Distribution and Its Applications in Reliability

A new negative binomial distribution (NBS) is proposed by introducing the concept of sparse connection. Finite Markov chain imbedding approach is used to obtain NBS for the cases of independent trials, first-order Markov dependent trials and high-order Markov dependent trials respectively. New compound negative binomial distributions called CNBS and CMNBS for independent and Markov cases respectively are derived by means of phase-type representations. The new compound negative binomial distributions can be applied in the shock models and start-up demonstration tests in the field of reliability. Numerical examples based on shock models are then presented to demonstrate the applicability of the new distribution.