A generalized censored $$\delta$$ δ -shock model for multi-state systems

In this paper I introduce a new generalized censored $$\delta$$ δ -shock models for both cases when the intershock times have discrete and continuous distributions. The system transits into a lower partially working state upon the occurrence of each interarrival time between two successive shocks greater than a critical threshold, say $$\delta$$ δ . The system fails when no shock occurs within $$k \ge 1$$ k ≥ 1 time periods of length $$\delta$$ δ . For arbitrary discrete intershock time distributions, it is shown that distribution of the system’s lifetime is of discrete compound negative binomial convolution type, and its probability generating function (pgf), probability mass function (pmf), mean and the variance are obtained. By considering that the intershock times have a discrete phase-type distribution, I derive in matrix form expression the pgf of system’s lifetime. For the binomial shock process, I obtain the joint pgf of system’s lifetime, the number of periods in which they do not appear shocks, and the number of shocks until the failure of the system. The marginal distributions of these distributions, as well as the distribution of the time spent by the system in a perfectly functioning state and the distribution of the total time spent by the system in a partially working state, are studied. Also, it is shown that the distribution of the system’s lifetime is directly linked with matrix-geometric distributions, and I obtain exact relations for evaluating its distribution. For arbitrary continuous intershock time distributions, it is shown that distribution of the system’s lifetime is of compound negative binomial convolution type, and its Laplace–Stieltjes transform, is derived. Using risk theory from actuarial science, lower and upper bounds for the survival function of the system are obtained. Also, approximations via matrix-exponential distributions for the survival function of the system are discussed. Finally, some numerical examples to illustrate our results, are also given.