Towards More Realistic Shock Models with Applications in Optimal Maintenance
Date
2023-02
Researcher
Goyal, Dheeraj
Supervisor
Hazra, Nil Kamal
Journal Title
Journal ISSN
Volume Title
Publisher
Indian Institute of Technology Jodhpur
Abstract
In this thesis we study some generalized shock models with applications in optimal
maintenance. Most of the systems used in reality are directly or indirectly a↵ected by some
harmful “instantaneous” events (shocks of di↵erent nature), which either cause the system’s
failure or decrease the system’s lifetime. Thus, the study of systems’ lifetimes subject to
external shocks is one of the important problems in reliability theory. In a shock modeling,
one has to answer two important questions, namely, “how the occurrences of shocks a↵ect
or decrease the lifetime of a system” and “how one can model the occurrences of shocks on
a system ”. In this thesis, we answer these questions in di↵erent setups.
Existing shock models are usually classified into four broad classes, namely, extreme
shock models, cumulative shock models, run shock models and !-shock models. The
!-shock model, which is an object of our study, is di↵erent in nature from other aforementioned
shock models. In all !-shock models developed so far in the literature, the recovery
time ! was assumed to be constant. However, this assumption is too restrictive and unrealistic
in describing many real-life scenarios. Indeed, ! can obviously depend on other
parameters, namely, magnitude of shocks, arrival times of shocks, etc. This motivates us
to introduce a new time-dependent !-shock model wherein the recovery time of a system
is assumed to be an increasing function of arrival times of shocks. For this model, we assume
that shocks occur according to the generalized P´olya process (GPP) that contains
the homogeneous Poisson process (HPP), the non-homogeneous Poisson process (NHPP)
and the P´olya process as particular cases. We further generalize this model to the general
!-shock model by considering the recovery time ! as the function of both arrival times and
magnitudes of shocks. We also consider a more general and flexible shock process, namely,
the Poisson generalized gamma process (PGGP) that includes the HPP, the NHPP, the
P´olya process and the GPP as the particular cases. With the same motivation, we study
a history-dependent mixed shock model which is a combination of the history-dependent
extreme shock model and the history-dependent !-shock model. As an application of the
aforementioned new shock models, we study the optimal replacement policy.
Although Poisson processes are widely used in various applications for modeling of
recurrent point events, there exist obvious limitations. Several specific mixed Poisson processes
(which are formally not Poisson processes any more) that were recently introduced
in the literature overcome some of these limitations. We define a general mixed Poisson process with the phase-type (PH) distribution as the mixing one. As the PH distribution is
dense in the set of lifetime distributions, the new process can be used to approximate any
mixed Poisson process. We study some basic stochastic properties of this new process and
discuss some relevant applications by considering the extreme shock model, the stochastic
failure rate model and the !-shock model.
We introduce and study a general class of shock models with dependent inter-arrival
times of shocks that occur according to the homogeneous Poisson generalized gamma process
(HPGGP). A lifetime of a system a↵ected by a shock process from this class is represented
by the convolution of inter-arrival times of shocks. This class contains many popular shock
models, namely, the extreme shock model, the generalized extreme shock model, the run
shock model, the generalized run shock model, specific mixed shock models, etc. For systems
operating under shocks, we derive and discuss the main reliability characteristics and
illustrate our findings by the application that considers an optimal mission duration policy.
Counting processes based on heavy-tailed distributions (namely, the fractional homogeneous
Poisson process (FHPP), the renewal process of matrix Mittag-Leffler type (RPMML),
etc.) have not yet been considered in the literature for modeling the occurrences
of shocks. Thus, we study some general shock models under the assumption that shocks
occur according to a renewal process with the matrix Mittag-Leffler (MML) distributed
inter-arrival times. As the class of MML distributions is wide and well-suited for modeling
the heavy tail phenomena, these shock models can be very useful for analysis of lifetimes
of systems subject to random shocks with inter-arrival times having heavier tails. Some
relevant stochastic properties of the introduced models are described. Moreover, two applications,
namely, the optimal replacement policy and the optimal mission duration are
discussed.
Lastly, we consider coherent systems subject to random shocks that can damage
a random number of components of a system. Based on the distribution of the number
of failed components, we discuss three models, namely, (i) a shock can damage any number
of components (including zero) with the same probability, (ii) each shock damages, at
least, one component, and (iii) a shock can damage, at most, one component. Moreover,
the arrivals of shocks are modeled using three important counting processes, namely, the
PGPP, the Poisson phase-type process (PPHP) and the RPMML. For the defined shock
models, we study some reliability properties of coherent systems. At the end, we discuss
the optimal replacement policy as an application of the proposed models.
Description
Keywords
Citation
Goyal, Dheera. (2023).Towards More Realistic Shock Models with Applications in Optimal Maintenance (Doctor's thesis). Indian Institute of Technology Jodhpur, Jodhpur. (Doctor's thesis). Indian Institute of Technology Jodhpur, Jodhpur.