A Semi-Markov Model with Geometric Renewal Processes

We consider a repairable system modeled by a semi-Markov process (SMP), where we include a geometric renewal process for system degradation upon repair, and replacement strategies for non-repairable failure or upon N repairs. First Pérez-Ocón and Torres-Castro studied this system (Pérez-Ocón and Torres-Castro in Appl Stoch Model Bus Ind 18(2):157–170, 2002) and proposed availability calculation using the Laplace Transform. In our work, we consider an extended state space for up and down times separately. This allows us to leverage the standard theory for SMP to obtain all reliability related measurements such as reliability, availability (point and steady-state), mean times and rate of occurrence of failures of the system with general initial law. We proceed with a convolution algebra, which allows us to obtain final closed form formulas for the above measurements. Finally, numerical examples are given to illustrate the methodology.

F the CDF of a new system with finite mean, The system deteriorates after each repair U i : lifetime of the system after the i-th repairs with F i as CDF: F i pxq " P pU i ď xq " F pa i xq, x ě 0 where a is the operational factor .
G the CDF of first repair duration with finite mean D i : repair duration after its i-th failures with G i as CDF: G i pxq " P pD i ď xq " Gpb i xq, x ě 0 where b the repair factor.
U i and D i are independent pU i q iPN and pD i q iPN are independent but not identically distributed sequences.

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A semi-Markov model with geometric renewal processes

Semi-Markov Kernel
In standard form, we have the Semi-Markov Kernel is Qpxq, x ě 0

Semi-Markov Kernel
From i 1 to state i 2 : the system fails externally and repaired.
Return to initial perfect state: the system has to be replaced by a new one, which means it fails with an internal failure or an non-repairable external failure.
To state i `11 , a repairman repairs the system.
After N repairs, we replace the system directly.
A semi-Markov model with geometric renewal processes

Embedded Markov Chain
The EMC pJ n q nPN gives the successive visited states by the SMP(Z t ) after n th jump.The transition matrix P " pP ij ; i, j P Eq is P 13/31 A semi-Markov model with geometric renewal processes

Reliability 14/31
A semi-Markov model with geometric renewal processes

Reliability
Initial law of SMP The distribution function of sojourn time in state i: H i ptq A semi-Markov model with geometric renewal processes The mean sojourn time in each state is For Up state, we have We also present the stationary distribution π of the SMP (Z t ) as follows π i :" ρ i m i m where the mean sojourn time of the system is m :" Here we will suppose that the semi-Markov kernel pQ ij q has derivatives (Radon-Nikodym) The Markov renewal function is expressed as

Reliability functions
The reliability function of the system is: Rptq " α 0 pI ´Q0 q p´1q ˚H0 ptq
A semi-Markov model with geometric renewal processes Mean Up Time M U T " If the system begins with perfect state, we have Reliability R  Example 2 The lifetime of system follows PH distribution with representation U i " P hpα, a i T q, and same distribution for repair time

Figure
Figure: System Transition ψptq :" pI ´Qq p´1q ptq If we put the Markov renewal function ψ in bloc matrix form, following the partition U and D of E, we have ψptq " ˆψ0 ψ 01 ψ 10 ψ 1 ˙ptq 7/31 A semi-Markov model with geometric renewal processes Figure: Example 1 4/31A semi-Markov model with geometric renewal processes Figure: Example 2 27/31 A semi-Markov model with geometric renewal processes The lifetime of system follows Weilbull distribution F " W pα, βq, same distribution for repair duration G " W pα 1 , β 1 q