A MARKOV TIME RELATED TO A PRIORITY SYSTEM

We consider a basic renewable duplex system characterized by cold standby and subjected to a priority rule. Apart from a general stochastic analysis presented in the previous literature, we introduce a Markov time called the recovery time of the system. In order to obtain the corresponding Laplace-Stieltjes transform, we employ a stochastic process endowed with transition measures satisfying generalized coupled differential equations. The solution is provided by the theory of sectionally holomorphic functions.


Introduction
Standby provides a powerful tool to enhance the reliability, availability, quality, and safety of operational plants, for example, [4,8,16].However, in practice, standby systems are often subjected to a priority rule.For instance, the external power supply station of a technical plant has usually overall priority in operation with regard to an internal (local) power generator kept in cold or warm standby, for example, [4].The local generator is only deployed if the external power station is down.
Cold or warm standby systems subjected to a priority rule and attended by a repair facility have received considerable attention in the previous literature, for example, [2, 3, 5, 6, 9-15, 17-19, 21, 23, 24].We consider a basic duplex system composed of a priority unit (the p-unit) and a nonpriority unit (the n-unit) kept in cold standby until the p-unit fails.The p-unit has overall (break-in) priority in operation with regard to the n-unit, that is, the n-unit is only deployed if the p-unit is down.In order to avoid undesirable delays in repairing failed units, we suppose that the entire system (henceforth called the T-system) is attended by two different repairmen.Each repairman has his own particular task.Repairman N is skilled in repairing the n-unit, whereas repairman P is an expert in repairing the p-unit.Both repairmen are jointly busy if, and only if, both units are down.Otherwise, at least one repairman is idle.Apart from a general stochastic analysis presented in the previous literature [19,21,23], we introduce a Markov time called the recovery time of the T-system.The recovery time is the total (random) time needed to restore the T-system from a prescribed risky state into the safe state (see the forthcoming formulation).In order to obtain the corresponding Laplace-Stieltjes transform, we employ a stochastic process endowed with transition measures satisfying generalized coupled partial differential equations.Our proposed transient equations are extending the steady-state equations presented by Vanderperre and Makhanov [23].The explicit solution is provided by a refined application of the theory of sectionally holomorphic functions.

Formulation
Consider the basic T-system satisfying the following conditions.The p-unit has a constant failure rate λ > 0 and a general repair time distribution R(•), R(0) = 0.The corresponding failure-free time and repair time are denoted by f and r.The operative nunit has a constant failure rate λ s > 0, but a zero failure rate in standby (the so-called cold standby) and a general repair time distribution R s (•), R s (0) = 0.The corresponding failure-free time and repair time are denoted by f s and r s .The random variables f , f s , r, r s are statistically independent.Any repair is perfect [7].The switch-over time from standby to the operative state is instantaneous.Characteristic functions are formulated in terms of a complex transform variable.For instance, 2) The corresponding Fourier-Stieltjes transforms are called dual transforms.Without loss of generality (cf.[21, page 361]) we may assume that both R and R s have bounded density functions (in the Radon-Nikodym sense) defined on [0,∞).
In order to analyse the random behaviour of the T-system, we employ a stochastic process {N t ,t ≥ 0} with arbitrary discrete state space {A, B,C,D} ⊂ [0,∞) characterized by the following events: (i) {N t = A}: "the p-unit is operative and the n-unit is in cold standby at time t;" (ii) {N t = B}: "the n-unit is operative and the p-unit is under repair at time t;" (iii) {N t = C}: "the p-unit is operative and the n-unit is under repair at time t;" (iv) {N t = D}: "both units are down at time t."State A is called the safe state.States B and C are called risky states and state D is called the system down state.The non-Markovian process {N t } is defined on a filtered probability space {Ω, B,P,F} where the history F := {F t , t ≥ 0} satisfies the Dellacherie conditions: (i) F 0 contains the P-null sets of B; (ii) for all t ≥ 0, F t = u>t F u , that is, the family F is right continuous.Consider the F-Markov time where Z t denotes the past repair time of the failed p-unit being under progressive repair at time t.Note that we take the instant of the first failure as time origin, that is, N 0 = B, Z 0 = 0, P-a.s.Thus, from t = 0 onwards, θ is the total amount of time needed to restore the (2.6) Note that, for instance, Notations 2.1.The real line and the complex plane are denoted by R and C, with obvious superscript notations such as C + , C − .For instance, C + := {ω ∈ C : Imω > 0}.
The indicator (function) of an event ε ∈ B is denoted by 1ε.The Heaviside unit-step function with the unit jump at t = t 0 > 0 is denoted by U t0 (t), t ≥ 0. Finally, let [t] be the greatest integer function.

Solution procedure
It should be noted that our differential equations are well adapted to a Laplace-Fourier transformation.In fact, the p-functions are locally integrable with respect to t and bounded on [0,∞).Consequently, the derivatives with respect to t are also locally integrable.
Moreover, the integrability of the p-functions and the repair time densities with respect to x, y on [0, ∞) implies that the corresponding partial derivatives are also integrable on [0,∞).A Laplace-Fourier transform technique applied to the equations, taking the initial condition into account, reveals that for Im ω ≥ 0, Imη ≥ 0, z > 0, where ) Note that z-independent Hilbert problems, related to reliability engineering, have been solved by the theory of sectionally holomorphic functions.See [20,21] for further details.
A similar approach shows that the z-dependent function is sectionally holomorphic in C, provided that the singular Cauchy integral is defined as a Cauchy principal value in a double sense, see [21, the Appendix].Finally, note that our statement holds for general repair time distributions!(cf.[21, Remarks, page 361]).

Conclusion
Our proposed priority system, subjected to general (bivariate) repair, can be analysed by elegant methods provided by the theory of sectionally holomorphic functions.However, the duplex system, subjected to general failure and repair time distributions, invokes an open (harsh) mathematical problem in the theory of statistical reliability engineering.The analysis of priority systems, subjected to arbitrary distributions, is far from complete.

1 −
−(ρ+λs)(t−k) λ s (t − k) e −(ρ+λs)(t−k−1) k n=0 ρ + λ s (t − k − 1) (iii) {(N t ,Y t )}, if N t = C, where Y t denotes the remaining repair time of the n-unit being under progressive repair at time t;(iv) {(N t ,X t ,Y t )}, if N t = D.The state space of the underlying Markov process with absorbing state A is given by T-system from the risky state B into the safe state A. θ is called the recovery time of the T-system.In addition, note that our priority rule implies that a transition from the safe state A into the risky state C is only possible via state D. A (vector) Markov characterization of the process {N t , t ≥ 0} is piecewise and conditionally defined by (i) {N t }, if N t = A (i.e., if the event {N t = A} occurs); (ii) {(N t ,X t )}, if N t = B, where X t denotes the remaining repair time of the p-unit being under progressive repair at time t; B (t,x)dx := P N t = B, X t ∈ dx , p C (t, y)dy := P N t = C, Y t ∈ dy , p D (t,x, y)dx,dy := P N t = D, X t ∈ dx,Y t ∈ dy .