A Time-Inhomogeneous Prendiville Model with Failures and Repairs

: We consider a time-inhomogeneous Markov chain with a ﬁnite state-space which models a system in which failures and repairs can occur at random time instants. The system starts from any state j (operating, F , R ). Due to a failure, a transition from an operating state to F occurs after which a repair is required, so that a transition leads to the state R . Subsequently, there is a restore phase, after which the system restarts from one of the operating states. In particular, we assume that the intensity functions of failures, repairs and restores are proportional and that the birth-death process that models the system is a time-inhomogeneous Prendiville process.


Introduction
Continuous-time Markov chains (CTMC) are usually used in various application fields related to queueing systems, mathematical biology, physics, and chemistry (cf., for instance, Anderson [1], Iosifescu and Tautu [2], Medhi [3], Bayley [4], van Kampen [5], Taylor and Karlin [6], Sericola [7]). In these cases, the stochastic process describes the evolution in continuous time of a Markov chain with a countable set of states that represent the number of customers in a queue, the number of molecules in a chemical reaction, the size of the population with births/deaths/immigrations/emigrations.
In the recent decades, particular attention has been paid to the study of these processes under the effect of random catastrophes that produce a sudden change of the state of a system. After such failure, one can think that the system is empty (total catastrophes) and then the dynamics immediately restart without delay (cf., for instance, Dharmaraja et al. [8], Giorno et al. [9][10][11], Di Crescenzo et al. [12], Economou and Fakinos [13,14], Chen et al. [15]). In more realistic cases, after a failure the system can be shipped for maintenance; in these cases, due to the extent of the failure, it is reasonable to assume random repair times. To introduce the effect of a catastrophe related to a failure of the system, one adds to the usual assumptions the existence of a non-zero probability of transition to an intermediate state from which the zero, or another operating state, can be reached at some randomly distributed instants (cf., for instance, Di Crescenzo et al. [16,17], Ye et al. [18], Mytalas and Zazanis [19], Krishna Kumar et al. [20]). In many cases, the times to failures and the times of repair are assumed to be exponential random variables. Some models consider the phase-type distributions for failure and repair times (see, for instance, Altiok [21][22][23], Dallery [24]).
Frequently, time-inhomogeneous Markov chains are used to model real dynamic systems. Research in this area are oriented to determine the transient and the limiting probability distribution, and to construct a continuous time diffusion approximation (cf., for instance,

Asymptotic Behavior of the System
Let q n = lim t→+∞ p j,n (t|t 0 ), j, n ∈ S, be the steady-state probabilities of the considered system.
Note that the last identity in (14) is the probability that the system is in an operating state n = 0, 1, . . . , in equilibrium regime.

Transient Behavior of the System
To determine the transient solution of system (7) with initial conditions (8), we denote by x 1 and x 2 the solutions of the following equation: Since

Proposition 2.
Under the assumptions (11), for t ≥ t 0 the following results hold: (7), with conditions (8), one has that p j,−2 (t|t 0 ) is solution of the second order differential equation to solve with the initial conditions: Similarly, for p j,−1 (t|t 0 ) one has to solve with the initial conditions: Results of theorem follow by using standard techniques to solve (16) and (18), with the initial conditions (17) and (19), respectively; then, recalling Equation (6), one determines P j (t|t 0 ).

Time of First Failure
We denote by the random variable that describes the time of first failure of the system, i.e. the time in which the chain enters in the state F for the first time, starting from the state be the density of the time of first failure.

Proposition 3.
Under the assumptions (11), for j ∈ {−1, 0, 1, . . . , } one has Proof. We consider a time-inhomogeneous Markov process { N(t), t ≥ t 0 } with state-space S obtained from N(t) by setting an absorbing boundary into the state −2, that corresponds to the failure state F of the system and we denote by the probability that the system is in state n at time t and that no failure has yet occurred. Since, Hence, to determine the density of the time of first failure, it is necessary to consider the following differential equations to solve with the initial conditions lim t↓t 0 p j,n (t|t 0 ) = δ j,n , j, n ∈ S, j = −2, lim t↓t 0 p −2,n (t|t 0 ) = 0, n ∈ S.
Let G j (z, t) = ∑ n=0 z n p i,n (t|t 0 ), j ∈ S (33) be the probability generating function (PGF) of the operating states of N(t). From (31) one has: to solve with the conditions (11) and (30), the PGF of the operating states of N(t) is

Proposition 4. Under the assumption
where Φ(t|t 0 ) is given in (12) and where Proof. The proof is given in Appendix A.
We remark that 0 ≤ b 1 (t|t 0 ) ≤ 1 and 0 ≤ b 2 (t|t 0 ) ≤ 1 for all t ≥ t 0 . Furthermore, we note that the function which appears to the right-hand sides of (36), is the PGF of the time-inhomogeneous Prendiville process N(t), characterized by the birth-death intensity functions λ n (t) and µ n (t), given in (30). The transition probabilities of N(t) are (cf. Zheng [43], Giorno and Nobile [49]): and the conditional mean and the conditional variance are: Under the assumptions (11) and (30), the probability that the system N(t) is in the zero-state at time t can be determined from (33): (42) where is obtained from (40). Similarly, the probability that the system N(t) is in the state n = 1 at time t follows from (36): (43) where, by virtue of (40), one has: For r ∈ N, let us introduce the r-th conditional moment of N(t): From (36), we have where E[ N(t)| N(t 0 ) = i] is given in (41).

Asymptotic Distribution of Operating States
To study the asymptotic behavior of the probabilities for the operating states, we assume that the intensity functions of N(t) are proportional. Specifically, in addition to the conditions (11), we suppose that with ϕ(t) positive, bounded and continuous function for t ≥ 0. Let be the asymptotic PGF of the operating states of N(t). From (34) one has to solve with the condition Proposition 5. Under the assumptions (11) and (46), the asymptotic PGF of the operating states is: Proof. The general solution of the differential Equation (48) is: where c is an arbitrary constant. Making use of the condition (49), we note that the term in square brackets at the right-hand side of (51) must vanish when z → 1, allowing to determine the constant c. Hence, from (51) we obtain (50).
The knowledge of the asymptotic PGF (50) allows to calculate the asymptotic probabilities of the operating states, as q 0 = G(0), q n = 1 n! d n G(z) dz n z=0 , n = 1, 2, . . . , , and the r-th asymptotic conditional moment of N(t): n r q n , r ∈ N.
(53) Proposition 6. Under the assumptions (11) and (46), one has: denotes the beta function and is the Gauss hypergeometric function.

Conclusions
In the present paper, we have considered a time-inhomogeneous CTMC with a finite space-state in which failures and repairs can occur at random times. In addition to the operating states, the space of the states includes two particular ones, denoted by F and R, representing the failure state and the repair one, respectively. The failures occur according to a non-stationary exponential distribution and they produce a transition from an operating state to F. Subsequently, a repair is required that involves a transition from F to R. Even the repair times are assumed to be random and occurring according to a non-stationary exponential distribution. After the reparation, the system restarts from one of the operating states.
Assuming that the failures, repairs and restores are characterized by proportional intensity functions, we determine the transition probabilities that, starting from an arbitrary state j at time t 0 , the system reaches the state F, or the state R, or one of the operating states at time t. The obtained results show that that the probability that the system is in an operating state at time t does not depend on the intensity functions related to the birth-death process without failures and repairs. In other words, the transition probabilities related to the states F, R, as well as the transition probability that the system occupies an operating state, are independent of the dynamics existing between the operating states. We determine the density of the time of first failure and the related average. Moreover, we focus on the transition probabilities of operating states by determining the PGF and the conditioned mean. Finally, under the assumption of proportional intensity functions, we analyze the asymptotic behavior for the probabilities of the operating states by calculating the asymptotic PGF and the asymptotic conditional mean.