LONG-RUN AVAILABILITY OF A PRIORITY SYSTEM: A NUMERICAL APPROACH

We consider a two-unit cold standby system attended by two repairmen and subjected to a priority rule. In order to describe the random behavior of the twin system, we employ a stochastic process endowed with state probability functions satisfying coupled Hokstadtype differential equations. An explicit evaluation of the exact solution is in general quite intricate. Therefore, we propose a numerical solution of the equations. Finally, particular but important repair time distributions are involved to analyze the long-run availability of the T-system. Numerical results are illustrated by adequate computer-plotted graphs.

In order to describe the random behavior of the T-system, we employ a stochastic process endowed with transition probability functions satisfying steady-state Hokstadtype differential equations. Unfortunately, the exact solution procedure is quite intricate (see, [21, page 359] and Remark 4.1). Therefore, we propose a numerical solution of the equations.
Finally, current repair time distributions (such as the Weibull-Gnedenko distribution) are involved to compute the long-run availability of the T-system. The results are illustrated by adequate computer-plotted graphs.

Formulation
Consider a T-system satisfying the usual conditions. The p-unit has a constant failure rate [15] λ > 0 and a general repair time distribution R(·),R(0) = 0, with mean ρ. The operative n-unit has a constant failure rate λ s > 0, but a zero failure rate in standby (the so-called cold standby state) and a general repair time distribution R S (·),R S (0) = 0, with mean ρ s . In order to describe the random behavior of the T-system, we introduce a stochastic process {N t , t ≥ 0} with arbitrary discrete state space {A, B,C,D} ⊂ [0,∞), characterized by the following mutually exclusive 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 simultaneously down at time t." State D is called the system-down state. Observe that the process {N t , t ≥ 0} is non-Markovian. A Markov characterization of the process is piecewise and conditionally defined by: where X t denotes the remaining repair time of the p-unit under progressive repair at time t, where Y t denotes the remaining repair time of the n-unit under progressive repair at time t, The state space of the underlying piecewise linear (vector) Markov process is given by  It can be demonstrated that the invariant measure exists for arbitrary R and R S with finite mean. However, in order to keep the analysis as simple as possible, we henceforth assume that R and R S have bounded densities on [0,∞), denoted by r and r s . Finally, we introduce the measures

Long-run availability
We recall that the T-system is only available (functioning) in states A, B, and C. Therefore, the long-run availability of the operational plant, denoted by Ꮽ, is given by

Differential equations
In order to determine the ϕ-functions, we first construct a system of coupled steady statetype differential equations based on a time-independent version of Hokstad's supplementary variable technique (see, e.g., [22, page 526] for further details). For x > 0, y > 0, we obtain

Numerical scheme
In order to construct an appropriate numerical procedure, we first remark that the ϕfunctions are vanishing at infinity irrespective of the asymptotic behavior of the repair time density functions! Therefore, a numerical procedure to solve the equations in the region (0,  Next, we calculate ϕ k+1 B,i and ϕ k+1 C, j by means of the first-order approximations of (4.2) and (4.3) given by ϕ k+1 ) at x = L is small, then (most likely) this particular L is appropriate. However, such a "brutal force" approach may require a large number of grid points and is therefore rarely applicable. We illustrate the phenomenon by comparing the exact and the numerical solution in the most simple case, that is, let where l D := λ s (λ + 1)λ/(λ s + λ + 2), l C := λ s λ/(λ s + λ + 2), l B := λ(λ + 2)/(λ s + λ + 2). Observe that, if L is not large enough, ε M does not decrease as ∆ decreases (see Figure 5.1). On the other hand, too large L (consequently, too large ∆) lead to large numerical errors. For instance, the error with L = 30 is larger than 2.5 · 10 −2 for any N ∈ [20,100], whereas the error with L = 4 is less than 2.5 · 10 −2 . There could be multiple options too. For instance, an error less than 2.5 · 10 −2 is achieved either with L = 4, N = 15, or L = 6, N = 22, or L = 10, N = 38.

Trial-and-error procedure
The complicated behavior of the numerical error requires an adaptive choice of ∆ and L. Therefore, we introduce the subordinate errors ε 1 := |ϕ ∆,L − ϕ ∆,L/2 | and ε 2 := |ϕ ∆,L − ϕ ∆/2,L |, where ε 1 characterizes the numerical error caused by truncation of the infinite region and ε 2 the numerical error related to the first-order approximants. In order to find the optimal pair (L,∆), we first specify the required accuracy ε. Next, we propose the following trial-and-error procedure: we vary L until ε 1 < ε and then ∆ until ε 2 < ε. Finally, we introduce the following.
First, we vary L, as shown in Table 7.1, until ε 1 < ε. Next, we vary ∆, as shown in Table 7.2, until ε 2 < ε. A spatial distribution of ε 1 and ε 2 is depicted in Figures 7.1 and 7.2.

Conclusion
An effective statistical analysis of the T-system requires the solution of coupled Hokstadtype differential equations. Our numerical solution procedure, endowed with a simple and robust algorithm, allows to compute and to analyze the long-run availability for a general class of current repair time distributions with tangible engineering applications.