Reliability of a capacitated system with parallel subsystems considering performance sharing mechanism

Performance sharing mechanism is widely applied in energy systems, computing systems and other types of industrial systems to increase system reliability. In reality, some systems may allow certain extent of performance deficiency instead of requiring every subsystem in the system satisfy its demand. Moreover, the subsystems in systems may combine with amounts of components, say that, a series of generators should be simultaneously used to provide the necessary energy for a given area. In this paper, we consider a system with performance sharing mechanism where the system remains functional as long as the summed weighted deficiency for subsystems after performance sharing is smaller than a predesigned reliable threshold. The amount of shared performance is further assumed to restricted by the bandwidth capacity of the system. We further consider the case where the surplus can be redistributed to maximize system reliability. Universal generating function technique is employed to model the reliability of the system and numerical examples are provided to illustrate the applications.


Introduction
Recently, there are amounts of studies concentrating on the reliability modelling of industrial systems with performance sharing mechanism (Xiao et al.,2015). Previous research widely assumed that the system will only function if all subsystems satisfy their demand individually (Peng et al.,2016). In fact, the surplus performance can be transmitted through a given bandwidth to the subsystems with deficiency for the sake of maximizing the reliability of whole system. Taking the energy system as an example, 1 there will be a series of subsystems (areas) in an energy system allocated with different weights. The IOP Publishing doi:10.1088/1757-899X/1043/2/022002 2 goal of the system is to minimize the summed weighted deficiency of all subsystems in the system. The system is regarded as functional as long as the summed weighted deficiency is lower than some prespecified threshold. Therefore, the surplus performance can be transmitted from low weight area to high weight area in order to make the system remain functional in case some subsystems suffer power deficiency. Similar examples can be found in production system, computing system and other types of engineering systems. Moreover, there will be amounts of components in the subsystem. For instance, in the energy system, areas should be equipped with amounts of generators to provide the necessary energy.
In light of this, this paper considers the reliability of a performance sharing system where the system fails if and only if the summed weighted deficiency of all subsystems is greater than a predesigned reliability threshold. Different from Wu et al. (2019) where each subsystem has only one component, the system is assumed to consist of several subsystems in parallel with different weights where each subsystem combines numbers of generators (components). The performance of a given subsystem equals to the summation of performance of all generators.
In addition, each generator has a random performance and each subsystem has a random demand to be satisfied. With the introduction of performance sharing bandwidth, the system will still be regarded as functional if the sum of weighted deficiency can be made to be lower than the threshold after certain types of redistribution strategies. Follow previous literature (M. El Falou et al,2009), we similarly employ universal generating function (UGF) to model the state distribution of generators, subsystems and system, based on which the system reliability is formulated.
The rest of this paper is organized as follows. Section 2 sets up the model. Section 3 introduces the application of UGF and discusses different performance sharing strategies. Section 4 presents numerical examples and Section 5 concludes.

Model Setup
Consider a system with n subsystems in parallel with different weights ,0 j W j n d . For each subsystem i , there will be i n generators simultaneously working to provide the necessary energy. For simplification, we assume each subsystem has a random discrete demand j D with given probability mass function (PMF) and each generator belonging to subsystem j has a random discrete performance ,0 jk j G k n d with given PMF. The predesigned reliability threshold of the system is assumed to be T and the bandwidth of the performance sharing mechanism is denoted as C with given PMF. To simplify the calculation, we use m to denote the amounts of subsystems that satisfy their demand through their own generators' performance and n m to represent the amounts of subsystems that suffer from performance deficiency. The system works as follows. If the minimal value between summed weighted deficiency of n m subsystems is smaller than or at least equal to T even without performance sharing mechanism, the system is reliable. Otherwise, if the summed weighted deficiency of n m subsystems can be made to be smaller than or at least equal to T after performance sharing, the system is also regarded as reliable. In this paper, the surplus redistribution mechanism proposed in paper Wu et al. (2019) is used for distributing the performance among subsystems. If the summed weighted deficiency of n m subsystems is smaller than or at least equal to T after surplus redistribution, the system is still reliable. If the summed weighted deficiency of n m components after surplus redistribution is still larger than T , the system is taken as failed. The performance redistribution mechanism is called surplus redistribution (SR) since the system redistributes the surplus performance of m component to compensate the deficiency of n m components. It is worth noting that the system will give priority to compensate the deficiency of components with the highest weight in performance sharing mechanism. Since deficiency can actually be taken as the negative sufficiency, we sort the subsystems such that the subsystem 1 to subsystem m are in descending order of deficiency and subsystem m+1 to subsystem n are in descending order of their weights.
1 m to component n now represent the unsatisfied components in ascending order. The weighted sufficiency of first m components can now be represented as WS and the weighted deficiency of next n m components can be denoted as WD . We should further note that at this time the total weighted surplus performance and total weighted deficiency of the system are actually equal to WS and WD , respectively. Further, the total sufficiency and deficiency can be represented as TS and TD , respectively. It is easy to show that Corresponding to four possible situations we mentioned, we now summarize the three cases theoretically.
A.The system is reliable if WD T d . Otherwise, go to cases B or C or D. B.The maximal redistribution performance of the system is limited to the bandwidth and can be obtained by min( , ) , say that, the minimal value between total sufficiency and bandwidth capacity can compensate the total deficiency to make the subsystems to have no deficiency. Otherwise, go to cases C or D.
C.When both WD T ! and min( , ) TS C TD , the surplus cannot compensate the deficiency. The system under this case can always give priority to satisfy subsystems with the highest weight and employ the total surplus to compensate the unsatisfied subsystems with higher weights first. Since the subsystems are already in descending order, there should exist an integer The system is still reliable if The reliability of the whole system can now be calculated by combining the first three cases as

Universal Generating Function Technique
Following the common assumption in UGF, we similarly assume the random demand of subsystems j D to take the value from a given set We further denote the random capacity of the bandwidth which takes the value from a given set , ' , ' , Here « » ¬ ¼ represents the maximum integer that is no bigger than the given number and mod( , ) a b represents the remainder of the division of a by b . The recursive procedure is commonly employed to obtain the UGF of the whole system and works as follows.
We should further note that the composition operator requires the multiplication of coefficients and the union of the exponents for every pair of terms of the two UGFs. Therefore, the UGF considering both the performance and demand (sufficiency/deficiency) can be obtained by To calculate the reliability of the whole system, we further design a reliability indicator function to depict the reliability distribution as follows.
To increase the reliability of the whole system, the surplus redistribution is now employed to reallocate the sufficiency to compensate the deficiency of component with the highest weight. This requires an updating UGF that eliminates all components that already satisfied their demand. Since the reliability indicator will equal to zero if the precondition is not held, we can multiply the one minus reliability indicator to the UGF to eliminate unnecessary terms. As a result, the updating UGF can be obtained by

R R
The updated UGF can be obtained by eliminating the unnecessary terms in initial UGF and denoted as: From the above results, performance sharing mechanism can improve system reliability.

Conclusions
In this paper, we consider capacitated system with parallel subsystems connected by a performance sharing mechanism. Different from previous literatures, we relax the assumption that the system cannot tolerate the deficiency. In our proposed model, only if the summed weighted deficiency is larger than a predesigned reliability threshold, the system fails. Otherwise, the system is regarded as reliable. We further consider the case where each subsystem combines amounts of components (generators in our energy system example) with random performance. The existence of bandwidth allows the redistribution of performance among all subsystems, making the surplus able to somehow compensate the deficiency and thereby increase the reliability. Our results show that surplus redistribution is efficient and effective in remaining a functional system. More works are needed. The system can actually employ the performance (not surplus) from the component with relative low weight to compensate the deficiency of component with relative high weight. It is also worth to analyze and compare other redistribution mechanisms.