Reliability estimation of a fault coverage distributed system with replacement options under four di

Abstract

Keywords: Availability; Expected Profit; Fault Tolerant Factor; Reliability; Sensitivity

Introduction

In many scenarios, computer system utilizes number of distributed networks to provide available and optimal network to the clients. The study of computer network system present its economic and technical feasibility as the best choice for the multipurpose network. However, with

the advancement in technology, availability of computer network happens to be subject of research and discussion. Meeting optimal level of availability is of paramount important in information, communication, military and institutional sector. Moreover, reliability could not attend its maximum level, computer network will be very poor.

*Corresponding author. *E-mail address: wujinbiao@csu.edu.cn This paper was recommended for publication in revised form by Editor in-Chief Ahmet Selim Dalkilic Published by Yıldız Technical University Press, İstanbul, Turkey Copyright 2021, Yıldız Technical University. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

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High computer system reliability is vital to industrial growth due to the fact that revenue mobilization is proportional to system performance. Due to its importance in industrial, domestic, institutional and manufacturing sector, literature study on dependability, reliability, maintainability and availability modelling of different computer network were developed. However, the developed models are used to address the computer network performance, subject to system failure. The technique of redundancy is thoroughly used to enhance reliability, dependability and availability of the system. In some computer network, the availability and dependability rely on the design of the system and strength of the units. To retain availability and dependability of complex computer network to an optimal level, the structure of the system and its components of optimal availability are required. Generally, system designers can develop technologies in a serial network to improve network availability, dependability and reliability. [1] Explore on performance analysis on computer network system that comprises of centralized database server, load balancer and distributed database server, [2] discuss on reliability metrics of network communication system having receiver, relay and transmitter. [3] Writes on reliability and dependability assessment of complex system having two subsystems on k-out-of-n working under G policy in subsystem 1 and four identical units in active parallel in subsystem 2. [4] Studied reliability of computer network base on genetic algorithm and the optimization technique were developed for better reliability. [5] Presents stochastic performance of computer based test having four subsystems arranged in series namely load balancer, clients, centralized server and database server. [6] Explore on reliability analysis of computer network which comprises of three subsystems: router, workstation and hub. [7] Investigate the performance measures of network with transparent bridge as follows 1-out-of-2: G, 2-out-of-3: F, a bridge unit and D 3-out-of-5: G schemes. [8] Dealt with an article on computer networking systems. [9] Publish an article on reliability enhancement in intuitionistic fuzzy space, [10] discuss on heterogeneity using rpc in client. [11] Estimate the coliform values of the Tekkekoy deep sea discharge system, which is chosen as an application area, by using a radial-based artificial neural network structure, [12] writes on production-distribution network system for a company, which is active in producing bottled natural spring water was established. [13] Develop models for the strength and performance analysis of computer network under different maintenance scenario. [14] Writes on reliability measures of database cluster, virtual router redundancy protocol and load balancer, the article analyzed availability by comparing the reliability if load balancer, virtual router redundancy protocol and high availability proxy were put in place, [15] investigated the impact of structure of the system reliability measures of software agent and client server. [16] Defined a Secure Simple Epidemic Algorithm (SSEA) for PSN where

a security condition controls the traffic. [17] Classification algorithms were used to classify electromyography and depth sensor data, [18] optimum CW size is defined through meta-heuristic optimization algorithms. [19] Published on OLTP applications with incremental repartitioning of shared-nothing distributed databases, [20] investigates the implication of load balancing of distributed system. [21] Writes on analysis of the FANET TCAs currently in use, along with a brand-new taxonomy of TCAs based on the FANET topology architectures and underlying mathematical models. [22] Explore on reliability analysis of computer network which comprises of three subsystems: router, workstation and hub. [23] Dealt with repairable system with reboot delay, one repair policy and imperfect coverage, [24] present work on the reliability measures of coverage factor with a standby system. [25] Investigate parallel system with three types of failure namely human failure, unit failure and major failure. [26] Consider a distributed system with five standby subsystems A (the clients), B (two load balancers), C (two distributed database servers), D (two mirrored distributed database serves) and E (centralized database server) is considered arranged as series-parallel system. [27] Analyzes the advantage of data center network topology by taking reliability and profit requirements into account, with distributed data center network topology having three components as follows: client applications, directory proxy server, and master servers were considered. This research work further improved the work of previous researchers were five subsystems were considered. Subsystem A consist of 2-clients, subsystem B comprises of a load balancer I, 2-fog node are in subsystem C, subsystem D comprises of load balancer II and lastly, subsystem E consist of 2-cloud server. However, analysis of the model in terms of fault tolerant, general repair and copula were thoroughly investigated. Reliability analysis measures such as availability, MTTF, sensitivity, cost analysis was carried out for different scenarios to check optimality of the entire system with respect both failure and repair rate. Moreover, some practical applications were considered. This work is structured as follows. Description, assumptions and nomenclatures on the system are presented in section 2, model formulation were discussed in section 3, section 4 consist of results and discussion and lastly section 5 which comprises of the conclusion. According to the literature review, little research articles on performance estimation of a fault coverage distributed system with replacement options under four different scenarios have been published. Motivated by this fact, we are interested to conducting a research on performance estimation of a fault coverage distributed system with replacement options under four different scenarios in this present work. The impact of the fault tolerance factor, in conjunction with the different scenarios, on the system availability, MTTF, sensitivity and profit were captured. The primary goal of this work is to determine how different scenarios will

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improve the availability and profit of the system under consideration, followed by a discussion and references, where the paper is concluded.

Notations, Assumptions And Model Description

Notations v0 failure rate of load balancer I v1 failure rate of clients v2 failure rate of fog nodes v3 failure rate of cloud servers v4 failure rate of load balancer II Repair rate of load balancer ξ0 Replacement rate of clients ξ1 Replacement rate of fog nodes ξ2 Replacement rate cloud servers ξ3 Repair rate of load balancer II ξ4 c Fault tolerant (probability of withstanding fault) δ0 = 1 - c ωi(t) Probability that a system is in a certain state at a given time. Avk At time t, the system is available Assumptions a. Failure of client is independent to the failure of fog node, load balancer and cloud server and vice vasa. b. Repair / Replacement is immediate. c. It is assumed that all the clients are active. d. Each failure is repairable. e. Rate of failure and repair obeys exponential distribution. f. Systems have redundant standby units g. Switching from standby to operation is perfect Model Description Subsystem A is made up of 2-clients in active parallel, subsystem B made up of load balancer I. 2-fog nodes in

Figure 2. Transition diagram of scenario 1 system with replacement at complete.

active parallel made up subsystem C, subsystem D made up of load balancer 2 and lastly, 2-cloud server in active parallel made up subsystem E. Moreover, the entire structure of the system, that is: Client, load balancer, fog node and cloud servers were configured as series-parallel, clients send request to the cloud server which in turn process the result and respond to the request. However, the two load balancers helps in utilization of the information required from the server, in Figure 1 (block diagram of the system), fog node serves as an intermediate between the clients, load balancers and cloud server. Table 1 provides a brief description of the states, while Figure 2 depicts all possible state transition for the model.

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The clients, fog nodes, load balancer and cloud servers are working.

Two fog nodes, One client failed, another client, load balancer and two cloud servers are working.

One fog node, another fog node, two clients, load balancer and two cloud servers are working.

One cloud server failed, another cloud server, two clients, two fog nodes are working

Subsystem A: System with Replacement at Complete Failure State Maintenance staff performs a perfect repair (repair as new) when a cloud server or client experiences a partial hardware failure. In the event of a complete failure over time, the component will be completely replaced. The system’s Markov chain-based state transitions are shown in Figure 2 below. Subsystem B: System with Replacement at Partial Failure and Complete State The analysis is carried out as follows: in the event a system component fails due to hardware failure, maintenance personnel are charged with the responsibility of replacing the problematic part of the system to ensure that the system can still function. Figure 3 below shows the markov chain transition diagram

Subsystem D: System with Detected Failure and Replacement at Complete State In this subsystem, the units were considered fault tolerant in the sense that even when a fault occurs in one or more host components, they continue to operate without malfunctioning. Fault tolerance device is the property that allows a system to continue operating properly on the

Subsystem C: System without Failure Detection and Replacement Repair at Complete State The underlying premise is that whenever a fault manifests itself, whether at the cloud server or client side, the failure detection device were not in place to verify the failed component, as a result the failed component is therefore being replaced in order to avoid the failure occurring again anytime soon. Figure 4 below shows a diagram of a Markov chain transition.

Figure 3. Transition diagram of scenario 2 system with replacement at partial failure and complete.

Figure 4. Transition diagram of scenario 3 system without failure detection and replacement repair at complete.

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From Figure 2, the corresponding set of differential difference equations for Subsystem 1 are

Figure 5. Transition diagram of scenario 4 system with detected failure and replacement at complete. occurrence of a failure. The fault tolerant system, however, cannot withstand catastrophic failures, which results in system failure and requires replacement. The system’s state transitions are shown in Figure 5 below using the Markov chain model.

Model Formulation

From Figure 1 above to derive the system of linear differential equation, the explicit expression of system availability can be obtained by solving the equations below. The results of the state probability equations for the system’s operational states can then be used to determine the system availability. In order to analyse the system availability of the system, we define the ωi(t) to be the probability that the system is in state i at time t and that we have be the probability row vector with initial conditions.

Using (4) to give the explicit expressions for the steadystate availability of Subsystem 1 given in (3) is now (5) where and . To evaluate the MTTF1, the rows and columns of the absorbing (failure) states from the above matrix were deleted and transposed to obtain the new matrix L1.

(1) The steady state probability of systems availability can be obtained from the solutions for State 0,1,2 and 3 are the only working states of all the scenarios in Figure 1, thus the steady state availability Avi(∞) at time bility that the system is in state i at time t and that we have is

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From Figure 3, the corresponding set of differential difference equations for Subsystem 2 are

Using the same argument above, availability expression of Subsystem 2 is

From Figure 4, the corresponding set of differential difference equations for Subsystem 3 are

Using the same argument above, availability expression of Subsystem 3 is

Using (8) to give the explicit expressions for the steadystate availability of Subsystem 2 given in (8) is now (10) Where

(14) Using (14) to give the explicit expressions for the steadystate availability of Subsystem 3 given in (14) is now

and To evaluate the MTTF2, the rows and columns of the absorbing (failure) states from the above matrix were deleted and transposed to obtain the new matrix L2.

Where and To evaluate the MTTF3, the rows and columns of the absorbing (failure) states from the above matrix were deleted and transposed to obtain the new matrix L3.

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(16) (17) From Figure 5, the corresponding set of differential difference equations for Subsystem 4 are

(19) Using (19) to give the explicit expressions for the steadystate availability of Subsystem 4 given in (19) is now (20) Where and, To evaluate the MTTF4, the rows and columns of the absorbing (failure) states from the above matrix were deleted and transposed to obtain the new matrix L4.

Using the same argument above, availability expression of Subsystem 4 is (18) as t → ∞ in steady state, to obtained

Results And Discussions

The objective of this section is to express numerical experiment so as to see effect of the parameters on the performance by the use of MATLAB software. The findings of availability, MTTF and profit for all the four (4) scenarios in terms of failure rates vo, v1, v2, v3, and v4 with repair rate ξo, ξ1, ξ2, ξ3, and ξ4 as follows: Tables 2, 3 and Figure [6 – 21] visually explain the detailed analysis of the availability,

Table 2. Variation of availability, MTTF and profit with respect to failure rate v0 for the four scenarios Availability

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Table 3. Variation of availability, MTTF and profit with respect to repair rate x0 for the four scenarios Availability

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20. Cost/availability versus repair rate.

MTTF, profit and cost benefit respectively in terms of vo and ξo. On the other hand, additional figures show an increasing pattern, highlighting the system’s robustness in reaction to variations in failure and repair rates v1 and ξ1 as shown in Tables 4, 5 and Figure [22 – 25] in terms of availability, Figure [26 – 29] for MTTF, Figure [30 – 33] in terms of profit and Figure [34 – 36] in terms cost benefit. Tables 6, 7 and Figure [37 -51] are relevant to availability, MTTF, profit and cost benefit in terms of v2 and ξ2. The graphical representations encapsulated in Tables 8, 9 and figure [52 – 66] serve as a visual exploration of the intricate dynamics between failure and repair rates v3 and ξ3 and their consequential impact on availability, MTTF, profit and cost benefit for four different scenarios. However, availability analysis, MTTF, profit analysis and cost benefit was carried out to the same scenarios in Tables 10, 11 and figure [67 -81] it was observed that availability increases with increase in all repair rates and decreases as the failure rate increases

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Table 4. Variation of availability, MTTF and profit with respect to failure rate v1 for the four scenarios Availability

Table 5. Variation of availability, MTTF and profit with respect to repair rate x1 for the four scenarios Availability

29. MTTF against scenario 4 for v1 and x1.

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Figure 36. Cost/MTTF versus failure rate. Table 6. Variation of Availability, MTTF and Profit with respect to failure rate v2 for the four Scenarios Availability

Table 7. Variation of Availability, MTTF and Profit with respect to repair rate x2 for the four Scenarios Availability

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Figure 51. Cost/MTTF versus failure rate. Table 8. Variation of Availability, MTTF and Profit with respect to failure rate v3 for the four Scenarios Availability

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Table 9. Variation of Availability, MTTF and Profit with respect to repair rate x3 for the four Scenarios Availability

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Table 11. Variation of Availability, MTTF and Profit with respect to repair rate x4 for the four Scenarios Availability

Table 10. Variation of Availability, MTTF and Profit with respect to failure rate v4 for the four Scenarios Availability

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Authorship Contributions

Muhammad Salihu Isa initiate the model and do all the writing and mathematical analysis while Jinbiao Wu helps in editing and supervision.

Conflict Of Interest

Authors have declared that there is no conflict of interest with regard to this research

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