Performance Analysis of Complex Series Parallel Computer Network with Transparent Bridge Using Copula Distribution

A network bridge is a computer networking device that creates a single aggregate network from multiple communication network or network segments. One of the type of network bridge is transparent bridge saddle with responsibility of checking incoming network traffic to identify media access control addresses. In this present research work on series parallel computer network performance, availability and cost analysis of complex computer network was considered to focus on a network that has four subsystems A, B, C and D and all the subsystems are arranged in series-parallel, subsystem A and B are working on 1-out-of-2: G and 2-out-of-3: F policy respectively, C subsystem behaved as a bridge with one unit and D subsystem has five units and are working in 3-out-of-5: G scheme. The system has two types of failure, degraded (partial failure) or complete failed states. The system was analyzed using supplementary variables techniques and Laplace transform, general distribution and copula family were employed to restore the partial failure and complete failure states. Computed results have been highlighted by the means of tables and graphs to investigate the performance of computer network. The result has shown that computer network with transparent bridge will be more reliable to filter incoming frame and forward it to media access control (MAC).


Introduction *
Many experts and researchers are constantly attempting to investigate the best approach of performance and effectiveness of series-parallel computer network due to its numerous applications in various areas such as industrials, health institutions, management, and financial institutions are always setting aside a special consideration to manage risk and explore a better performance computer network. The benefits of seriesparallel complex systems in various disciplines, as well as their requirements in determining availability and reliability, are now becoming an important issue in system performance. System availability is the percentage of time that the system is available to operate. Failure or degradation of the system results in cost, * salihu.muhd.isa@gmail.com danger, decreased production and profit, and even the loss of lives. To improve the system reliability and availability of a series-parallel computer network, redundant components must be implemented, with some units working while others remaining reserves for immediate action; this operational system style is knownas the k-outof-n: G/F scheme. In this approach, k units must work from the system's domain n in order for it to function; failure of more than k units results in the system's complete failure. Among the vast amount of literature on reliability theory model that exists. Chauhan and Malik [1] investigated the reliability of series-parallel systems for arbitrary parameter values. Fadi and Sibai [2] investigated series-parallel photovoltaic module modeling and output power estimation. Hu et al. [3] analyzed availability analysis and design optimization in a repairable series-parallel system with failure dependencies. Khatab et al. [4] studied selective maintenance optimization for series-parallel systems with alternating missions and scheduled breaks of stochastic duration. Mohammad et al. [5] investigated the impacts of a method for distributing reliability in series-parallel systems while accounting for common cause failure. Mustafa [6] concentrated on improving the reliability of a series-parallel system using a modified Weibull distribution. Peng et al. [7] published their research work on reliability analysis and optimal structure of seriesparallel phased mission systems subject to fault level coverage. Xie et al. [8] focused on the reliability and barrier assessment of series-parallel systems subject to system failure. Muhammad et al. [9] focused mainly on the cost benefit analysis of three different series parallel dynamo configurations, whereas Zhang [10] examined computer network reliability analysis using an intelligent cloud computing method. Potapov et al. [11] investigated reliability in the context of an information system with a client-server architecture. Kovalev et al. [12] investigated the reliability of distributed computer systems with clientserver architecture. System performance is determined by system configuration and repair dynamics. Different authors have adopted various types of failure and repair; many of them have considered general repair, while many have adopted copula [13] which is accepted as an efficient approach for better performance results because it operates with more than one repair system compared to general repair. Abubakar and Singh [14] focused on assessment and performance of industrial systems using the Gumbel Hougaard copula approach. Kabiru et al. [15] have focused on reliability assessment of complex system with two subsystems using joint distribution. Ibrahim et al.
[16] have analyzed the performance analysis of multicomputer system with three subsystems in series. Singh and Monika [17] examined on reliability analysis of n client's system under star topology. Pratap et al [18] have examined on the assessment of complex system with two subsystems and multi types failure and repair. Muhammad et al. [19] published their work on sensitivity analysis of three different series parallel dynamo configurations. Kabiru et al. [20] published their research work on availability and cost analysis of complex tree topology of computer network with multi-server using gumbel hougaard family copula approach, Druhv et al. [21] worked on reliability prediction of distributed system with homogeneity in software and serverusing joint probability distribution viacopula approach. From the previous research of computer network, little or no attention is paid on the reliability and performance analysis of computer network with transparent bridge, in this research work performance analysis of complex series parallel computer network with transparent bridge using copula distribution is studied.
Gumbel Hougaard family distribution is one of the types of copula family distribution that takes into account more than one repair. In this paper, system of first order linear partial differential difference equations were to obtained and solved using copula approach to analyze a complex series-parallel computer network system with a transparence bridge, which had four subsystems named A,B,C, and D. Subsystem A operates under the 1-out-of-2:G scheme, Subsystem B operates under the 2-out-of-3:F scheme, Subsystem C has a single unit that serves as a bridge, and failure of the bridge results in system failure, and Subsystem D operates under the 3-out-of-5:M scheme. The objective of this research work are three. First is to obtain the expressions of reliability, availability, MTTF, cost functions. Secondly, is perform sensitivity analysis using MTTF. Thirdly, is capture the impact of time and other system parameters on reliability, availability, MTTF and cost function. The system works in both series and parallel modes, and the Gumbel Hougaard family copula distribution is used for computation and illustration. Finally, S1, S2, S3, S4, S5, S6, S7, S8, S9, S10 represents the states of operation in degraded/partial failure while S11, S12, S13, and S14 are completely failed states and S0 is at perfect operational state. Degraded points have been repaired using general repair, and completely failed states have been repaired using the Gumbel Hougaard family copula. Supplementary variables and the laplace transformation were used to analyze the system, and tables and graphs are used to represent reliability measurements such as availability, reliability, and MTTF, as well as cost analysis.

State
State Description

S0
The state S0 represents a perfect state in which all the subsystems are in good working condition. S1 S1, represents a degraded state/minor partial failure in the subsystem D, as the result of the failure of first unit of the subsystem.

S2
State S2 amount to another degraded state/minor partial failure in the subsystem D, due to the failure of the second units of the subsystem D.

S3
S3 represents a degraded state/minor partial failure in the subsystem B, when the failure of the first unit of the subsystem B occurred.

S4
S4, shows the degraded state or minor partial failure in the subsystem A, due to the failure of first unit of the subsystem.

S5
S5, represents a degraded state/ minor failure, due to the failure of one unit each in subsystems A and D.

S6
S6, represents a degraded state/ minor failure, as the result of failure of the one unit each in subsystems A and B.

S7
This state accounts for a degraded state / minor partial failure, due to the failure of one unit each in subsystems B and D.

S8
S8, represents a degraded state with minor failure, due to the failure of the one unit in subsystem D and bridge (subsystem C).

S9
This state accounts for a degraded state/ minor partial failure, due to the failure of one unit in each of the subsystems A and B. S10 S10 ,reveals a degraded state/ minor partial failure, due to the failure of one unit of the subsystem D. S11 S11, shows the complete failure of the system, due to failure of more than two units in subsystem D and the system is under repair using copula S12 S12, indicates total failure of the system, as the result of the failure of more than one unit in subsystem B. And the system is under repair using copula S13 The state S13 revealed another complete failed state of the system, when two units failed in subsystem A and goes to repair using copula.

S14
The state S14 represents a complete failed state of the entire system as the result of failed stage of bridge (subsystem C) and considered to be under repair using copula distribution.
The state description highlights that, S0 is a state where the system is in a perfect state where all the subsystems are in good working condition. S1, S2, S3, S4, S5, S6, S7, S8 S9 and S10 are the states where the system is in degraded/ minor partial failure but operational mode and repair is being invited named as general distribution. The states S11, S12, S13, and S14 are allin complete failed mode state, but repair is deployed to the complete failed state as Gumbel-Hougaard family copula distribution (Fig. 1).

Assumptions
The following assumptions are taken throughout the discussion of the model: (i) Initially, S0 is the state where all units in the systems are in its perfect good state. (ii) The subsystems A and B are attached in working condition, with two and three units respectively, and failure of one unit in subsystem A tends the system to a partial failure (degraded) state, and at least two units most work in subsystem B otherwise degraded state, and both follows general distribution for repair, hence the complete failed state system is restore using copula.
(iii) The subsystem C served as a bridge between subsystems D and attached A and B, failure of the subsystem C amount the entire system to complete failed state and restore by copula (iv) At least three units most work in the subsystem D, if more than three failed, the system leads to complete failure state, all complete failure stage are repair using copula distribution. (v) It is assumed that a repaired system works like a new and no damage appears during repair. (vi) As soon as the failed unit gets repaired, it performs its task normally. (vii) All failure rates are constants and follow a negative exponential distribution.
Indicate full working state Indicate partial working state Indicate failure state Repair rates for complete failed states. , : The probability that the system is in Si state at instant's' for i =0 to 12. : Laplace transformation of state transition probability P (t). Expected profit during the time interval [0, t).

K1, K2:
Revenue and service cost per unit time in the interval [0, t) respectively. Standard repair distribution function = The expression of joint probability (failed state Si to good state S0) according to Gumbel-Hougaard family copula is given as where as a parameter, 1≤ ≤ ∞.

Formulation of Mathematical Model
The following steps of the differential difference equation are derived for the probability of considerations.

Solution of the Model
By taking the Laplace transformation of equations (1) to (29) we obtain the following results where:

Analytical Analysis of the model for particular cases
This section deals with the reliability models formulation. Models such reliability, availability, MTTF, sensitivity and cost analysis were obtained.

Mean Time to Failure (MTTF):
Taking all repairs zero in equation (***), and the limit, as s tends to zero we obtain the expression for MTTF.

Sensitivity Analysis
The sensitivity in MTTF of the system is studied through the partial differentiation of MTTF concerning the failure rates of the system. By employing the set of parametric values of the failure rates after partial differentiation of MTTF with respect of failure rates and then varying βA=0.01, βB=0.02, βC = 0.03, and βD = 0.04, in resulting expression, one can calculate the MTTF sensitivity as shown in Table 4 and the corresponding graphs shown in Figure 5.  Table 5 and graphical representation in Figure 6. Table 5. Expected Profit in [0,t) t=0,1 ,2, 3, 4,…,9 Figure 6. Profit with respect to failure rates. Table 1 and Table 2 provide a simple description of how the performance of the repairable series system changes with respect to time t as failure rates are set at different values. When failure rates are at lower values = 0.01, = 0.02, = 0.03 and = 0.04 availability and reliability of the system decreases gradually as the value of increases, with the passage of time and ultimately become steady to the value zero after a long interval of time. For this reason, the future behavior of the repairable device can be accurately predicted at any point for any given set of parametric values. This is obvious from the graphical design of the platform. It is observed that when the repair is provided the system performance is far better than when the repair is not provided. The corresponding values of availability are greater than the values of reliability as evident from table 1 and table 2. This simulation suggests that regular repair should be invoked to improve system performance. Furthermore, Table 3 and figure 3 yields the MTTF of the system with respect to variation in , , , respectively when other parameters are kept constant. Table 4 and the corresponding figure 4 displays the trends of cost function against the time when the revenue cost per unit time is fixed at1, service cost = 0.6, 0.5, 0.4, 0.3, 0.2, 0.1. It is evident in this table and figure that the cost increases with respect to time when the service cost decreases. The computed cost in table shows that = 0.1 is the maximum and = 0.6 is the minimum. Finally, it has been observed that as service cost decreases, the cost increases with variation of time. In general, for low service cost = 0.1 the cost function is high compare to high service cost = 0.6 . This study will serve as a guide to engineers and computer system designers to design more critical systems to improve efficiency and reduce operational costs. As a result, existing work should integrate repairs and replacement for partial and full failure under the free renewal warranty. The future work will study the reliability and performance analysis of computer network using different type of transparent bridge.