RELIABILITY ANALYSIS OF COMMUNICATION NETWORK SYSTEM WITH REDUNDANT RELAY STATION UNDER PARTIAL AND COMPLETE FAILURE

The purpose of this paper is to study the performance of a communication network system consisting of a transmitter, two relay stations and a receiver arranged in series parallel. Through the transition diagram, the partial differential equations are derived, and Laplace transforms are then taken on these equations to derive system reliability, availability, the mean time to system failure (MTTF) and cost function. It is assumed that failure rates are constant and follows exponential distribution, repair rates of partial failure state are assumed to follow general distribution and complete failure states are repaired through Gumbel-Hougaard family copula. The system is analyzed through supplementary variable technique and Laplace transform. Different measures of testing system effectiveness which include reliability, availability, mean time to failure (MTTF) and profit function have been calculated for particular values of time, failure and repair rates. From the study, it is clear that time and failure rates of both transmitter, relay stations and receiver influence the reliability, availability, MTTF and profit function. Mathematical models developed in this paper can aid plant management for proper maintenance and system safety, avoiding incorrect reliability, availability and profit assessment and leading to inadequate maintenance decision making, which may result in unnecessary expenditures and reduction of safety standards.


INTRODUCTION
System reliability and availability are vital towards system performance, quality, production output, expected revenue as well as industrial growth. To this end, reliability and availability of a system may be enhanced by maintenance, fault tolerance components as well as adequate system design. Many researchers have studied comparison and performance evaluation problem of different systems. Munial and Singh [12] analyzed the reliability of a complex system with subsystems connected in parallel. Negi and Singh [13] analyzed non repairable system reliability with serial subsystems. Niwas et al [14] presents probabilistic analysis models of a single system with preventive maintenance. Niwas and Kadyan [15] modelled the reliability of a system with warranty. Kakkar et al [4] studied reliability of two unit parallel industrial system. Kakkar et al [5] presents reliability of two dissimilar parallel unit in the present of preventive maintenance. Kumar and Malik [6] studied reliability modelling of a computer system where hardware repair has priority over replacement. Kumar and Malik [7] presents reliability measures of a computer system where preventive maintenance has priority over hardware repair. Kumar et al [8] presents reliability optimization of complex system through cuckoos search algorithm. Kumar et al [9] studied availability and cost analysis of a system with subsystems in series. Gahlot et al [2] analyzed performance measure for a serial system under different types of failure. Gulati et al [3] investigated the performance modelling and assessment of two unit system with different types of failure. Lado et al [11], Singh et al [20] and Singh et al [21]. Large volumes of literature exist on the issue relating to prediction of performance communication network. Cetinkaya and Sterbenz [1] present network challenges through past and potential and the correlation with taxonomy events. Ioan et al [10] studied the damage to the network leading to its malfunction leading to the network failure. Neumayer and Modiano [16] present network reliability with geographically correlated failures. Rafiee and Shabgani [17] presents fault tree model in addressing reliability evaluation of communication network. Sterbenz et al [22] presents three resilience and mechanism to redundancy, multilevel resilience, and diversity for disruption tolerance. Rawal et al [18] studied modeling of internet data center with various maintenance policies. Sadeghi and Roghanian [19] analyzed reliability of warm standby repairable system with two cases of imperfect switching mechanism.

RELIABILITY ANALYSIS OF COMMUNICATION NETWORK SYSTEM
From the literature study above, little or no attention is paid on the reliability analysis of communication network system with standby relay stations. The present paper is aim at reliability modelling and analysis of a communication network system with transmitter, two relay stations in parallel and a receiver. Explicit expressions for reliability, availability, mean time to failure and coat function have been obtained. The objectives of this paper are twofold: First is to derive the explicit expressions for the reliability, availability, mean time to failure and profit function.
Second is to capture the effect of both passage time and failure rates on reliability, availability, MTTF and profit.
The organization of the paper is as follows. Section 2 presents the notations of the study. Section 3 contains a description of the system under study. Section 4 presents formulations of the models.
The results of our analytical comparison between the systems are presented in section 5. Numerical examples are presented in section 5. Finally, we make some concluding remarks in Section 6.

S0
Initial state, transmitter, two consecutive relays from subsystems 2 and 3, receiver are working, one relay each from subsystems B and C are on standby. The system is working.

S1
Transmitter, two consecutive relays from subsystems B and C, receiver are working, one relay has failed in subsystems B and one relay is on standby in subsystem C. The system is working S2 Transmitter, two consecutive relays from subsystems B and C, receiver are working, one relay has failed in subsystems C and one relay is on standby in subsystem B. The system is working

S3
Transmitter and one relay each from subsystems B and C have failed, receiver is idle. The system is down.

S4
Receiver has failed, transmitter and relay stations are ide. The system is down S5 Transmitter, two consecutive relays from subsystems C, receiver are working, two relays have failed in subsystems B and one relay is on standby in subsystem C. The system is working

S6
Relay stations in both subsystems B and C have failed. The system is down.

S7
Two relays one each from subsystems B and C have failed, transmitter, relay stations from subsystems B and C, and receiver are working. The system is working.

S8
Two relays one each from subsystems B and C and receiver have failed. The system is down.

MATHEMATICAL MODELS FORMULATION
By probability of considerations and continuity arguments, the following set of difference differential equations are associated with the present mathematical model.

SOLUTION OF THE MODEL
Taking Laplace transform of (1) to (17) using the initial condition above, to obtained      Table 3 and graphical representation in Figure 4.     Table 5 below.

COST ANALYSIS
If the service facility is always available, then expected profit during the interval [0, t] is   hand, availability, profit and mean time to failure are higher higher value of repair rates and lower value of failure rates. Table 4 and the corresponding Figure 5  This sensitivity analysis implies that preventive and major maintenance should be invoked to the receiver, relay stations and the transmitter to minimize the system break down, prolong MTTF and maximizes the system reliability, availability as well as expected profit. Tables 5 and Figure 6 displayed the variation of sensitivity analysis with respect to change in  Table 6 and Figure 7 displayed the results on revenue cost per unit time.
From the table and the figure it is evident that expected profit increases with respect to time when 2 k . The expected profit is lower when 2 0.6 k = and higher for 2 0.1 k = . To achieve high quality, minimum production losses, higher production output as well as expected revenue, there should be failure free network and highest system reliability, availability and mean time to failure (MTTF).