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This paper deal with the study of reliability of a complex system consists of three subsystems, (subsystem-1, subsystem-2 and subsystem-3) connected in series configuration. The subsystem-1 has three units and working under the policy 2-out-of-3: F policy, subsystem-2 has two units in parallel configuration and is working under 1-out-of-2: G policy and subsystem-3 is connected with subsystem-2. All three subsystems are connected in series configuration. The system has two types of failure, partial failure and completely failure. It is assumed that the partial failure bring the system to degraded state, while the complete failure stop functioning in the system. All failure rates are constant and assumed to follow exponential time distribution, but the repair follow two types of distribution, i.e., general distribution and exponential distribution. It is assumed that the system in partially failed/degraded states are repaired by employing general time distribution but the completely failed states are repaired by Gumbel–Hougaard family copula distribution. The system is analyzed by supplementary variable technique. Some crucial measures of reliability such as Availability, mean-time-to-failure (MTTF) and profit analysis have been discussed. Some computations are taken as special case by evaluating Availability, MTTF and profit by taking different failure rates.

Purpose – The purpose of this paper is to find the reliability measures of Linux operating system connected in local area network (LAN). Design/methodology/approach – The system has two different topologies, i.e. star topology (subsystem-1) and bus topology (subsystem-2), are placed at two different places, and connected to a server through a hub. Both the topologies have n clients. The system has partial failure and complete failure. The partial failure is of two type’s namely minor partial and major partial. The minor partial failure degrades the system whereas the major partial failure brings the system to a break down mode. The system can completely fail due to failure of server hacking and blocking. Findings – By using supplementary variable technique and Laplace transformation, by taking different types of failure and two types of repairs the availability, non-availability, mean time to failure and cost analysis (expected profit) of the design system have been obtained. Originality/value – In this research, a mathematical model of Linux operating system has been discussed from which one can check the behavioral analysis of the designed system.

In the present paper authors have focused on the study of complex system consisting two subsystems, in series configuration and handling by a human operator. The subsystem (1) has three units at super priority, priority and ordinary & the subsystem (2) has one unit in series configuration with the subsystem-1. The whole system is operated by a human operator and human failure can also appear at different state where system is in operational mode. Initially super priority unit starts function and failure during of super priority unit the priority unit start functioning and super priority unit goes under repair. The primitive resume repair policy is employed for repair of subsystem-1. The all failure rates are assumed to constants and follow exponential distribution but repair follow general and Gumbel-Hougaard family copula distribution. The system is studied by supplementary variable technique and Laplace transform. Various measure of reliability such as availability, state transition probabilities, mean time to system failure(M.T.T.F) and profit function has been discussed for available maintenance cost for all time and profit incurred by unit time for given interval. Some particular cases have been discussed for different values of different rates.

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