Research Article | Open Access
Reetu Malhotra, Gulshan Taneja, "Stochastic Analysis of a Two-Unit Cold Standby System Wherein Both Units May Become Operative Depending upon the Demand", Journal of Quality and Reliability Engineering, vol. 2014, Article ID 896379, 13 pages, 2014. https://doi.org/10.1155/2014/896379
Stochastic Analysis of a Two-Unit Cold Standby System Wherein Both Units May Become Operative Depending upon the Demand
The present paper analyzes a two-unit cold standby system wherein both units may become operative depending upon the demand. Initially, one of the units is operative while the other is kept as cold standby. If the operative unit fails or the demand increases to the extent that one operative unit is not capable of meeting the demand, the standby unit becomes operative instantaneously. Thus, both units may become operative simultaneously to meet the increased demand. Availability in three types of upstates is as follows: (i) when the demand is less than or equal to production manufactured by one unit; (ii) when the demand is greater than whatever produced by one unit but less than or equal to production made by two units; and (iii) when the demand is greater than the produces by two units. Other measures of the system effectiveness have also been obtained in general case as well as for a particular case. Techniques of semi-Markov processes and regenerative processes have been used to obtain various measures of the system effectiveness.
In the literature of reliability, extensive studies have been made on different types of one-unit or two-unit standby redundant systems owing to their frequent use in modern business and industrial systems. There are two major types of redundancy—parallel and standby. In parallel redundancy, the redundant units form part of the system from the start, whereas in a standby system the redundant units do not form part of the system from the start (until they are needed). Standby units can be classified as cold, warm, or hot. A cold standby is completely inactive and since it is not hooked up, it cannot fail until it is replacing the primary unit. A warm standby has a diminished load because it is only partially energized. A hot standby is fully active in the system (although redundant).
A lot of work has been done on reliability and cost analysis of various systems by various researchers including [1–17] who have analyzed these systems by considering various concepts like the Erlangian repair time, operating and rest periods, hardware/software faults, congestion of calls, availability, two types of repair facility, human failure, regenerative point technique, priority repair discipline, instruction, accident, patience time, chances of nonavailability of expert repairman, one big unit and two small identical units with priority for operation/repair to big unit, patience time, partial failures, and optimized maintenance of the diesel system in locomotives. In all such studies, the demand was fixed. There may be situations where demand may vary and hence it affects the operability of the units of a system. The concept of variation in demand has been studied by [18, 19]. This concept of variation in demand was considered for single unit systems, where the system either is in working state on some demand or is put to shut down mode on no demand. However, the demand may be much more than whatever produced by a single unit system and hence there is need of having one additional unit to meet the demand. Study of the concept of variation in demand for the two-unit system thus becomes more important.
Keeping the above observations in view, we, in the present paper, develop a reliability model for a two-unit standby system working in a cable manufacturing plant wherein cold standby may become operative depending on demand. Information of such a system was gathered on visiting a cable manufacturing plant in H.P., India. It was observed that there were two units in the plant which were used to manufacture polyvinyl chloride (PVC) wires as per the demand in the market. If the demand is less than whatever produced by one unit, only one unit is kept operative, whereas if demand is greater than the production from a single unit, both units are put to operative mode. Various measures of the system effectiveness are analyzed by making use of semi-Markov processes and regenerative point techniques.
The rest of the paper is organized as follows. In Section 2, we develop the proposed semi-Markov model and presented its description. In Section 3, we find the steady-state probabilities and mean sojourn times. Section 4 deals with the computation of steady-state measures, such as mean time to system failure (MTSF), availability in three types of upstates (i.e., when demand is less than or equal to production made by one unit; when demand is greater than production made by one unit, but less than or equal to production made by two units; when demand is greater than production made by two units), busy period of the repairman for repair, and expected number of visits by the repairman. In Section 5, cost-benefit has been obtained as a general case. On the basis of the data/information estimated from the cable manufacturing plant, a particular case has been discussed in Section 6 and various graphs have been plotted in Section 7. Final conclusions along with some future directions are presented in Section 8.
2. Model Description and Assumptions
Figure 1 depicts the state transition model which we developed for a two-unit standby system working in a cable manufacturing plant wherein cold standby may become operative depending on demand. If one or both units are in working mode, then the system is in operative state. When both units are not working, that is, one is under repair and the other is waiting to be repaired, the system will stay in the failed state.
Various assumptions for the model are as follows.(1)The units are similar and statistically independent.(2)Demand cannot be decreased further, when at the most one unit is operative.(3)Each unit is new after repair.(4)If a unit is failed, standby unit takes over automatically.(5)The system becomes inoperable when both units fail in a two-unit system.(6)All the random variables are independent.(7)Switching is perfect and instantaneous.(8)Failure and repair time follow exponential and general time distribution, respectively.
3. Transition Probabilities and Mean Sojourn Times
The transition diagram showing the various states of the system is shown as in Figure 1. The epochs of entry into states , , , , , and are regeneration points and thus are regenerative states. States , , and are failed states. The transition probabilities are Clearly, for the model to be accurate, it is important to estimate accurately the model parameters (i.e., mean sojourn times and the steady-state transition probabilities).
The nonzero elements are obtained as By these transition probabilities, it can be verified that The mean sojourn time () in state is Analysis carried out in this paper depends only on the mean sojourn time and is independent of the actual sojourn time distributions for the semi-Markov processes states. If we were to carry out a transient analysis of the semi-Markov processes, this will no longer be true. The unconditional mean time taken by the system to transit for any state when it is counted from epoch of entrance into state is mathematically stated as Thus, where .
4. Measures of System Effectiveness
4.1. Mean Time to System Failure
To determine the mean time to system failure (MTSF) of the system, we regard the failed states as absorbing states. By probabilistic arguments, we obtain the following recursive relations for , where are given by Taking Laplace-Stieltjes Transform (L.S.T.) of the relations given by (7) and solving them for , we obtain where Now, the reliability of the system at time is given as Now, the mean time to system failure (MTSF) when the system starts from the state “0” is Using L’Hospital rule and putting the value of from (8), we have where
4.2. Availability Analysis When Demand Is Less Than or Equal to Production Made by One Unit
Letting where as the probability that the system is in upstate when demand is not less than production made by one unit at instant given that it entered the state at and using the arguments of the theory of the regeneration process, the availability is seen to satisfy the following recursive relations: where and Taking Laplace transforms of above equations (14) and solving them for , we get where In steady-state, the availability of the system is given by where
4.3. Availability Analysis When Demand Is Greater Than the Production Made by One Unit and Less Than or Equal to Production Made by Two Units
Letting where as the probability that the system is in upstate when demand is greater than the production made by one unit and less than or equal to production made by two units at instant given that it entered the state at and proceeding in the similar fashion as in 5.2, in steady-state, the availability is given by where and is already specified.
4.4. Availability Analysis When Demand Is Greater Than Production Made by Two Units
Letting where as the probability that the system is in upstate when demand is greater than the production made by two units at instant given that it entered the state at and proceeding in the similar fashion as in Section 4.2, the availability in steady-state is given by where and is already specified.
4.5. Busy Period Analysis of the Repairman
The total fraction of the time for which the system is under repair of the repairman, in steady-state, is given by where and is already specified.
4.6. Expected Number of Visits by the Repairman
The expected number of visits per unit time by the repairman, in steady-state, is given by where and is already specified.
5. Cost-Benefit Analysis
The expected profit can be calculated by expected total revenue in minus expected total repair in minus expected cost of visits by repairman in . Hence the total profit in is given by