Comparative study between a single unit system and a two-unit cold standby system with varying demand

The concerned paper illustrates the comparison of two stochastic models of a cable manufacturing plant with varying demand. Here, it shows the comparison between a single unit system (Model 1) and a two-unit cold standby system (Model 2). In Model 1, the system is either in working state on some demand or put to shut down mode on no demand. In Model 2, at initial stage, one of the units is operative while the other is kept as cold standby. At times when the operative unit stops working due to some breakdown/failure, the standby unit instantaneously becomes operative while the repairman repairs the failed unit. In this working model, only one unit remains operative at a time. However, there may be a state when both the units fail. The comparison of systems is done by means of MTSF (mean time to system failure), steady state availability and profit function using Laplace transforms and software package Code-Blocks 13.12. Different graphs have been plotted to discover which model is superior to the other model under the given conditions. The system is analysed by making use of semi-Markov processes and regenerative point technique.

perfect or imperfect switching, Erlangian repair time and three types of repair facilities. Taneja and Naveen (2003), Ke and Chu (2007), Wang and Chen (2009) and Yusuf (2014) compared two models considering different situations such as expert repairman, redundant repairable system and switching failures. Zhang et al. (2012) and Dessie (2014) studied the modeling of diesel system in locomotives and HIV/AIDS dynamic evolution.
The demand has been kept fixed in most of these studies systems. However, in some practical situations there may be fluctuation in demand, such as the General Cable Energy System, where demand of the product varies. General cable energy system (Taneja and Malhotra 2013) is a cable manufacturing plant where different types of cables are produced. Two extruders of diameter 65 and 120 mm are available which are put to operation on the basis of demand. Hence, variation in demand plays an important role in the functioning of such systems. Malhotra and Taneja (2013a, b) studied the cost-benefit analysis of a single unit system (Model 1) while introducing variation in demand. In this model, initially demand is greater than or equal to the production. If the operative unit ceases working, a repairman repairs the failed unit. If there is fall off in demand, the system moves to a state in which demand is less than the production. Moreover, if demand declines further, the system will be in a shut down state. Malhotra and Taneja (2015) compared two single units with varying demand. Malhotra and Taneja (2014) developed a model for a two-unit cold standby system, without considering a shut down state where both the units may become operative simultaneously depending upon the demand.
In practical situations however, out of the two units being studied (in two-unit cold standby systems), only one unit remains operative at a time and the other unit is kept standby. Standby unit works only when the first unit fails. Information of such systems was collected on visiting a cable manufacturing plant in H.P., India and the authors developed a new model (Model 2) in which besides studying the above behavior, a concept of a new state generated was observed when there is no demand and system goes in shut down mode.
Depending on the situation, a model can be better or worse, therefore the comparative study becomes more important. Taking this into consideration, comparison is done graphically between the concerned models (Models 1 and 2) by computing various measures of system effectiveness using Laplace transforms and software package Code-Blocks 13.12.

Methods
The probabilistic analysis of the two models is analyzed by making use of semi-Markov processes and regenerative point technique.

Op
Unit is in operative state d ≥ p, d < p Demand is not less than production, demand is less than production C S , D

Description of Model 2
The transition diagram shows the various states of the system in Fig. 1. In this diagram, S 0 , S 1 , S 2 , S 6 and S 8 are regenerative states. S 3 , S 7 and S 9 are non-regenerative states. States S 4 and S 5 are failed states. In Model 2 ( Fig. 2), initially (state S 0 ) demand is not less than the production and one of the units is operative while the other is kept as cold standby. If the operative unit stops working, repairman repairs the failed unit and standby unit becomes operative instantaneously (state S 1 ). When both the units are not working i.e. one is under repair and other is waiting to be repaired, the system will stay in the failed state S 4 . If demand gets decreased, system goes to state S 2 . After this state, three possibilities are there: (1) if demand gets increased, system goes back to state S 0 , (2) if demand further gets decreased, system goes to down state S 8 and (3) if demand remains constant but operative unit fails, system goes to state S 6 . It moves to state S 7 as the demand increased and failed unit is being repaired. From state S 3 four possibilities are there (1) if failed unit gets repaired, then system moves to state S 2 , (2) if other unit also gets failed, system moves to state S 5 (one unit is under repair and other is waiting to get repaired, (3) if demand gets increased, system goes back to state S 1 and (4) if demand gets decreased further, system put to shut down state S 9 .
Model 2 differs from the model presented in Fig. 3 (Malhotra and Taneja 2014) because the earlier paper did not include the case of shut down on very less or no demand. Also, in the earlier paper, the two units may become operative simultaneously when the demand is very high leading to increase in the failure rate of the system as compared to Model 2 but on the other hand high demand is met sooner in the earlier model. So, a comparative study between these two models is also reported here in the conclusion revealing which and when one model is better than the other.
Various assumptions for the Model 2 are as follows: 1. The units are similar and statistically independent. 2. Once the down state is reached, system will not be operative till all the units become operable, irrespective of the nature of demand. 3. A single repair facility is available. Each unit is new after repair. 4. All the random variables are independent. Switching is perfect and instantaneous. 5. Each unit is assumed to have an exponential distribution of the time to failure while the distribution of repair time is taken as arbitrary.

Measures of system effectiveness of Model 2
Using semi-Markov processes and regenerative point technique, various recursive relations are solved and measures of system effectiveness for Model 2 are evaluated. From transition probabilities q ij , the steady state probabilities pij of reaching different states and mean sojourn times have also been computed. The non-zero elements p ij obtained as 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 ф i (t) where i = 0, 1, 2, 6, 8: Taking Laplace-Steltjes Transform (L.S.T.) of these relations and solving them for φ * * 0 (s), we obtain The reliability R(t) of the system at time t 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 φ * * 0 (s) from Eq.
(2), we have where Letting AD2(t) as the probability that the system is in upstate when demand is not less than production at instant t given that it entered the state i at time t. AP2(t) as the probability that the system is in upstate when demand is less than production at time t. B2(t) as the probability that the repairman is busy to repair the failed unit at instant t given that it entered the regenerative state i at any time t. V2(t) as the expected number of visits by the server in (0, t] given that the system entered the regenerative state i. DT2(t) as the probability that the system is in down state at instant t given that the system entered regenerative state i at any time t. These measures of system effectiveness (AD2, AP2, B2, V2, DT2) have been obtained, in steady state, by using the probabilistic arguments in the similar fashion as shown for MTSF except the fact that here the failed state is not considered as the absorbing state. For more elaboration of these measures, Malhotra and Taneja (2014) may be referred to.

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(15) Figure 4 depicts the behaviour of the availabilities (AD1, AD2) when demand is not less than the production with respect to the failure rate (λ). It can be interpreted from the graph that AD2 is < or = or >AD1 according as λ> or = or <0.3903. So far as the behaviour of the availabilities (AP1, AP2) when demand is less than the production with respect to the failure rate (λ) is concerned, it has been observed that Model 1 is better than that of Model 2, whatever the values of λ may be. Figure 5 depicts the behaviour of the differences of profits (P1-NP2) with respect to revenue (C 0 ) per unit up time for different values of loss (C 4 ) per unit time during the system remains down. It can be seen that this difference decreases with increase in the values of C 0 and has lower values for higher C 4 . Figure 6 depicts the behaviour of the differences of profits (P1-NP2) with respect to cost (C 2 ) for different values of cost (C 3 ). It can be seen that this difference increases with increase in the values of C 2 and has higher values for higher C 3 . Figure 7 depicts the behaviour of profits (P1, NP2) with respect to revenue (C 1 ) per unit up time. It is clear from the graph that NP2> or = or <P1 according as C 1 is > or = or <202.594. Figure 8 depicts the behaviour of profits (P1, NP2) with respect to installation cost (ICA) for additional unit. It is clear from the graph that NP2> or = or <P1 according as ICA is < or = or > 1632.46. Here, we wish to report the comparison between the Model 2 and that discussed in Malhotra and Taneja (2014) also as under: 1. MTSF in case of Model 2 is greater than that of the earlier paper, irrespective of the value of λ. However, availability in case of the former is greater or lesser than that the latter according as the demand is lesser or greater than production. 2. So far as the profit of the system is concerned, Model 2 is better or worse than that discussed in Malhotra and Taneja (2014) according as the value of revenue per unit up time is lesser or greater than that at cut-off point as shown in Fig. 9. 3. Behaviour of the profit with respect to loss per unit down time reveals that Model discussed in Malhotra and Taneja (2014) is better or worse than Model 2 if the loss per unit down time is greater or lesser than the value at cut-off point as shown in Fig. 10.

Conclusion
The semi-Markov process is applied to show the comparison of two stochastic models of a cable manufacturing plant with varying demand. The comparison of systems is done by means of MTSF, steady state availabilities and profit function.
In the terms of availability (when demand is not less than production), Model 2 is more profitable than Model 1, provided the failure rate does not exceed the calculated cut-off value. It can also be observed that all the availabilities decrease with increase in the value of failure rate. The lower limits (cut-off points) for the revenue per unit up time (when demand is not less than the production; when demand is less than production) for positive profit have been obtained, which may be quite useful for the system manufacturers/engineers/system analysts to check which model is best. Cut-off points for cost of engaging the repairman for repair and installation cost for additional unit has also been obtained. If these costs exceed corresponding cut-off values, Model 1 should be preferred.
Thus cut-off points on taking various numerical values for different parameters (rates/ costs) prove to be helpful in taking important decisions so far as the reliability and the profitability of the system is concerned. These also help to conclude that which model gives more profit as compared to other under favourable conditions.
Which and when one model is better than the other has been presented in the Table 1. The comparison between Model 2 and the model discussed in Malhotra and Taneja (2014) also reveals that none of these is always better than the other. One is better than the other for some values of failure rate/loss/revenue and worse for other values of these parameters.