Comparative study between a single unit system and a two-unit cold standby system with varying demand
© Malhotra and Taneja. 2015
Received: 31 July 2015
Accepted: 29 October 2015
Published: 19 November 2015
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.
Mathematics Subject Classification
The manufacturing of tools and special equipments is an inevitable part of our modern society. When equipment fails, production falls instantaneously. To maintain production, one has to keep tools/equipment available at all times, in order to run the systems. Since this can be expensive, availability and profit of an industrial system are becoming increasingly important. Indeed, profit will increase when the availability of a system increases.
System reliability has also been one of the major factors in most of the system performance-related studies. Though many researchers made meaningful contributions a lot in the field of system reliability modelling, fewer studies have reported the comparative analysis of different types of systems. Tuteja et al. (1991), Alidrisi (1992), Mokaddis et al. (1994), Pan (1997), Chandrasekhar et al. (2004) and Xu et al. (2005) analysed the reliability and availability of standby systems by studying various parameter viz. partial failures, 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.
The probabilistic analysis of the two models is analyzed by making use of semi-Markov processes and regenerative point technique.
Notations used for the description of models
Unit is in operative state
- d ≥ p, d < p:
Demand is not less than production, demand is less than production
- CS, D:
Unit is in cold standby state, down unit
- Fr/Fw :
Failed unit is under repair/waiting for repair
- FR :
Repair of failed unit continuing from previous state
- λ, α:
Failure rate, repair rate of the operative unit
- γ1 :
Rate of decrease of demand < production
- γ2 :
Rate of increase of demand ≥ production
- γ3 :
Rate of going from upstate to downstate
- γ4, β1 :
Rate of reaching the state of no demand from some demand in Model 1, Model 2
- p1 :
Probability that during the repair time demand ≥ production
- p2 :
Probability that during the repair time demand < production
c.d.f. of first passage time from regenerative state i to a failed state
- ADi, APi :
Availability that the system is in upstate when demand ≥ production and when demand < production for each Model i where i = 1, 2
- Bi :
Busy period analysis of the repairman for each Model i where i = 1, 2
- Vi :
Expected number of visits of repairman for each Model i where i = 1, 2
- DTi :
Expected down time for each Model i where i = 1,2
Profit incurred to the system for Model 1
Net profit [total profit incurred in Model 2—installation cost for addition unit (ICA)]
- μi :
Mean sojourn time in regenerative state i before transiting to any other state
Symbol for Laplace transforms
Symbol for Laplace Stieltjes transforms
- qij(t), Qij(t),:
p.d.f and c.d.f of first passage time from a regenerative state i to a regenerative state j or to a failed state j without visiting any other regenerative state in (0, t]
- g(t), G(t):
p.d.f. and c.d.f. of repair time for the unit
Description of Model 2
The units are similar and statistically independent.
Once the down state is reached, system will not be operative till all the units become operable, irrespective of the nature of demand.
A single repair facility is available. Each unit is new after repair.
All the random variables are independent. Switching is perfect and instantaneous.
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
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.
Results and discussion
Comparison with respect to
Which model is better (according to different situations)
Model 1 is better if
Model 2 is better if
Both the models are equally good
γ1 = 0.008/hr, γ2 = 0.235/hr, γ3 = 0.353/hr, γ4 = 0.4213/hr, α = 0.05/hr, β1 = 0.002/hr, p1 = 0.665, p2 = 0.335, C2 = INR 500, C1 = INR 700, C3 = INR 400, ICA = INR 500, λ = 0.003/hr
Availability (AD) when d ≥ p
λ > 0.3903
λ < 0.3903
λ = 0.3903
C4 = 100
C0 < 728.173
C0 > 728.173
C0 = 728.173
C4 = 600
C0 < 574.291
C0 > 574.291
C0 = 574.291
C4 = 1100
C0 < 352.178
C0 > 352.178
C0 = 352.178
λ = 0.003/hr, α = 0.05/hr, γ1 = 0.008/hr, γ2 = 0.23/hr, γ3 = 0.353/hr, γ4 = 0.4213/hr, β1 = 0.002/hr, p1 = 0.665, p2 = 0.335, C0 = INR 7000, C1 = INR 700, C4 = INR 400, ICA = INR 500
C3 = 200
C2 > 574.248
C2 < 574.248
C2 = 574.248
C3 = 10,200
C2 > 478.789
C2 < 478.789
C2 = 478.789
C3 = 20,200
C2 > 391.432
C2 < 391.432
C2 = 391.432
λ = 0.003/hr, α = 0.05/hr, γ2 = 0.235/hr, γ3 = 0.353/hr, γ4 = 0.4213/hr, β1 = 0.002/hr, p1 = 0.665, p2 = 0.335, C2 = INR 500, C0 = INR 7000, C3 = INR 400, C4 = INR 400, ICA = INR 500, γ1 = 0.008/hr
C1 < 202.594
C1 > 202.594
C1 = 202.594
λ = 0.003/hr, α = 0.05/hr, γ1 = 0.008/hr, γ2 = 0.235/hr, γ3 = 0.353/hr, γ4 = 0.4213/hr, β1 = 0.002/hr, p1 = 0.665, p2 = 0.335, C2 = INR 500, C0 = INR 7000, C3 = INR 400, C4 = INR 400, C1 = INR 700
ICA > 1632.46
ICA < 1632.46
ICA = 1632.46
It has been observed that the MTSF for the Model 2 is greater than that of Model 1 irrespective of the values of failure rate (λ). However, availability of one may be greater or lesser than that of the other depending upon the values of λ as discussed below.
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.
Comparative analysis between Model 2 and the Model discussed in Malhotra and Taneja (2014)
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.
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.
RM did literature review, collected the data sets, designed the model, calculated various measures of system effectiveness, plotted different graphs to draw various conclusions and wrote the article. GT analyzed the data, helped in designing the model and finding results and drawing conclusions. Both authors read and approved the final manuscript.
Our appreciation also goes to staff of Cable Manufacturing plant, Baddi for providing us the requisite data/information.
The authors declare that there is no competing interest regarding the publication of this manuscript.
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