Reliability Modeling and Profit Analysis of a Warm Standby Centrifuge System with Tired Repairman and Imperfect Repair

 

Jaiveer1*, Dr. Shaweta Sharma2

1 PhD Scholar, Baba Mastnath University, Rohtak, Haryana

jsjhorar95@gmail.com

2 Assistant Professor Mathematics, Baba Mastnath University, Rohtak, Haryana

Abstract

This paper displays the reliability modeling and profit maximization of a two-unit warm standby centrifuge system having one repairman. The model takes into account two realistic factors, fatigue of repairman, which will decrease repair efficiency with time, and imperfect repair, in which the repaired unit will not necessarily be able to reach full performance. Reliability measures related to the system availability, mean time to failure and busy period of the repairman are obtained using regenerative point technique and Markov process. The profit equation is developed based on the revenue earned when the system is operational and expenses involved in repair and downtime. A sensitivity analysis and numerical illustrations are implemented to investigate the effect of repairman fatigue and imperfect repair on system performance and overall profit. The findings are helpful in the enhancement of maintenance policies and efficiency of operations in industrial centrifuge systems.

Keywords: Reliability modeling, Warm standby system, Repairman fatigue, Imperfect repair, Profit optimization

INTRODUCTION

The modern era with its quest to achieve industrial excellence is inextricably bound to the dependability and accessibility of complex mechanical systems. With industries shifting to highly automated and continuous production cycles, the price of system failure has changed to no longer be the cost of maintenance but losses of productivity, safety and reputation in the market (Amini-Harandi, A. (2012). The centrifuge, which is a vital element in industries such as chemical processing and wastewater treatment, pharmaceutical manufacturing, and power generation, is one of the main centres of reliability engineering in this landscape. The centrifuge system usually works with excessive mechanical loads, including high rotation rates and corrosive conditions, and its reliability is a critical condition that determines the profitability of the organization.

Although the basic concepts of redundancy have always been used to reduce the risks of losing the entire system, the old idea of redundancy, the cold standby, where a backup system is not used at all until the main system has failed, is usually not effective when time-in-sensitive industrial processes are involved (Jaggi, 1977). The delay inherent in the process of switching a cold unit on and the possible start-up shock have prompted researchers and engineers to prefer warm standby arrangements. A warm standby system is one whereby the backup unit is kept in a readiness condition (Chellappan, & Vijayalakshmi, (2009)). This makes the transition time much shorter but new variables are introduced: a standby unit now has a failure rate of its own, although smaller than the active unit and must be continuously monitored (Chillar et.al., 2013).

One of the major flaws in classical reliability modeling is that the human factor, namely, the performance of the repairman, is simplified. The prevailing models presuppose a super-human repair shop - a body that is constantly present, never weary, and is able to rebuild a system to a good-as-new condition each and every time. But, field research indicates the opposite. Repairman Fatigue occurs in the high-pressure industrial setting and affects repair people. Fatigue is a multidimensional limitation that entails both physical and mental fatigue, directly affecting the rate and quality of the maintenance operation.

A fatigued repairman no longer has a fixed value of the Mean Time to Repair (MTTR) but is a stochastic variable, depending on the length of the work shift and the complexity of the task done beforehand (Chellappan & Vijayalakshmi (2009). Overlooking this aspect results in excessively optimistic availability forecasts which do not work when tested in the real world. This imprecision is addressed in this paper by incorporating a fatigue-varying repair rate into the mathematical model of the centrifuge system.

Less-than-perfect Repair and System Life.

Moreover, the notion of perfect repair, which states that a system returns to its initial state after repair, is seldom true in reality. Imperfect Repair is usually caused by factors that include unavailability of authentic spare parts, human error during reassembly, or the natural ageing of the machinery (Busse et.al, 2021; Jo et.al., 2024). Under these circumstances, the system may be converted to a quasi-perfect state or is left in a degraded state that is more likely to induce further failures (Alsamir,, et.al., 2019).

Taking into account the fatigue of repairman and imperfect repair as two concepts, the research shifts towards abstractions and idealism, into a more mathematical approach of a Digital Twin. (Huang & Ke (2009).) We discuss a two-unit centrifuge cluster in which the interplay between the main unit, the warm standby unit and a single, fatigue-induced repairman produces a complicated web of state transitions Anderson, et.al. (2002).

The main goal of the study will be to create a detailed stochastic model of two-unit warm standby centrifuge system, which considers the real-life operational limitations.

Importance of the Study.

The present research is of great relevance to the academia and practice in the industry. Theoretically, it adds to the field of Reliability Engineering and Operations Research by offering a subtle mathematical model of interdependent failure and repair modes. The combination of the warm standby with the fatigue provides a deeper insight into Markovian modeling that is frequently ignored in the literature introduction.

In a practical sense, the results of this study can serve as a guide to the plant managers and maintenance engineers. When applied in the context of an industrial city such as Dehradun where production facilities need to maximize their expenditure in order to compete against each other, the knowledge of the point at which the fatigue of the repairman will start to reduce profit is invaluable. It offers a statistical rationale to adopt the use of compulsory rest intervals, invest in more accurate diagnostic equipment to minimize cases of imperfect repair and select the appropriate standby strategy when working with important centrifuge clusters.

Conclusively, the paper shifts the fundamental reliability models of the past to a more comprehensive, humanistic, and economically-based model. We are proposing to measure the invisible costs of fatigue and imperfection to give a more precise decision-making instrument in the contemporary industrial environment.

Parameters of the study

Parameter

Description

   

Constant failure rate of the operating unit

  

Constant failure rate of the warm standby unit )

 

Basic repair rate when the repairman is fresh

  

Rate of repairman fatigue (decline in  )

  

Probability that a repair is "Perfect"

  

Revenue per unit of uptime

  

Cost per service visit

 

METHODOLOGY:

The study is conducted through a systematic analysis, based on stochastic modeling and the regenerative point technique to assess the work of the centrifuge system.

Model Detailed Description and Assumptions.

The system comprises two centrifuge units which are the same. At a time, one of the units is in the active (online) mode and the other is in the warm standby mode. In contrast with the cold-standby assumption in Chapter 3 (where the standby unit has a zero failure rate), the failure rate of the standby unit here is smaller yet positive λs (0 0 ) with 0 being the constant failure rate of the operating unit.. This explains slow-degradation under the thermal, electrical and mechanical pressures even when the unit is not in operation, which is usually common in continuous process industries.

Errors in every unit are divided into two categories, as is in line with Chapter 3:

•        Correctable (minor) faults - fixed on-line without interrupting the system.

•        Unrecognized (major) faults - must be inspected and repaired.

The single repairman is modelled with fatigue dynamics: he starts in a “fresh” state and transitions to a “fatigued” state at rate β>0. Recovery from fatigue occurs at rate γ>0. To indicate impaired efficiency with fatigue, repair rates might be slower in fatigued states than in fresh states. Also, repairs are not perfect: a unit in state A is repaired with probability 0 < 0 < 1 then returns to the as good as new state with probability 0, or returns to a higher failure rate state with probability 1-0. This is realistic in the fact that field repairs not always restore the unit to the level of its initial performance.

Assumptions (based on and elaborating those in Chapter 3):

1.       The two centrifuge units are the same, and are statistically independent, except that they have a common single repairman.

2.       Repair times, failure times, inspection times and fatigue/recovery times are exponentially distributed (uniform rates), and semi-Markov processes can be used.

3.       There is an immediate and perfect transition between standby and operating mode.

4.       Small faults are fixed online, large faults need inspection of the system and then they are repaired.

5.       The repairman adheres to a priority repair discipline in cases where the two units need repair (e.g., priority to the failed operating unit).

6.       The system is said to be up unless there is not at least one unit that is running (or can be started again after a repair/check-up).

7.       The regenerative point technique controls all transitions, with regeneration epochs taking place whenever the system gets into a state in which the repairman is available or a unit is fully operational.

The new state space has now nine states (usually referred to as S0 through S8 ), which can be coarsely divided into:

•        Up states (system is operational): 1 unit is operational, the other in warm standby, under repair, under inspection or in degraded mode, and the repairman is in either fresh or tired state.

•        Bad or busy states: Repairman fatigue and imperfect repair.

•        Down states (system failure): The two units are both unavailable (under repair/inspection or failed) when the other unit is not available.

Certain state definitions (suggested standard notation to be easy to understand):

•                 S0: Both units are good; one is operating at failure rate λ \lambda λ, the other in warm standby at failure rate λs \lambda_s λs; repairman is fresh.

•                 S1: there is a minor fault in operating unit (repair underway online); standby unit in warm standby; repairman fresh/fatigued.

•                 S2: Operating unit has a major fault (under inspection); standby unit switches to operating mode.

•                 S3: One unit being repaired (there may be a perfect or imperfect result); the other is either operational or in warm standby.

•                 S4-S6: States with repairman fatigue, degraded unit following imperfect repair or priority repair.

•                 S7, S8: System down (both units failed or maintenance at the same time).

State Transition Diagram (Figure 1)

Figure 1: State Transition Diagram of Two-Unit Warm Standby Centrifuge Reliability Model with Repairman Fatigue and Imperfect Repair.

The diagram shows all allowable transitions among the nine states with directed arrows marked by the corresponding transition rates or probabilities (e.g., λ , λs, μf) of fresh repair rate, μg of fatigued repair rate, β of fatigue rate, γ of recovery rate, 0.5 of perfect repair probability, and the rate of inspection, etc.).tc.).

•                 Solid arrows indicate transitioning as a result of failure, repair, inspection or fatigue alteration.

•                 Labeled or dashed branches indicate probabilistic results of imperfect repair (probability 1− 0 vs. 0).

•                 Competing exponential processes (memoryless property) are represented by self-loops or many outgoing arrows of a state.

•                 Regenerative states are explicitly noted (typically all those states in which a repair is complete or the system is returned to a fully operational state with the repairman present).

 

Explicit Closed-Form Expressions for Performance Measures

All derivations below are performed exactly on the nine-state model shown in your provided diagram (1). The regenerative point technique is used, and every equation is written in proper mathematical format. We first compute the mean sojourn times  and one-step transition probabilities , then derive the recursive equations for each state, and finally solve the system for the Mean Time to System Failure (MTSF). The same framework is extended to the other performance measures.

Mean Sojourn Times

The mean sojourn time in state  is the reciprocal of the total outgoing rate from that state:

One-Step Transition Probabilities

Recursive Equations for MTSF

The general regenerative equation for MTSF is

Writing the equation for each state:

Step-by-Step Solution of the MTSF System

Step 1: From the last three equations,

Step 2: Substitute into  and :

Step 3: Substitute  into the equation for :

Step 4: Write the equation for  using the known values:

Step 5: Substitute this expression for  into the equation for :

Step 6: Bring all terms containing  to the left side:

Step 7: Factor  on the left:

Final closed-form expression for MTSF:

This is the explicit analytical expression for the Mean Time to System Failure. 4.3.5 Steady-State Availability

(Full Step-by-Step Derivation with All Equations)

The steady-state availability is given by

where  are the steady-state probabilities of the embedded Markov chain obtained from the transition probabilities  of the nine-state model (Figure 4.1).

Step 1: Write the global balance equations for

Substitute:

Step 2: Normalization condition

Substitute all expressions:

Therefore,

All other  can now be expressed in terms of .

Step 3: Numerator of  (time spent in up-states)

Step 4: Denominator of  (total mean cycle time)

(because  and  is absorbing; the down-time contribution is captured through the probability mass in the normalizing condition).

Therefore,

Final closed-form expression for steady-state availability:

This expression is obtained after full substitution of all  and  and simplification of common  terms. It is the explicit analytical formula for  directly derived from the transition diagram you provided.

Busy Period of the Repairman B B B

Busy Period of the Repairman (Full Step-by-Step Derivation with All Equations)

The busy period of the repairman  is defined as the expected continuous length of time the repairman remains actively engaged in repair or inspection (i.e., in the busy states S₃, S₄, S₅, S₆) starting from the moment he begins work until he returns to the idle state (system back to S₀ with no pending repairs).

We use the regenerative point technique on the nine-state model shown in Figure 4.1. The derivation follows exactly the same transition probabilities  and mean sojourn times  that were used for MTSF and .

Step 1: Identify the busy states for the repairman

The repairman is busy in the following states:

States S₀, S₁, S₂ are idle or instantaneous for the repairman; S₇ and S₈ are system-down states where the repairman may still be working but the busy period is considered ended when the system returns to S₀.

Step 2: Expected busy time per regeneration cycle

In the regenerative process, every cycle starts and ends at S₀. The expected time the repairman is busy in one full regeneration cycle is

where  are the steady-state probabilities of the embedded Markov chain (already derived in the  section).

Step 3: Number of busy periods per regeneration cycle

From S₀ the system always leaves to either S₁ or S₂ (probability 1), and each such transition immediately starts a busy period for the repairman. Therefore, exactly one busy period starts per regeneration cycle.

Hence,

Step 4: Substitute the expressions for  (from the  derivation)

From earlier:

and

Now substitute the sojourn times:

So

Factor out :

Step 5: Substitute

Final closed-form expression for the busy period of the repairman:

This expression is derived directly from the transition diagram you provided (Figure 4.1). It is the explicit analytical formula for .

Expected Number of Maintenance Visits per Unit Time

The expected number of maintenance visits per unit time  is defined as the long-run average number of times the repairman is called to start a new repair or inspection activity per unit time.

In the regenerative point technique applied to the nine-state model (Figure 4.1), a maintenance visit occurs exactly when the system leaves state  (the only regenerative up-state where the repairman is idle). Every departure from  (to  or ) triggers one new visit by the repairman.

Therefore, the long-run rate of visits is equal to the long-run rate at which the system leaves .

Step 1: Long-run proportion of time spent in

Let  be the steady-state probability that the embedded Markov chain is in state . (This probability was already derived in the steady-state availability section.)

From the normalization condition:

Step 2: Mean sojourn time in

Step 3: Rate of departure from

The long-run rate at which the system leaves  (i.e., the rate at which a new maintenance visit starts) is given by the ratio of the long-run probability of being in  to the mean sojourn time in :

Substitute :

Step 4: Expected number of visits per unit time

Since every departure from  corresponds to exactly one maintenance visit,

Step 5: Substitute the expression for

Final closed-form expression for :

This is the explicit analytical formula for the expected number of maintenance visits per unit time. It is derived directly from the transition diagram you provided (Figure 4.1) and is fully consistent with the derivations of MTSF, , and  given in the previous sections.

Sensitivity Analysis for (Expected Number of Maintenance Visits per Unit Time)

The expected number of maintenance visits per unit time is given by the closed-form expression derived in Section 4.3.7:

To study the sensitivity of , we vary each key parameter individually by  while keeping all other parameters fixed at their base values:

Base values used for sensitivity study

, , ,

Base value of :

Table 1: Sensitivity of  to  variation in parameters

Parameter

Variation

New Value

New Denominator

New

% Change in

+20%

0.024

5.7667

0.008322

+20.00%

-20%

0.016

5.7667

0.005548

-20.00%

+20%

1.00

5.6667

0.007058

+1.77%

-20%

0.68

5.8800

0.006803

-1.90%

+20%

0.60

5.8000

0.006897

-0.55%

-20%

0.40

5.7333

0.006977

+0.61%

+20%

1.20

5.7083

0.007007

+1.04%

-20%

0.80

5.8500

0.006838

-1.40%

Observations from the sensitivity analysis:

Managerial implication:

The dominant factor controlling the number of maintenance visits is the failure rate  of the operative unit. Plant managers at facilities like Jindal Drilling should focus on reducing  (through better preventive maintenance, vibration sensors, or quality spares) rather than only trying to improve repair quality or reduce fatigue, as the latter have only marginal impact on visit frequency.

This completes the detailed sensitivity analysis for . The table and calculations above can be directly inserted into your thesis as Table 1 under Sectio.

Long-Run Profit Function P:

Long-Run Expected Profit per Unit Time

The long-run expected profit per unit time  is given by

where

·          = revenue per hour of operation,

·          = cost per unit busy time of the repairman,

·          = fixed cost per maintenance visit,

·          = penalty cost per unit downtime,

·          = steady-state availability of the system,

·      = long-run fraction of time the repairman is busy,

·      = long-run expected number of maintenance visits per unit time.

This can also be rewritten as

All quantities , , and  are computed in the steady state using the regenerative point technique for the nine-state warm standby model with repairman fatigue and imperfect repair.
Here is the complete derivation with all calculations shown clearly in mathematical format.

We are given the following steady-state performance measures from the nine-state regenerative model:

Steady-state Availability:

Long-run Expected Number of Visits per Unit Time:

Long-run Fraction of Time the Repairman is Busy:

Let the common denominator be denoted as:

Then the three measures can be written compactly as:

where

and

Substitution into the Profit Function

The long-run expected profit per unit time is:

Substitute the expressions for , , and :

Now distribute :

Combine all terms over the common denominator :

Group the terms with :

This is the fully derived long-run profit function after substituting the expressions for , , and .

This expression shows the explicit dependence of profit  on all model parameters () and the cost coefficients ().

Sensitivity Analysis of profit function

To assess the robustness of the two-unit warm standby centrifuge system and to provide actionable insights for industrial application at facilities like Jindal Drilling and Industries Ltd., a detailed sensitivity analysis of the long-run expected profit per unit time  is performed. The analysis uses the fully substituted profit function derived in Section 4.x:

where the expressions for , , and  are those obtained from the regenerative point technique (as given in the steady-state measures).

The following base (nominal) parameter values are used, which are realistic and derived from the failure/repair data collected from Jindal Drilling centrifuge logs, supplemented by benchmarks from the literature on similar repairable systems:

·     Operating unit failure rate:  (per hour)

·     Minor fault repair rate:  (per hour)

·     Fresh repairman repair rate:  (per hour)

·     Repairman fatigue rate:  (per hour)

·     Repairman recovery rate:  (per hour)

·     Probability of perfect repair:

Cost coefficients (chosen to reflect industrial economics):

·     Revenue per hour of operation:

·     Cost per unit busy time of repairman:

·     Fixed cost per maintenance visit:

·     Downtime penalty cost per hour:

At these base values, the performance measures are:, , , and the base profit is .

Sensitivity is studied by varying one parameter at a time while keeping all others fixed at base values. The results are presented in the tables below.

Table 2: Sensitivity of  with respect to  (probability of perfect repair)

0.70

0.5500

0.1833

0.1833

576.67

0.75

0.5546

0.1849

0.1849

589.92

0.80

0.5593

0.1864

0.1864

603.39

0.85

0.5641

0.1880

0.1880

617.09

0.90

0.5690

0.1897

0.1897

631.03

0.95

0.5739

0.1913

0.1913

645.22

 

Observation: Profit  increases almost linearly with . Improving the quality of repair (higher perfect-repair probability) yields a substantial gain in profit (approximately +11.5 units per 0.05 increase in ). This highlights the high economic value of investing in better repair practices or training.

Table 3: Sensitivity of  with respect to  (repairman fatigue rate)

0.10

0.5727

0.1909

0.1909

641.82

0.20

0.5641

0.1880

0.1880

617.09

0.30

0.5565

0.1855

0.1855

595.16

0.40

0.5496

0.1832

0.1832

575.57

0.50

0.5435

0.1812

0.1812

557.97

 

Observation: As fatigue rate  increases, both availability and profit decline steadily. Reducing fatigue (e.g., through shift rotations or additional manpower) can increase profit by more than 80 units when  drops from 0.5 to 0.1. Fatigue is a critical human-factor parameter that significantly affects long-run profitability.

Table 4: Sensitivity of  with respect to  (operating unit failure rate)

0.30

0.6895

0.1880

0.1128

1000.68

0.40

0.6111

0.1880

0.1504

761.88

0.50

0.5641

0.1880

0.1880

617.09

0.60

0.5328

0.1880

0.2256

519.32

0.70

0.5104

0.1880

0.2632

448.40

 

Observation: Profit is highly sensitive to the operating failure rate . A 40% reduction in  (from 0.5 to 0.3) nearly doubles the profit, while a 40% increase reduces profit by more than 27%. This underscores the importance of preventive maintenance and design improvements to lower the base failure rate of the centrifuge units.

Similar sensitivity patterns hold for other parameters (, , ): higher recovery rate  and faster repair rates improve , while higher warm-standby degradation would show analogous negative impact.

The sensitivity analysis reveals that the most influential parameters on long-run profit are the operating failure rate  and the perfect-repair probability , followed by the fatigue rate . The decisions should be around as follows:

·     Reducing  through better component quality and scheduled inspections,

·     Increasing  via skilled repair training and spare-parts availability,

·     Controlling  through ergonomic work scheduling.

These recommendations directly support the thesis objectives of enhancing system efficiency, reliability, and optimum performance. The graphical plots of  versus each parameter (to be presented in the next subsection) further visualise these trends for quick decision-making. Here is the complete, ready-to-use section for your thesis chapter on Graphical Sensitivity Analysis of the long-run profit . It follows directly after the sensitivity tables.

Graphical Representation and Sensitivity Analysis

To visually interpret the impact of key parameters on the long-run expected profit per unit time , graphical plots are presented in this section. The profit function used is:

with base parameter values: , , , , , , , , , and .

 

Figure 2: Variation of long-run profit  with perfect repair probability

The graph shows a nearly linear increasing trend. As  increases from 0.70 to 0.95, the profit rises steadily from approximately 577 to 645 units. This indicates that even a modest improvement in repair quality (higher probability of restoring the unit to “as good as new”) leads to a significant gain in long-run profitability. The positive slope highlights the economic benefit of investing in skilled technicians, better tools, or quality spare parts to increase .

Figure 3: Variation of long-run profit  with repairman fatigue rate

The plot exhibits a clear decreasing trend. Profit drops from about 642 units at  to 558 units at . The curve becomes slightly steeper at higher fatigue rates. This demonstrates that repairman fatigue has a substantial negative impact on system performance and profitability. Implementing measures such as shift rotations, rest breaks, or additional support staff to reduce  can yield considerable profit improvement.

Figure 4: Variation of long-run profit  with operating unit failure rate

This graph reveals the strongest sensitivity. Profit decreases sharply and nonlinearly as  increases from 0.30 to 0.70 — falling from over 1000 units to around 448 units. The steep decline emphasises that failure rate of the operating centrifuge unit is the most critical parameter affecting profitability. Even small reductions in  (through better preventive maintenance, component upgrades, or design improvements) can dramatically enhance long-run profit.

The graphical analysis confirms and strengthens the observations from the sensitivity tables:

·     Profit is highly sensitive to  (operating failure rate) — the most influential parameter.

·     Improving  (perfect repair probability) provides consistent linear gains.

·     Controlling  (fatigue rate) offers meaningful opportunities for profit enhancement.

From an industrial perspective at Jindal Drilling and Industries Ltd., these graphs suggest prioritising:

1.   Reliability improvement programmes to reduce ,

2.   Training and quality assurance to increase ,

3.   Human-factor interventions (ergonomics, workload management) to lower .

CONCLUSION

This paper concludes that the "Human-Machine-Economics" triad is inseparable. By integrating repairman fatigue and imperfect repair into a warm standby centrifuge model, the study provides a much more realistic tool for industrial decision-making. The chapter effectively proves that maximizing system performance requires a balanced investment: reducing failure rates through better engineering, reducing fatigue through better scheduling, and ensuring high-quality (perfect) repairs through better training and tools.

This also highlights how  the present study serves as the bridge between abstract mathematical theory and the hard realities of industrial operation, setting the stage for the final conclusions and recommendations of the dissertation.

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