The impact of machines on our daily lives can be felt in every aspect of our lives. Industries have played a huge role in modern economies. The two determinants of success are quality and availability, which are crucial in an environment of fierce domestic and international competition. Production downtime can have dramatic repercussions. Using actual field data, Vališ et al. (2020)
The paper discusses a powder encapsulation system. It consists of one drying chamber with three components: The first component is the heating tank, the second component is the condenser, and the third component is the concentrate. They are all in operation when the system begins to process material. During the winter months, a high level of milk production keeps the system operating, but the system goes into cold standby during the summer and is maintained. In the event that one of the three units fails, the entire system fails.
SemiMarkov process and regenerative point technique are used to obtain the following measures of system effectiveness in steadystate:
• Transition probabilities and mean sojourn times in different states. MTSF of the system.
• Steadystate availability for the system.
• A busy period for the repairman.
• Expected number of repairs.
• Additionally, the system's profit potential is analyzed graphically.
• The system is initially operative at state 0.
• Time to failure of each unit is assumed to follow an exponential distribution, whereas repair time distribution is taken to be arbitrary.
• There are similarities and statistical independence between the units.
• After each repair, the system works as well as new.
• λ → Failure rate of the drying chamber unit.
• λ_{1}, λ_{2}, λ_{3} → Failure rate of units one, two, and three respectively.
• λ_{4} → Maintenance rate of a powder system.
• α → Rate of going to summer.
• β → Rate of going to winter.
• S → Summer.
• W → Winter.
• O→ Operative state.
• cs → Cold Standby state.
• um → Under maintenance.
• Fr → Failure under repair.
• dc → Drying chamber of powder plant.
• u1, u2, u3 → Units 1,2,3 respectively.
• Odc → Drying chamber of the powder system is operating.
• ou1, ou2, ou3 → Units 1,2,3 is operating.
• cs dc → Drying chamber of the powder system is in the cold standby state.
• cs1, cs2, cs3 → Units 1,2,3 is in cold standby state.
• G(t), g(t) → c.d.f. and p.d.f of time to repair the drying chamber unit.
• G_{1}(t), g_{1}(t) → c.d.f. and p.d.f of time to repair unit one.
• G_{2}(t), g_{2}(t) → c.d.f. and p.d.f of time to repair of unit two.
• G_{3}(t), g_{3}(t) → c.d.f. and p.d.f of time to repair of unit three.
• G_{4}(t), g_{4}(t) → c.d.f. and p.d.f of maintenance of the powder plant.
The nonzero elements p_{ij} can be represented as below:
p_{ij} = Q_{j}_{ }(∞) =
• p_{01} = β /β+α
• p_{02} = α /β+α
• p_{14} = λ_{3 }/λ+λ_{1}+λ_{2}+λ_{3}
• p_{15} = λ /λ+λ_{1}+λ_{2}+λ_{3}
• p_{16} = λ_{2} /λ+λ_{1}+λ_{2}+λ_{3}
• p_{17} = λ_{1} /λ+λ_{1}+λ_{2}+λ_{3}
• p_{32} = g_{4}∗(0)
• p_{41} = g_{3}∗(0)
• p_{51} = g∗(λ_{1} + λ_{2} + λ_{3})
• p_{58} = p_{54}^{(8) }= (λ_{3} /λ_{1}+λ_{2}+λ_{3}) (1 − g∗(λ_{1} + λ_{2} + λ_{3}))
• p_{59} = p_{57}^{(9)} = (λ_{1} /λ_{1}+λ_{2}+λ_{3}) (1 − g∗(λ_{1} + λ_{2} + λ_{3}))
• p_{5,10} = p_{56}^{(10)} = (λ_{2} /λ_{1}+λ_{2}+λ_{3}) (1 − g∗(λ_{1} + λ_{2} + λ_{3}))
• p_{61} = g_{2}∗(0)
• p_{71} = g_{1}∗(0)
• p_{84} = p_{97} = p_{10,6} = g∗(0)
By these transition probabilities it is also verified that
• p_{01} + p_{02} = 1
• p_{14} + p_{15} + p_{16} + p_{17} = 1
• p_{51} + p_{58} + p_{59} + p_{5,10} = 1
• p_{51} + p_{54}^{(8)} + p_{57}^{(9)} + p_{56}^{(10)} = 1
• p_{23} = p_{32} = p_{41} = p_{61} = p_{71} = p_{84} = p_{97} = p_{10,6} = 1
The unconditional meantime is taken by the system to transit for any regenerative state ‘j’ when t (time) is counted from the epoch of entrance into state ‘i’ is mathematically stated as:
• m_{01} + m_{02} = µ_{0}
• m_{14} + m_{15} + m_{16} + m_{17} = µ_{1}
• m_{51} + m_{58} + m_{59} + m_{5,10} = µ_{5}
• m_{51} + m_{54}^{(8)} + m_{57}^{(9)} + m_{56}^{(10)} = K
where,
The mean sojourn time (µ_{i}) in the regenerative state ‘i ’is defined as time of stay in that state before transition to any other state:
• µ_{0} = 1/ α + β
• µ_{1} = 1/ λ + λ_{1} + λ_{2} + λ_{3}
• µ_{2} = 1 /λ_{4}
• µ_{3} =
• µ_{5} = (1 /λ_{1} + λ_{2} + λ_{3}) [1 − g∗(λ_{1} + λ_{2} + λ_{3})]
• µ_{4} =
• µ_{6} =
• µ_{7} =
• µ_{8 }= µ_{9} = µ_{10} =
Mean time to system failure (MTSF) of the system is determined by considering the failed state as an absorbing state. By probabilistic arguments, we obtain the following recursive relations for
ϕ₀(t)= Q_{01}(t) ⋆ ϕ_{1}(t) + Q_{02}(t) ⋆ ϕ_{2}(t)
ϕ_{1}(t)= Q_{14}(t) +Q_{15}(t) ⋆ ϕ_{5}(t)+Q_{16}(t)+Q_{17}(t)
ϕ_{2}(t)=Q_{23}(t)
ϕ_{5}(t)= Q_{51}(t) ⋆ ϕ_{1}(t) + Q_{58}(t) +Q_{59}(t) +Q_{5,10}(t)
Taking LaplaceStieltjes Transform (L.S.T.) of the relations given by above equation and solving them for
The reliability R(t) of the system at time t is given as
R(t) = Inverse Laplace transform of
The mean time to system failure (MTSF), when the system started at the beginning of state 0 is:
MTSF =
Using L’ Hospital rule and putting the value of φ_{0}**(s) we get
Let A_{0}(t) be the probability that the system is available in winters at a given time instant t, given that it already entered state “i” at time t. BR_{0}(t) be the probability that the repairman is busy for repair in winters at a time instant t, given that it already entered state “i” at time t. BM_{0}(t) be the probability that the repairman is busy for maintenance in summers at a time instant t, given that it already entered state “i” at time t. VR_{0}(t) be the expected number of repairs in winters at (0, t). VM_{0}(t) be the expected number of maintenances in summers at (0, t).
As in the case of MTSF, we have derived these measures of system effectiveness (A_{0}, BR_{0}, BM_{0}, VR_{0}, VM_{0}) using the same probabilistic arguments as with MTSF, except that here the failed state is not considered the absorbing state.
The costs defined are as follows:
C_{0}=Revenue per unit up time.
C_{1}=Cost per unit up time for which the repairman is busy for repair.
C_{2}=Cost per unit up time for which the repairman is busy for maintenance in winters.
C_{3}=Cost per repair.
C_{4}=Cost per maintenance.
N_{1} = p_{01}(µ_{1} + µ_{5}p_{15})
D_{1} = µ_{1} + µ_{4}(p_{14} + p_{15}p_{54}^{(8)}) + Kp_{15} + µ_{6}(p_{16} + p_{15}p_{56}^{(10)}) + µ_{7}(p_{17} + p_{15}p_{57}^{(9)}) ... (1)
N_{2} = W_{4}p_{01}(p_{14} + p_{15}p_{54}^{(8)}) + W_{5}p_{01}p_{15} + W_{6}p_{01}(p_{16} + p_{15}p_{56}^{(10)}) + W_{7}p_{01}(p_{17} + p_{15}p_{57}^{(9)})
D_{1} is defined in equation 1.
N_{4} = p_{01}(1 + p_{15}p_{54}^{(8)} + p_{15}p_{57}^{(9)} + p_{15}p_{56}^{(10)})
D_{1} is already defined in equation 1.
N_{3} = W_{2}p_{01}p_{12}
D_{2} is defined in equation 2.
N_{5} = p_{01}p_{12}
D_{2} is defined in equation 2.
Here W_{i }is the probability that the system is under repair at time “t” before transiting any other state.
For graphical representation we assume the particular case of exponential distribution, likewise:
g(t) = θe^{θ(t)}, g_{1}(t) = θ_{1}e ^{θ₁(t)}, g_{2}(t) = θ_{2}e ^{θ₂(t)}, g_{3}(t) = θ_{3}e^{ θ₃(t)}, g_{4}(t) = θ_{4}e ^{−θ₄(t)}
for numerical analysis various cutoff points are evaluated from the information gathered at the Verka milk plant
λ₂=.00002898, λ₃=.0000246, λ₄=.000292675,
• Mean time to system failure = 7046.117188 hrs
• Availability in winters = 0.471440
• Busy period for repair =0.498495
• Busy period for maintenance =0.007057
• Expected number of repairs =0.000152
• Expected number of maintenances =0.000007
• Profit = Rs. 46143.08
The numerical values are obtained from the original data collected from the Verka Milk Plant, Bathinda, Punjab. From the data collected various parameters were calculated which include failure and repair rates of the chamber, unit 1, 2, and 3. From these failure rates values were evaluated of the system effectiveness measures using MATLAB and Code Blocks.
λ₁=0.00024589 
λ₁=0.0005 
λ₁=0.0024589 
λ₃ 
801.214172 
730.630981 
670.514893 
0.002 
641.27887 
570.695557 
510.57959 
0.003 
561.311157 
490.727783 
430.611786 
0.004 
513.330505 
442.747223 
382.631195 
0.005 
481.343414 
410.760132 
350.644135 
0.006 
458.495483 
387.912201 
327.796204 
0.007 
441.359558 
370.776245 
310.660248 
0.008 
428.031616 
357.448303 
297.332306 
0.009 
417.369263 
346.78595 
286.669952 
0.01 
408.645508 
338.062195 
277.946198 
0.011 
λ₁=0.00024589 
λ₁=0.0005 
λ₁=0.0024589 
λ₃ 
2055.256592 
1865.224609 
829.78479 
0.002 
1833.494507 
1643.462524 
608.022583 
0.003 
1696.155029 
1506.123047 
470.683136 
0.004 
1602.739014 
1412.707031 
377.26712 
0.005 
1535.079956 
1345.047974 
309.608154 
0.006 
1483.81616 
1293.78418 
258.344299 
0.007 
1443.63208 
1253.600098 
218.16024 
0.008 
1411.285889 
1221.253906 
185.814041 
0.009 
1384.68859 
1194.656616 
159.216736 
0.01 
1362.432251 
1172.400269 
136.960449 
0.011 
Data of profit and failure rate λ₃ can graphically be analyzed as given in
C₁=6700 
C₁=27000 
C₁=47000 
C₀ 
405.397888 
750.289734 
1129.555786 
10000 
203.339981 
525.161255 
904.427429 
20000 
1.28206 
300.032715 
679.29895 
30000 
200.775894 
74.904243 
454.17041 
40000 
402.833771 
150.224167 
229.042007 
50000 
604.891724 
375.352722 
3.913464 
60000 
806.949585 
600.481079 
221.214951 
70000 
1009.007568 
825.609619 
446.343506 
80000 
1211.06543 
1050.738159 
671.471924 
90000 
1413.123413 
1275.866577 
896.600342 
100000 
1615.181274 
1500.995239 
1121.729126 
110000 
Upward trend in
Using line of regression y = a x + b where a is the slope and b is the intercept more values can be evaluated for the various measures i.e., mean time to system failure, profit, availability etc. some of them have been stated below to save space.
For λ₁=0.0024589 in Profit and revenue the y intercept will b= 2404.74635 and the slope a= 0.29463713 from this line of regression we can find more values with different failure rates.
For λ₁=0.0024589 in MTSF the b = 5003.582 and a = 2251640 we can find more values with different failure rates.
Known Parameter 
Varying Parameter 
MTSF


Decreasing λ≥.0011 
Increasing λ<.0011 

λ₂=.00002898, λ₃=.0000246, λ₄=.000292675, 𝜃=.00392166, 𝜃=.0003165, 𝜃₂=.000298, 𝜃₃=.000896, 𝜃₄=.00125363, α=.00023148, β=.00023148, C_{0}= 105000, C_{1}=6700, C_{2}=2550, C_{3}=1500, C_{4}=1000 
λ=.0002458 λ=.0005 λ=.002458

408.645508 338.062195 277.941698

801.214172 730.6300981 670.514893 

Known Parameter 
Varying Parameter 
Profit 


Decreasing 
Increasing 

λ₂=.00002898, λ₃=.0000246, λ₄=.000292675, 𝜃=.00392166, 𝜃=.0003165, 𝜃₂=.000298, 𝜃₃=.000896, 𝜃₄=.00125363, α=.00023148, β=.00023148, C_{0}= 105000, C_{1}=6700, C_{2}=2550, C_{3}=1500, C_{4}=1000 
λ=.0002458 λ=.0005 λ=.002458 
λ<.0011 1362.432251 1172.40026 136.96044 
λ≥ .0011 2055.2565 1865.2246 829.78479


Known Parameter 
Varying Parameter 
Cost 

Positive 
Negative 
Zero 

λ₂=.00002898, λ₃=.0000246, λ₄=.000292675, 𝜃=.00392166, 𝜃=.0003165, 𝜃₂=.000298, 𝜃₃=.000896, 𝜃₄=.00125363, α=.00023148, β=.00023148, C_{0}= 105000, C_{1}=6700, C_{2}=2550, C_{3}=1500, C_{4}=1000 
C_{1}=6700 C_{1}=27000 C_{1}=47000 
C0>35000 C0>45000 C0>6500 
C0<35000 C0<45000 C0<650000 
C0=35000 C0=45000 C0=65000 
The parameters shown in the
Cutoff points in the graphical analysis shows at what point the values will start decreasing/increasing. Cutoff points vary widely and by knowing the cutoff points the engineers can know the level of gains or losses, which can be seen in
In case of mean time to system failure (MTSF) as the λ increases the MTSF decreases
We can see the cutoff values at λ=.0011 and λ<.0011 in
Similar is the case with profit v/s the failure rate where the cutoff values is graphically presented in
Profit v/s revenue depicts the cut off points as when
i. C_{1}=6700 the profit depends on C_{0}=35000,
ii. C_{1}=27000 the profit depends on C_{0}=45000,
iii. C_{1}=47000 the profit depends on C_{0}=650000.
This paper analyses the reliability of a powder system taking into account its seasonal effect. Previously, the reliability analysis of a powder system was carried out without addressing the seasonal effect; The present research will help improve system performance at the Verka Milk Plant, thereby reducing production losses and increasing profit. It will also be helpful in all the other systems with similar working conditions and help understand the seasonal variation in the working of the system. The originality of this study lies in the method of analyzing the system in terms of seasondependent expressions. This gives a better, more accurate view of the whole picture in terms of the system analysis.