Consider A Coffee Vending Machine That Breaks Down Regularly

Consider A Coffee Vending Machine That Breaks Down Accord

Consider a coffee vending machine that breaks down according to a Poisson process with an average of once a year. When it fails, it must complete a repair procedure consisting of four phases (Phases 1, 2, 3, and 4 in that order). The time required to complete each phase follows an exponential distribution with rates: μ₁ = 365 (one day), μ₂ = 52 (one week), μ₃ = 730 (half a day), and μ₄ = 26 (two weeks).

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Paper For Above instruction

The reliability analysis of systems such as coffee vending machines is critical in determining operational efficiencies and maintenance scheduling. This paper models the breakdown process of a coffee vending machine as a continuous-time Markov chain (CTMC), analyzes its equilibrium behavior, and computes the proportions of time the machine spends in operational and failed states. The problem involves the stochastic nature of machine failures and repairs, utilizing exponential distributions for phase durations and Poisson processes for failure events.

Modeling the Breakdown and Repair Process as a Continuous-Time Markov Chain

The system states are defined based on whether the machine is operational or undergoing repairs in one of the four phases. Specifically, states are:

• State 0: The machine is functioning normally.
• States 1, 2, 3, 4: The machine is in repair phases 1 through 4 respectively.

Failures occur according to a Poisson process with a mean rate of once per year, or λ = 1/365 per day. The repair phases are sequential, with durations following exponential distributions specified by rates μ₁ = 365, μ₂ = 52, μ₃ = 730, and μ₄ = 26. These rates correspond to average durations of one day, one week, half a day, and two weeks, respectively.

Part A: Graphical Representation of the Markov Chain

The state transition diagram exhibits a cycle starting from the operational state (0). When the machine fails, it transitions to state 1 (phase 1) at rate λ. The process then moves sequentially through repair phases: 1 → 2 → 3 → 4 → 0, with each transition governed by the respective exponential rates. The diagram is as follows:

• From state 0 to state 1 at rate λ = 1/365.
• From state 1 to state 2 at rate μ₁ = 365.
• From state 2 to state 3 at rate μ₂ = 52.
• From state 3 to state 4 at rate μ₃ = 730.
• From state 4 back to state 0 at rate μ₄ = 26.

Transitions occur only in these directions, with no other transitions such as regressions or skipping phases, ensuring the Markov chain is a birth-death process with a cyclic repair sequence.

Part B: Equilibrium Equations and Steady-State Probabilities

Let p₀, p₁, p₂, p₃, p₄ denote the stationary probabilities of states 0 through 4. The balance equations are derived via the global balance method, equating flow into and out of each state.

1. State 0 (Operational): The inflow occurs when the machine completes repairs from state 4, and the outflow occurs due to failure:

• Inflow: p₄ × μ₄
• Outflow: p₀ × λ

Balance equation: p₄ μ₄ = p₀ λ

• λ p₀ = μ₁ p₁

• μ₁ p₁ = μ₂ p₂

• μ₂ p₂ = μ₃ p₃

• μ₃ p₃ = μ₄ p₄

From these, we can express all p₁, p₂, p₃, p₄ in terms of p₀:

p₁ = (λ / μ₁) p₀

p₂ = (μ₁ / μ₂) p₁ = (λ / μ₂) p₀

p₃ = (μ₂ / μ₃) p₂ = (λ / μ₃) p₀

p₄ = (μ₃ / μ₄) p₃ = (λ / μ₄) p₀

Using the normalization condition p₀ + p₁ + p₂ + p₃ + p₄ = 1, we find:

p₀ + (λ / μ₁) p₀ + (λ / μ₂) p₀ + (λ / μ₃) p₀ + (λ / μ₄) p₀ = 1

p₀ [1 + (λ / μ₁) + (λ / μ₂) + (λ / μ₃) + (λ / μ₄)] = 1

Thus, the steady-state probability of being operational:

p₀ = 1 / [1 + (λ / μ₁) + (λ / μ₂) + (λ / μ₃) + (λ / μ₄)]

Part C: Proportion of Time the Machine is Shut Down

The machine is considered shut down during all repair phases (states 1-4). Therefore, the proportion of time spent in failure states is:

P_failure = p₁ + p₂ + p₃ + p₄ = p₀ [ (λ / μ₁) + (λ / μ₂) + (λ / μ₃) + (λ / μ₄) ]

Substituting the values: λ = 1/365, μ₁=365, μ₂=52, μ₃=730, μ₄=26, we compute the sum inside brackets to evaluate P_failure accordingly.

Part D: Proportion of Time the Machine is Functional

The solution is straightforward:

P_operational = p₀ = 1 / [1 + (λ / μ₁) + (λ / μ₂) + (λ / μ₃) + (λ / μ₄)]

This represents the long-run proportion of time the machine is operational, reflecting the system's reliability over time.

Conclusion

This stochastic modeling approach enables detailed insights into the performance of maintenance-dependent systems such as vending machines. The use of Markov chain analysis provides a quantitative understanding of operational availability and failure durations, essential for optimizing maintenance schedules and improving system reliability. By calculating the equilibrium probabilities, operators can better forecast downtime and plan for resource allocations to mitigate operational disruptions.

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