1. Series System

System Reliability:

R_sys = R₁ × R₂ × R₃ × ... × Rₙ

General Form:

R_sys = ∏(i=1 to n) R_i

If all components identical:

R_sys = Rⁿ

Example: If R₁ = R₂ = R₃ = 0.95
R_sys = 0.95³ = 0.857 (85.7%)

2. Parallel System

System Reliability:

R_sys = 1 - (1-R₁)(1-R₂)(1-R₃)...(1-Rₙ)

General Form:

R_sys = 1 - ∏(i=1 to n)(1-R_i)

If all components identical:

R_sys = 1 - (1-R)ⁿ

Example: If R₁ = R₂ = R₃ = 0.90
R_sys = 1 - (0.1)³ = 0.999 (99.9%)

3. Parallel System (2-out-of-3 Required)

System Reliability (2-out-of-3):

R_sys = 3R²(1-R) + R³

Simplified Form:

R_sys = 3R² - 2R³

General k-out-of-n Form:

R_sys = Σ(i=k to n) C(n,i)Rⁱ(1-R)^n-i

Example: If R₁ = R₂ = R₃ = 0.90
R_sys = 3(0.9)² - 2(0.9)³ = 0.972 (97.2%)

Note: Less reliable than 1-out-of-3 (99.9%) but more reliable than 3-out-of-3 (72.9%)

4. Complex System (Series-Parallel Combination)

System Configuration:

A → (B || C) → D

Step 1: Parallel Subsystem (B and C)

R_BC = 1 - (1-R_B)(1-R_C)

R_BC = 1 - (0.1)(0.1) = 0.99

Step 2: Overall System (Series)

R_sys = R_A × R_BC × R_D

R_sys = 0.95 × 0.99 × 0.95

Solution:
R_sys = 0.894 (89.4%)

Reliability Block Diagrams

1. Series System

2. Parallel System

3. Parallel System (2-out-of-3 Required)

4. Complex System (Series-Parallel Combination)