Essential System Performance and Reliability Formulas

1. Throughput

X = C / T

• C = Completed jobs
• T = Observation time

2. Arrival Rate

λ = A / T

• A = Arrivals

3. Utilization

U = B / T or U = XS or Uk = XkSk or Uk = XDk

4. Service Time

S = B / C

5. Little's Law

N = XR or R = N / X

6. Interactive Response Time Law

R = N / X − Z

7. Forced Flow Law

Xk = XVk

8. Service Demand

Dk = VkSk

9. Utilization using Demand

Uk = XDk

10. Bottleneck Law

Maximum Throughput: Xmax = 1 / Dmax

11. Minimum Response Time

Rmin = NDmax − Z or Rmin = max(Dtot, NDmax − Z)

12. HDD Formulas

• I/O Time: TIO = Tseek + Trot + TOH + TT
• Rotational Latency: Trot = 30000 / RPM (ms)
• Full Rotation: Trev = 60000 / RPM (ms)
• Transfer Time: TT = Data / TransferRate
• Locality: T = TT + (1 − L)(Tseek + Trot)
• Multiple Contiguous Sets: T = TT + #Sets(Tseek + Trot)

13. RAID Capacity

• RAID 0: Capacity = ND
• RAID 1: Capacity = (N/2)D
• RAID 5: Capacity = (N − 1)D
• RAID 6: Capacity = (N − 2)D

14. RAID Read Throughput

• Sequential: NT
• Random: ≈ NT

15. RAID Reliability

• Failure Rate: λ = 1 / MTTF
• RAID 0: MTTDL = MTTF / N
• RAID 1: MTTDL = MTTF² / (2MTTR)
• RAID 10 (1+0): MTTDL = MTTF² / (NMTTR)
• RAID 0+1: MTTDL = 2MTTF² / (N²MTTR)
• RAID 5: MTTDL = MTTF² / [N(N−1)MTTR]
• RAID 6: MTTDL = MTTF³ / [N(N−1)(N−2)MTTR²]

16. Availability

A = MTTF / (MTTF + MTTR)

17. Reliability

R(t) = e^(−t/MTTF)

Failure Probability: F(t) = 1 − R(t)

18. Series Availability

A = A1 × A2 × A3 × ...

19. Parallel Availability

A = 1 − Π(1 − Ai)

20. Replica Probability

P = 1 − (1 − p)^n

Find number of replicas: n = ln(1 − P) / ln(1 − p)

21. PUE

PUE = Facility Power / IT Power

22. Mean Value Analysis (MVA)

• Initialize: Qk(0) = 0
• For n = 1 ... N:
  • Response Time: Rk = Dk(1 + Qk)
  • Throughput: X = n / (Z + ΣRk)
  • Queue Length: Qk = XRk
• Repeat until n = N.