Multiprocessor Computing System Reliability Analysis ✓ Solved

Multiprocessor computing system reliability analysis A multip

A multiprocessor system is composed of two computing modules: CM1 and CM2. Each of them contains one processor (P1 and P2, respectively), one memory module (M1 and M2), and two disks: a primary disk (D11 and D21, respectively) and a backup disk (D12 and D22, respectively). Initially, the primary disk is accessed by the corresponding computing module, while the backup disk contains the copy of the primary disk’s data, and it is accessed only periodically for updating operations. If the primary disk fails, it is replaced in its function by the backup disk.

In terms of reliability, the disks are identical; they are characterized by the same failure rate or reliability cumulative distribution function (CDF). The computing modules are connected by means of the bus B; moreover, P1 and P2 are energized by the power supply PS: the failure of PS forces P1 and P2 to fail.

M3 is a spare memory replacing M1 or M2 in the case of failure. If both M1 and M2 are operational, M3 is just kept alive or in warm standby to maintain the data stored, but it is not accessed to read or write any data. When M1 or M2, or both, fail, M3 substitutes the failed unit. In order to properly work, the multiprocessor computing system requires that at least one computing module (CM1 or CM2), the PS, and the bus B operate correctly.

A computing module (CM1 and CM2) is operational if the processor (P1 and P2, respectively), one between the local memory (M1 and M2) and the shared memory M3, and one disk (D11 or D21 for CM1 and D12 or D22 for CM2) are not failed. Assuming that all the components have a failure time exponentially distributed and the memory module M3 has different failure rates when it is in warm standby or active, compute the system reliability function and the MTTF. Failure rates in Table 1 are expressed in failures in time (FIT), i.e., the number of faults per billion device hours. Moreover, compute the system availability assuming that the system is reparable and the component repair rates are as in Table 1 (Remark: a repair rate equal to 0 means the component is reliable).

Component Failure rate Repair rate (FIT) (repairs/h):

  • B: 2, 0
  • P1, P2: 0.85 · 10–2, 6000
  • M1, M2: 30, 4.00 · 10–2
  • M3 (active): 30, 4.00 · 10–2
  • M3 (standby): 25, 4.00 · 10–2
  • D: 0.45 · 10–2, 0.45 · 10–2

Table 1: System parameter values

Paper For Above Instructions

The reliability analysis of multiprocessor computing systems is crucial for understanding their performance, availability, and interruptions due to component failures. This paper focuses on the reliability function and mean time to failure (MTTF) of a multiprocessor system made up of two computing modules, CM1 and CM2, each with its components that include processors, memory modules, and disks.

System Configuration and Failure Modes

The multiprocessor system is designed with redundancy, comprising two computing modules, each with primary and backup disks, memory, and processors. If a primary disk fails, the system is designed to switch to a backup disk. The same redundancy applies to memory modules, where an additional memory M3 serves as a backup for M1 and M2.

Reliability Models

Reliability calculations are often modeled using the exponential distribution for the time until failure of components. For each component, we determine the reliability function R(t) as:

R(t) = e-λt,

where λ represents the failure rate. For example, if we consider the failure rate of the bus (B) as 2 FIT, its reliability function over a given time t can be calculated using the above formula, while also considering the repairs that can occur to improve overall system reliability.

Calculating MTTF

The Mean Time to Failure (MTTF) can be computed based on the individual failure rates of each component. The overall system MTTF can be approximated in a combined manner, taking into account the operational conditions of the components.

MTTF for an individual component is calculated as:

MTTF = 1/λ.

Considering components with different failure rates, one can create a combined MTTF for the entire system accounting for all primary components. M1 and M2, each with a failure rate of 30 FIT, will relate to an MTTF as follows:

MTTF (M1 or M2) = 1/30 hours = 0.0333 hours or 120 seconds.

System Availability Analysis

To compute system availability, the formula used is:

Availability = Uptime / (Uptime + Downtime)

Uptime is directly affected by reliability, and Downtime is affected by repair rates. The availability of a component, given its reliability and repair rate, can be calculated, which contributes to the overall system availability. Given that the repair rate for component B is 0, it is considered reliable, whereas other components can be repaired, so their impact on system availability becomes significant.

Conclusion

In conclusion, reliability analysis of multiprocessor systems reveals essential insights into how the various components interact and support prolonged operations through redundancy. Understanding the MTTF and availability ensures that systems are designed and operated efficiently. Adopting exponential models for failure and incorporating repair rates aids in critical infrastructure considerations.

References

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