k-out-of-m Network

The k-out-of-m network is another form of redundancy in which at least к units out of a total of m active units must work normally for the successful operation of the system/network. The block diagram of a k-out-of-ш unit system/network is shown in Figure 3.4. Each block in the diagram represents a unit.

The parallel and series networks are special cases of this network for к = 1 and k = m, respectively.

By using the binomial distribution, for identical and independent units, we write down the following expression for reliability of k-out-of-m unit network/system shown in Figure 3.4:

where

R_k/m is the k-out-of-m network reliability.

R is the unit reliability.

For constant failure rate of the identical units, using Equations (3.11) and (3.31), we obtain

where

A is the unit constant failure rate.

Rk/m (0 is •he k-out-of-m network reliability at time t.

By substituting Equation (3.33) into Equation (3.12), we get where

MTTF_k/m is the mean time to failure of the k-out-of-m network/system.

Example 3.8

Assume that a system has five active, independent, and identical units in parallel. For the successful operation of the system, at least four units must operate normally. Calculate the mean time to failure of the system if the unit constant failure rate is 0.0005 failures per hour.

By inserting the given data values into Equation (3.34), we obtain

Thus, the mean time to failure of the system is 900 hours.

Standby System

This is another network or configuration in which only one unit operates and m units are kept in their standby mode. The total system contains (m + 1) units, and as soon as the operating unit fails, the switching mechanism detects the failure and turns on one of the standby units. The system fails when all the standby units fail. The block diagram of a standby system with one functioning and m standby units is shown in Figure 3.5. Each block in the diagram denotes a unit.

Using Figure 3.5 block diagram for independent and identical units, perfect switching mechanism and standby units, and the time-dependent unit failure rate, we write the following expression for the standby system reliability [10]

where

R_ss (/) is the standby system reliability at time t.

A(f) is the unit time-dependent failure rate/hazard rate.

FIGURE 3.5 Block diagram of a standby system containing one operating unit and m standby units.

For unit’s constant failure rate (i.e., A(f) = A),Equation (3.35) yields where

A is the unit constant failure rate.

By inserting Equation (3.36) into Equation (3.12), we obtain

where

MTTF_SS is the standby system mean time to failure.

Example 3.9

Assume that a standby system contains two independent and identical units (i.e., one operating and other on standby). The unit constant failure rate is 0.002 failures per hour.

Calculate the standby system reliability for a 100-hour mission and mean time to failure, assuming that the switching mechanism is perfect and the standby unit remains as good as new in its standby mode.

By inserting the stated data values into Equation (3.36), we get

Similarly, by substituting the given data values into Equation (3.37), we get

Thus, the standby system reliability for a 100-hour mission and mean time to failure are 0.9824 and 1,000 hours, respectively.

Bridge Network

Sometimes units in systems may form a bridge network/configuration, as shown in Figure 3.6. Each block in the figure represents a unit, and all five units are labeled with numerals.

For independently failing units of the bridge network shown in Figure 3.6, reliability is expressed by [11]

where

R_his the reliability of the bridge network.

R, is the reliability of unit i, for i' = 1,2,3, 4, 5.

For identical units, Equation (3.38) simplifies to

where

R is the unit reliability.

For constant unit failure rate using Equation (3.11) and Equation (3.39), we obtain where

R_h (/) is the reliability of the bridge network at time t.

A is the unit constant failure rate.

By inserting Equation (3.40) into Equation (3.12), we obtain where

MTTF_h is the bridge network mean time to failure.

Example 3.10

Assume that a system has five independent and identical units forming a bridge network/configuration and the constant failure rate of each unit is 0.0005 failures per hour.

Calculate the bridge network/configuration reliability for a 250-hour mission and the mean time to failure.

By substituting the given data values into Equation (3.40), we obtain Similarly, by inserting the specified data value into Equation (3.41), we get

Thus, the bridge network/configuration reliability and the mean time to failure are 0.9700 and 1633.33 hours, respectively.

Problems

• 1. Describe the bathtub hazard rate curve and write down the equation that can be used to represent it.
• 2. Write down the general formulas for the following two functions:
  • • Reliability function
  • • Hazard rate function
• 3. Write down three formulas to obtain mean time to failure.
• 4. Assume that a system has five identical and independent subsystems, and the constant failure rate of a subsystem is 0.0002 failures per hour. All five subsystems must operate normally for the system to operate successfully. Calculate the following:
  • • System mean time to failure
  • • System reliability for an eighteen-hour mission
  • • System failure rate
• 5. Assume that a system has three identical, independent, and active units. At least one of these units must operate normally for the system to operate successfully. Calculate the following:
  • • The system reliability for an eight-hour mission if the constant failure rate of a unit is 0.0002 failures per hour.
• 6. What are the special case networks of the k-out-of-w network?
• 7. Assume that a standby system contains three identical and independent units (i.e., one operating, the other two on standby). The unit constant failure rate is
• 0.008 failures per hour. Calculate the standby system reliability for a 200-hour mission and mean time to failure, assuming that the switching mechanism is perfect and the standby units remain as good as new in their standby modes.
• 8. Compare the k-out-of-m network with the standby system.
• 9. Prove Equation (3.34) step-by-step by utilizing Equation (3.33).
• 10. Assume that a system has five identical and independent units forming a bridge network and the unit constant failure rate is 0.0007 failures per hour. Calculate the system reliability for a 100-hour mission and the mean time to failure.

References

• 1. Layman, W.J., Fundamental consideration in preparing a master plan. Electrical World, Vol. 101, 1933, pp. 778-792.
• 2. Smith, A., Service reliability measured by probabilities of outage, Electrical World, Vol. 103, 1934. pp. 371-374.
• 3. Dhillon, B.S., Power System Reliability, Safety, and Management, Ann Arbor Science Publishers, Ann Arbor, Michigan, 1983.
• 4. Dhillon, B.S., Human Reliability: With Human Factors, Pergamon Press, New York, 1986.

• 5. Dhillon, B.S., Mechanical Reliability: Theory, Models, and Applications, American Institute of Aeronautics and Astronautics, Washington, DC, 1988.
• 6. Kapur, K.C., Reliability and Maintainability, in Handbook of Industrial Engineering, edited by G. Salvendy, John Wiley, New York, 1982, pp. 8.5.1-8.5.34.
• 7. Dhillon, B.S., Design Reliability: Fundamental and Applications, CRC Press, Boca Raton, Florida. 1999.
• 8. Dhillon. B.S.. Life distributions, IEEE Transactions on Reliability, Vol. 30, 1981, pp. 457-459.
• 9. Shooman, M.L., Probabilistic Reliability: An Engineering Approach, McGraw-Hill, New York, 1968.
• 10. Sandler, G.H., System Reliability Engineering, Prentice Hall, Englewood Cliffs, New Jersey, 1963.
• 11. Lipp, J.P., Topology of switching elements versus reliability, Transactions on IRE Reliability and Quality Control, Vol. 7, 1957, pp. 21-34.