Models for Performing Internet Reliability and Availability Analysis

There are many mathematical models that can be used for performing various types of reliability and availability analysis concerning the reliability of Internet-related services [12,42-45]. Two of these models are presented below.

6.9.1 Model I

This mathematical model is concerned with evaluating the reliability and availability of an Internet server system and it assumes that the server system can either be in an operating or a failed state. In addition, its (i.e., Internet server system) outage/failure and restoration/repair rates are constant, and all its outages/failures occur independently and the repaired/restored server system is as good as new.

The internet server system state space diagram is shown in Figure 6.4, and the numerals in rectangles represent system states.

The following symbols were used for developing equations for the model:

i is the /'th Internet server system state shown in Figure 6.4 for i = 0 (Internet server system operating normally), i = l (Internet server system failed).

An. is the Internet server system constant outage/failure rate.

/r is the Internet server system constant restoration/repair rate.

P, (t) is the probability that the Internet server system is in state / at time /, for

/ = 0, 1.

Using the Markov method, we write down the following equations for the diagram shown in Figure 6.4 [12].

By solving Equations (6.24)-(6.25), we obtain the following probability equations: where

AVis (?) is the Internet server system availability at time t.

UAis (?) is the Internet server system unavailability at time t.

As time t becomes very large, Equations (6.26) and (6.27) reduce to

For q,( = 0, Equation (6.27) reduces to where

Rjs (?) is the Internet server system reliability at time t.

Thus, the Internet server system mean time to failure is given by [12]

where

MTTFix is the Internet server system mean time to failure.

Example 6.5

Assume that the constant failure and repair rates of an Internet server system are 0.006 failures/hour and 0.05 repairs/hour, respectively. Calculate the Internet server system unavailability for a 6-hour mission.

By substituting the given data values into Equation (6.27), we get

Thus, the Internet server system unavailability for the stated mission time is 0.0305.

6.9.2 Model II

This model is concerned with evaluating the availability of an Internetworking (router) system composed of two independent and identical switches. The model assumes that the switches form a standby-type configuration and that the system fails when both the switches malfunction. Furthermore, the switch failure and restoration (repair) rates are constant. The state space diagram of the system is shown in Figure 6.5, and the numerals in boxes represent system states.

The following symbols were used for developing equations for the model:

i is the ith state shown in Figure 6.5 for: i = 0 (system operating normally [i.e., two switches functional: one operating, other on standby]), i = 1 (one switch operating, the other failed), i = 2 (system failed [both switches failed]).

FIGURE 6.5 System state-space diagram.

p is the probability of failure detection and successful switchover from switch failure.

/,. is the switch constant failure rate.

ps is the switch constant restoration/repair rate.

psl is the constant restoration/repair rate from state 2 to state 0.

Pj (/) is the probability that the Internetworking (router) system is in state / at time f; for i' = 0, 1,2.

Using the Markov method, we write down the equations for the diagram shown in Figure 6.5 [12,46]:

At time t = 0.PQ (0) = (0) = 0, and P2 (0) = 0.

The following steady-state probability solutions are obtained by setting derivatives

2

equal to zero in Equations (6.32)-(6.34) and using the relationship У.

i'=0

where

where

^ is the steady-state probability that the Internetworking (router) system is in state i, for / = 0, 1,2.

The Internetworking (router) system steady-state availability is given by

where

A Vjss is the Internetworking (router) system steady-state availability.

Problems

• 1. What are the main causes of computer system failures?
• 2. Discuss at least five classifications of computer failures.
• 3. What are the sources of computer hardware and software errors?
• 4. Make a comparison between hardware and software reliability.
• 5. What is fault masking?
• 6. Compare the Mills model with the Musa model.
• 7. List at least ten Internet outage categories.
• 8. Describe the pinpoint method.
• 9. Prove Equations (6.26) and (6.27) by using Equations (6.24) and (6.25).
• 10. Prove Equation (6.39) by using Equations (6.32), (6.33), and (6.34).