An analytical model is built to answer the five questions posed in the previous section. The machine failure rate is denoted by l and indicates the rate at which machines move from the operational state to the failed state. Thus, each machine stays in the operational state 1/l time units, on average. This value is called the Mean Time to Failure (MTTF). It is assumed here that machines fail independently from one another and that all machines fail at the same average rate. (Note: If machines are observed to fail at different rates, a multiclass model can be constructed. In this case, machines in class r fail at rate l The system of Fig. 7.1 can be modeled by the Markov Chain (see Chapter 10) of Fig. 7.2, where state k represents the number of failed machines. A transition from state k to state k + 1 happens when a machine fails. A transition from state k to state k 1 occurs when a machine is repaired. At state k, there are M k machines in operation and each has a failure rate of l. Thus, the aggregate failure rate, l
## Figure 7.2. Markov chain model for the data center.
The aggregate repair rate, m
Solving this Markov Chain means finding the steady state probabilities, p
where p
where p
The average aggregate rate at which machines fail, , which is also equal to the average aggregate rate at which machines are repaired, can be computed from the probabilities p
The Interactive Response Time Law (see Chapter 3) can be used to compute the MTTR (i.e., the average time spent by a machine from when it fails until it is back in operation). The analogy between Fig. 3.4 and Fig. 7.1 is as follows: client workstations correspond to machines in operation, the average think time Z corresponds to the MTTF (1/l), the average response time R corresponds to the MTTR, and the system throughput X
The average number of failed machines, N
Similarly, the average number of operational machines, N
which is Little's Law applied to the upper portion of Fig. 7.1. The Markov Chain model described in this section is implemented in the MS Excel work-book |

Performance by Design: Computer Capacity Planning By Example

ISBN: 0130906735

EAN: 2147483647

EAN: 2147483647

Year: 2003

Pages: 166

Pages: 166

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