The system of Fig. 7.1 can be modeled equivalently by a two-device QN. One device is a delay server representing the machines in operation. The other device is a load-dependent server representing the repair people (See Fig. 7.5). The delay server is used to represent machines in operation. Once a machine is fixed, it goes into operation immediately without queuing. Moreover, the time a machine stays in operation does not depend on the behavior of other machines, but only upon its mean time to failure (i.e., 1/l).
Figure 7.5. QN model of the data center.
Now consider the load dependent repair server. Given that there are N repair people, the collective rate at which machines are repaired depends on two things: 1) the number of failed machines, k, (including those waiting to be repaired and those being repaired) and 2) the number of repair people, N. If k km. However, if k > N, all repair people are busy and the collective repair rate is Nm (i.e., its maximum repair rate value). Thus, the service rate m(k) of the load-dependent device is given by
To solve this model, the MVA method with load-dependent devices can be used. Load-dependent MVA is presented in Chapter 14. In particular, the single-class algorithm of Fig. 14.5 can be used.
Before solving the QN model, the service rate multipliers a(k) (k = 1, ···, M) are required. These multipliers are defined in Chapter 14 as a(k) = m(k)/m(1). Thus, from Eq. (7.5.14), it follows that
The solution of this MVA model with M customers (i.e., M machines circulating from being operational to being repaired) yields the average throughput and the average residence time at each server. The average residence time at the load-dependent (LD) device, R'LD, is the Mean Time to Repair (MTTR).
Applying Little's Law to the LD device (including its queue), the average number of failed machines, Nf, is computed as
Therefore, the average number of machines in operation, No, is just
The solution of the MVA model also provides the probability distribution of the number of customers at the LD device (see Chapter 14). If desired, a multiclass MVA model with LD devices can be constructed and used to model situations in which the set of machines have different failure rates. In this case, the machines exhibiting different failure rates would be grouped into different customer classes in the model.