15.8 Exercises

1. Derive Eqs. (15.2.4) and (15.2.5). (Hint: where is the second moment of a random variable and its mean.)

2. Consider the example of Section 15.3.2.

   • Use the open multiclass model of Chapter 13 to solve the unconstrained model.

   • Compute the values of line 1 of Table 15.2.

3. Consider the example of Section 15.4.1. Assume that consumer requests have priority at the CPU over corporate requests. Compute the resulting response time.

4. An iterative approach for modeling priorities was presented by Lazowska et al. [23]. In their approach they suggest that one model with P shadow CPUs be built. The service demands have to be inflated properly in the same way as discussed in the algorithm given in Figure 15.7. However, this requires knowing the values of the utilizations for the various classes. Since the utilization U_cpu,r is equal to X_0,r x D_cpu,r, the algorithm starts by assuming an initial value of zero for all throughputs. MVA can now be used to solve the QN and obtain new values of the throughputs, which are used to compute the utilizations and therefore new values of the service demands for the shadow CPUs. This process continues until the throughput values converge within a given tolerance. Write a program that implements this algorithm and use it to solve the example given in Section 15.4.2.