Computing various performance measures

So far, our focus has been on computing the stationary probabilities in the RFB/GFB process. In this section, we discuss how the stationary probabilities can be used to compute other performance measures. In Section 3.8.1, we consider the distribution of the number of jobs in the system and its moments. In Section 3.8.2, we consider the distribution of response time and its moments. Since the computation of these performance measures depends on particular multiserver systems and details of modeling, we will illustrate the approach via an example in the case of an M/M/2 queue with two preemptive priority classes (recall Figure 3.3), and briefly discuss how this can be applied to other multiserver systems.

Figure 3.28: Markov chains whose stationary probabilities are computed via DR in the analysis of an M/M/2 queue with two preemptive priority classes.

(a) high priority jobs

(b) low priority jobs

Figure 3.28 shows the Markov chains whose stationary probabilities are computed via DR in the analysis of an M/M/2 queue with two preemptive priority classes. Figure 3.28(a) shows the background process, where state (level) $\ell$ corresponds to the state with $\ell$ high priority jobs for $\ell\geq 0$. Let $\pi_\ell^{(H)}$ be the stationary probability in state (level) $\ell$ of the background process. Observe that $\pi_\ell^{(H)}$ is the probability that there are $\ell$ high priority jobs in the system for $\ell\geq 0$. Figure 3.28(b) shows the 1D Markov chain reduced from the 2D Markov chain (FB process), where level $\ell$ consists of the states with $\ell$ low priority jobs for $\ell\geq 0$. Let $\Vec{\pi_\ell}^{(L)}$ be the stationary probability vector in level $\ell$ of the 1D Markov chain shown in Figure 3.28(b). Observe that $\Vec{\pi_\ell}^{(L)}\Vec{1}$ is the probability that there are $\ell$ low priority jobs in the system for $\ell\geq 0$.