Distribution and moments of the number of jobs in the system

Deriving the distribution of the number of (high priority or low priority) jobs in the system and its moments is straightforward. Let $N_H$ (respectively, $N_L$) be the number of high priority (respectively, low priority) jobs in the system. Then,

$${\rm Pr}(N_H=n) = \pi_n^{(H)} \quad\mbox{and}\quad {\rm Pr}(N_L=n) = \Vec{\pi_n}^{(L)} \Vec{1}$$

for $n\geq 0$. Also, their $r$-th moments can be computed via

$$\mbox{{\bf\sf E}}\left[ (N_H)^r \right] = \sum_{\ell=0}^\inf... ...ht] = \sum_{\ell=0}^\infty \ell^r \Vec{\pi_\ell}^{(L)} \Vec{1}$$

for $r\geq 1$. As in (3.6), the infinite summation can be reduced to a finite summation, since the QBD processes in Figure 3.28 repeat after level $\hat\ell=2$.

In general, the distribution of the number of jobs in the system that is modeled as an RFB/GFB process can be derived similarly, as long as the number of jobs is well defined for each state. For example, the number of jobs is well defined for each state in all the examples of RFB/GFB processes that we have provided in Section 3.4. Also, the distribution of the number of jobs in the queue (number of jobs that are waiting and not in service) and its moments can be computed similarly. However, note that the error in the higher moments are larger as we will see in Section 3.9, since we match only the first three moments of the ``busy period'' in DR.