Distribution and moments of response time

Let $T_H$ (respectively, $T_L$) be the response time of the high priority jobs (respectively, low priority jobs). The mean response time follows immediately from $\mbox{{\bf\sf E}}\left[ N_H \right]$ and $\mbox{{\bf\sf E}}\left[ N_L \right]$ via Little's law:

$$\mbox{{\bf\sf E}}\left[ T_H \right] = \frac{\mbox{{\bf\sf E}... ...right] = \frac{\mbox{{\bf\sf E}}\left[ N_L \right]}{\lambda_L}$$

where $\lambda_H$ (respectively, $\lambda_L$) is the arrival rate of the high priority jobs (respectively, low priority jobs). However, as we discussed in Section 2.2, the distributional Little's law does not apply to per-class response time in an M/M/2 queue with two priority classes, since (high priority or low priority) jobs do not necessarily leave the system in the order of their arrivals. In fact, the higher moments of response time depend on details of which low priority job is preempted when a high priority job arrives and sees two low priority jobs in service: we assume that the low priority job that started to receive service later is preempted by the high priority job. Below, we illustrate a way to compute the distribution of per-class response time and its moments.