Benefits of cycle stealing: wide range $\rho _B$

Figure 6.4: The mean response time for beneficiary jobs and donor jobs as a function of $\rho _B$ under cycle stealing and dedicated servers. In all figures $X_B$ and $X_D$ are exponential with mean 1; switching times ( $K\equiv K_{sw}=K_{ba}$) are exponential with mean 0 or 1 as labeled.

Mean response time: low-to-medium load: $\rho_D=.5$

beneficiary

response time

donor

response time

(a) $\mbox{{\bf\sf E}}\left[ K \right]=0$

(b) $\mbox{{\bf\sf E}}\left[ K \right]=1$

Mean response time: medium-to-high load: $\rho_D=.8$

beneficiary

response time

donor

response time

(c) $\mbox{{\bf\sf E}}\left[ K \right]=0$

(d) $\mbox{{\bf\sf E}}\left[ K \right]=1$

Cycle stealing is always a win when $\rho_B \geq 1$, but does not pay when $\rho_B \leq 0.5$.

Figure 6.4 shows the mean response time for beneficiary jobs (rows 1 and 3) and donor jobs (rows 2 and 4) as a function of $\rho _B$, where $\rho_D$ is low-to-medium ($\rho_D=0.5$; top half) and medium-to-high ($\rho_D=0.8$; bottom half). When $\rho_B \geq 1$ (and $\rho_D < 1$), cycle stealing can provide infinite gain to beneficiary jobs over dedicated servers, with comparatively little pain for the donor jobs. This is because the stability region for the beneficiary jobs under cycle stealing is much greater than under dedicated servers. While factors such as increased switching times and increased $\rho_D$ do increase the mean response time of the beneficiary jobs, the gain is still infinite, and these factors are less important. We also see that the mean response time of donor jobs is bounded by the mean response time for an M/GI/1 queue with setup time $K_{ba}$. When $\rho_B \leq 0.5$, there is so little gain to the beneficiary jobs that cycle stealing with non-zero switching overhead does not pay. We therefore primarily focus the rest of the results section on the remaining case: $0.5< \rho_B < 1$.