Effect of thresholds

The thresholds $N_B^{th}$ and $N_D^{th}$ have very different effects.

In this section, we study the effect of threshold settings on performance. We will see that increasing $N_B^{th}$ helps alleviate donor pain given nonzero switching time, without appreciably diminishing beneficiary gain. Thus, the optimal value of $N_B^{th}$ tends to be well above 1. By contrast, increasing $N_D^{th}$ increases beneficiary gain substantially (by increasing their stability region), but also increases donor pain. Overall, the impact of changes in $N_D^{th}$ is more dramatic than the impact of changes in $N_B^{th}$.

Figure 6.8 shows the mean response time for beneficiary jobs (rows 1 and 3) and the mean response time for donor jobs (rows 2 and 4) as a function of $\rho _B$ for different threshold values. In the top half of the figure we study the effect of changing $N_B^{th}$ from $1$ to $10$ as we hold $N_D^{th}$ fixed at $1$. In the bottom half of the figure we study the effect of changing $N_D^{th}$ from $1$ to $10$ as we hold $N_B^{th}$ fixed at $1$. Throughout, $X_B$ and $X_D$ are exponential with mean 1, $K_{sw}$ and $K_{ba}$ are exponential with mean 0 or 1, and we fix $\rho_D=0.5$.

Figure 6.8: The mean response time for beneficiary jobs and donor jobs as a function of $\rho _B$. Graphs show the case of (i) $N_B^{th}=N_D^{th}=1$, (ii) $N_B^{th}=10$ and $N_D^{th}=1$, and (iii) $N_B^{th}=1$ and $N_D^{th}=10$. $X_B$ and $X_D$ are exponential with mean 1. Switching times are exponential with mean 0 or 1 as labeled, where $K\equiv K_{sw}=K_{ba}$. $\rho_D=0.5$.

Effect of $N_B^{th}$

Beneficiary

response time

Donor

response time

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

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

Effect of $N_D^{th}$

Beneficiary

response time

Donor

response time

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

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

As $N_B^{th}$ is increased from $1$ to $10$, Figure 6.8 shows only slightly higher response times for the beneficiary jobs. Recall that increasing $N_B^{th}$ does not change the beneficiary stability region, although the beneficiary queue is helped less frequently. In fact, under longer switching times, the effect of raising $N_B^{th}$ on beneficiary mean response time is even more negligible, since the decreased frequency of helping beneficiary jobs is counteracted by the positive benefit of wasting less time on switching. We also see that increasing $N_B^{th}$ creates less penalty for the donor, as the donor does not have to visit the beneficiary queue as frequently. Observe that when the switching times are nonzero, the donor mean response time is always bounded above by the mean response time for a corresponding M/GI/1 queue with setup time $K_{ba}$, and this bound is tight for all $N_B^{th}$ values as $\rho _B$ reaches its maximum, since the beneficiary queue always exceeds $N_B^{th}$ in this case. We conclude that $N_B^{th}$ has somewhat small impact; however higher values of $N_B^{th}$ are more desirable for the system as a whole under longer switching time.

By contrast increasing $N_D^{th}$ from $1$ to $10$ has dramatic effects. In general (assuming non-zero switching time) increasing $N_D^{th}$ can drastically improve beneficiary response time. This result is not obvious, since increasing $N_D^{th}$ allows the donor to spend more time at the beneficiary queue before leaving, but also means that when the donor leaves the beneficiary queue, the donor will be absent for a longer time (since more time is needed to empty the donor queue). Another positive effect of increasing $N_D^{th}$ is less switching overall. In the end, it is the enlargement of the stability region due to higher $N_D^{th}$ which substantially improves the beneficiary response time when the switching times are large and beneficiary load is high. When switching times are very short, increasing $N_D^{th}$ only slightly worsens the mean response time for beneficiary jobs, as beneficiary jobs experience longer intervals between help. In all cases evaluated, increasing $N_D^{th}$ results in much higher mean response times for donor jobs, since, for $N_D^{th}>1$, the donor job arriving at an empty queue must wait for another $N_D^{th}-1$ jobs to arrive before being served. We conclude that increasing $N_D^{th}$ can have large impact, positive for the beneficiary jobs, but negative for the donor jobs. Thus setting $N_D^{th}$ is much trickier than $N_B^{th}$.

Finally we seek to determine good values for the thresholds, $N_B^{th}$ and $N_D^{th}$, as a function of the system parameters. Above, we have already observed some characteristics of $N_B^{th}$. (i) Increasing $N_B^{th}$ leads to lower gain for the beneficiary jobs and lower pain for the donor jobs. (ii) Perhaps less obvious, the relative drop in gain for the beneficiary jobs is slight compared to the drop in pain for the donor jobs. This points towards choosing a higher value of $N_B^{th}$. Thus, if the switching time is zero, the optimal $N_B^{th}$ is 1 (or 0), since there is never any pain for the donor jobs. Figure 6.9 (a) and (b) show optimal values of $N_B^{th}$ for minimizing overall mean response time (over all jobs) as a function of $\rho _B$ and $\rho_D$ under various switching times when $N_D^{th}=1$. The numbers on the contour curves show the optimal $N_B^{th}$ at each load. For clarity we only show lines up to $N_B^{th} = 14$. The following additional characteristics of $N_B^{th}$ are implied by the figure: (iii) the optimal $N_B^{th}$ is an increasing function of $\rho_D$ and a decreasing function of $\rho _B$; (iv) increasing the switching time increases the optimal $N_B^{th}$.

Figure 6.9: Optimal values of $N_B^{th}$ and $N_D^{th}$ with respect to overall mean response time, where $X_B$ and $X_D$ have an exponential distribution with mean 1.

Selecting optimal $N_B^{th}$ ($N_D^{th}=1$)

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

(b) $\mbox{{\bf\sf E}}\left[ K \right]=2$

Selecting optimal $N_D^{th}$ ($N_B^{th}=1$)

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

(d) $\mbox{{\bf\sf E}}\left[ K \right]2$

Figure 6.9 (c) and (d) show optimal values of $N_D^{th}$ for minimizing overall mean response time when $N_B^{th}=1$. First observe that (i) under low $\rho_D$ or low $\rho _B$ the optimal $N_D^{th}$ is 1. When $\rho_D$ is low and $N_D^{th}>1$, the pain for donor jobs is so huge that the optimal $N_D^{th}$ is always 1. When $\rho _B$ is low, the beneficiary gains little from increasing $N_D^{th}$, while the donor can have nonnegligible pain, which increases with $N_D^{th}$; hence the optimal $N_D^{th}$ is always 1. The following characteristics of $N_D^{th}$ are also implied by the figure: (ii) the optimal $N_D^{th}$ is not a monotonic function of $\rho_D$, but is an increasing function of $\rho _B$; (iii) increasing the switching time increases the optimal $N_D^{th}$. Note that although the range of the optimal values of $N_D^{th}$ is smaller than $N_B^{th}$ in Figure 6.9, Figure 6.8 tells us that the performance effect of changing $N_D^{th}$ on the mean response time of both beneficiary jobs and donor jobs is more significant than changing $N_B^{th}$.