Under the Dedicated policy, the queues are stable iff and . The stability region becomes wider under SBCS-ID and SBCS-CQ, as is characterized in the next theorem.
Proof:We first consider the stability conditions for SBCS-ID. Let (respectively, ) denote the load at the long server (respectively, short server). The queues are stable iff and .
We first deduce . The PASTA (Poisson arrivals see time averages) principle implies that the fraction of the short jobs that are dispatched to the long server is , assuming . Thus,
Next we deduce . The PASTA principle implies that the fraction of the short jobs that are dispatched to the short server is , assuming . Thus,
The stability conditions for SBCS-CQ follow immediately via the
law of large numbers. The long
jobs (the number of the long jobs in the central queue) are stable
iff , and the short jobs are stable
iff
. width 1ex height 1ex depth 0pt
The restriction on for stability under each of the three task assignment policies is shown in Figure 5.2 as a function of ( is necessary for all the three policies). Observe the advantage of cycle stealing in extending the stability region. When is near zero, can be as high as about (golden ratio) under SBCS-ID and close to under SBCS-CQ.