5 Final Remarks 4.2 Advanced-notice Limit Law Theorem 17 Suppose That for I = 1; 2; : : :, We Have a Distribution a I with Compact Support, and Suppose That a I Has Atness Bound

Table 1: Reservation probabilities for A uniform on 0; a] and unit intervals. The ^ P(a;) are values from simulations. Algorithmics is yet another obvious avenue of research. The greedy rule is a theme of this paper, but in general it is unlikely to be an optimal policy in the sense of maximizing the reservation probability. As a simple example, consider unit reservation intervals and advance notices that are uniform over some large interval. An intriguing question is: When, if ever, is it better to reject a request even though the interval it requests is available? Acknowledgments The authors are pleased to acknowledge many helpful discussions with Proof: It suuces to prove K(t; x) K(t; y) when 0 x y x + 1. Applying the left half of Lemma 18 with P = P x;y shows that we have the desired inequality sample by sample. Remark: As in the preceding section, one can consider a fuzzy boundary at 0 instead of x, and fuzzy boundaries at both 0 and x. By using the same combinatorial arguments as in Lemma 18 and adapting the notation in the obvious way, one can easily obtain the respective There are several avenues for further research in reservation systems. Expanding the class of models leading to exact results, or even good estimates, is one such avenue. However, even incremental extensions to the models of Sections 2 can lead to diicult problems. We give just two examples where this seems to be the case. First, suppose we extend the slotted system to include reservation intervals covering 2 slots as well as 1. The analysis must then account for the fact that patterns of reservations in one period of time can eeect the patterns in much later periods. One way to avoid this dependence is to impose anàlignment' on the length-2 intervals; e.g., suppose requests are such that length-2 reservation intervals begin only at even integer times. Then the reservation patterns in i; i+2], i even, are independent of the reservation patterns elsewhere. Assuming k = 2 copies of the resource is another important incremental extension. With multiple copies, the protocol is as given in Section 2: a request is denied if and only if, at some time, its reservation interval overlaps two currently reserved intervals. We have seen that this extension is easy to handle in the slotted model of Section 2, but it is not …