Threshold policies for single-resource reservation systems

Requests for a resource arrive at rate ¦Ë, eachrequest specifying a future time interval, called a <i>reservationinterval,</i> to be booked for its use of the resource. The<i>advance notices</i> (delays before reservation intervals are tobegin) are independent and drawn from a distribution<i>A</i>(<i>z</i>). The durations of reservation intervals aresampled from the distribution <i>B</i>(<i>z</i>) and areindependent of each other and the advance notices. We let A and Bdenote random variables with the distributions <i>A</i>(<i>z</i>)and <i>B</i>(<i>z</i>) (the functional notation will always allowone to distinguish between our two uses of the symbols <i>A</i> and<i>B</i>). The following greedy reservation policy was analyzed in [3]: Arequest is immediately accepted (booked) if and only if theresource will be available throughout its reservation interval,i.e., the resource has not already been reserved for a time periodoverlapping the requested reservation interval. In [3], the authorscompute an efficiency measure, called the <i>reservationprobability,</i> which is the fraction of time the resource is inuse. This paper studies the reservation probability for a moregeneral greedy policy of threshold type that is defined by twoparameters <i>s</i> and ¦Ó. If a request has anadvance notice less than <i>s</i> or a duration exceeding¦Ó, then the threshold policy makes an attempt tobook it under the greedy rule; otherwise, it is rejected even if itcould have been accommodated. Our main result is an expression forthe asymptotic reservation probability as <i>s</i> ¡ú¡Þ and the advance-notice distribution becomesprogressively more spread out. The above result relates asymptotics of reservation policies toasymptotics of interval packing policies, a connection firststudied in [3]. In the interval packing problem [1], intervalsarrive randomly in <b>R</b>₊² according to aPoisson process in the two dimensions representing arrival times<i>t</i> and the left endpoints of the arriving intervals. Intervallengths are i.i.d., and since we will map them to reservationintervals, we let their distribution also be denoted by<i>B</i>(<i>z</i>). The intensity is 1, i.e., an average of oneinterval arrives per unit time per unit distance. For a given<i>x</i> > 0, an arriving interval is packed (or accepted) inthe 'containing' interval [0, <i>x</i>] under the greedy algorithmif and only if it is a subinterval of [0, <i>x</i>] and it does notoverlap an interval already accepted. The problem is to find, or atleast estimate, the function <i>K</i>(<i>t, x</i>), which is theexpected total length of the intervals accepted by the greedypolicy during [0, <i>t</i>], assuming that none has yet beenaccepted by time 0 ([0, <i>x</i>] is initially empty). Estimates of <i>K</i>(<i>t, x</i>) were obtained in [3] from itsLaplace transform <i>K</i>(<i>t, u</i>); these results are specialcases of the corresponding results for the <i>threshold</i> packingpolicy with parameters <i>s,</i> ¦Ó The thresholdpacking policy extends greedy interval packing much as we extendedthe greedy reservation policy: An interval is processed by thegreedy packing algorithm if its length is at least ¦Óor if it arrives no sooner than <i>s</i>; otherwise, it isrejected. The next section exhibits the Laplace transform of<i>H</i>_¦Ó(<i>s, t, x</i>), the expectedtotal length of the intervals accepted during [0,<i>t</i>],<i>t</i> ¡Ý <i>s,</i> by the threshold packing policywith parameters <i>s,</i> ¦Ó. Note that thresholdpacking reduces to simple greedy packing if ¦Ó = 0 orif <i>s</i> = 0. The formulas in the next section will verify that<i>K</i>(<i>t,x</i>) = <i>H</i>₀(<i>t,t,x</i>). As noted in [3], there are many potential applications coveredby models like ours. However, relatively new applications inexisting and proposed communication systems, e.g., teleconferencingand video-on-demand systems, have given a fresh impetus to researchon reservation systems. Previous work in the communications fieldis quite recent and focuses more on engineering problems thanmathematical foundations; past research has dealt with theimplementation issues of incorporating distributed advance-noticereservation protocols in current networks, and with the algorithmicissues concerned with well utilized resources in reservationsystems (see [3, 4, 5] for many references). For the analysis ofmathematical models different from our own, see the work of Virtamo[5] and Greenberg, Srikant, and Whitt [4].