Requests for a resource arrive at rate ¦Ë, each request specifying a future time interval, called a reservation interval, to be booked for its use of the resource. The advance notices (delays before reservation intervals are to begin) are independent and drawn from a distribution A ( z ). The durations of reservation intervals are sampled from the distribution B ( z ) and are independent of each other and the advance notices. We let A and B denote random variables with the distributions A ( z ) and B ( z ) (the functional notation will always allow one to distinguish between our two uses of the symbols A and B ). The following greedy reservation policy was analyzed in [3]: A request is immediately accepted (booked) if and only if the resource will be available throughout its reservation interval, i.e., the resource has not already been reserved for a time period overlapping the requested reservation interval. In [3], the authors compute an efficiency measure, called the reservation probability, which is the fraction of time the resource is in use. This paper studies the reservation probability for a more general greedy policy of threshold type that is defined by two parameters s and ¦Ó. If a request has an advance notice less than s or a duration exceeding ¦Ó, then the threshold policy makes an attempt to book it under the greedy rule; otherwise, it is rejected even if it could have been accommodated. Our main result is an expression for the asymptotic reservation probability as s ¡ú ¡Þ and the advance-notice distribution becomes progressively more spread out. The above result relates asymptotics of reservation policies to asymptotics of interval packing policies, a connection first studied in [3]. In the interval packing problem [1], intervals arrive randomly in R + 2 according to a Poisson process in the two dimensions representing arrival times t and the left endpoints of the arriving intervals. Interval lengths are i.i.d., and since we will map them to reservation intervals, we let their distribution also be denoted by B ( z ). The intensity is 1, i.e., an average of one interval arrives per unit time per unit distance. For a given x > 0, an arriving interval is packed (or accepted) in the 'containing' interval [0, x ] under the greedy algorithm if and only if it is a subinterval of [0, x ] and it does not overlap an interval already accepted. The problem is to find, or at least estimate, the function K ( t, x ), which is the expected total length of the intervals accepted by the greedy policy during [0, t ], assuming that none has yet been accepted by time 0 ([0, x ] is initially empty). Estimates of K ( t, x ) were obtained in [3] from its Laplace transform K ( t, u ); these results are special cases of the corresponding results for the threshold packing policy with parameters s, ¦Ó The threshold packing policy extends greedy interval packing much as we extended the greedy reservation policy: An interval is processed by the greedy packing algorithm if its length is at least ¦Ó or if it arrives no sooner than s ; otherwise, it is rejected. The next section exhibits the Laplace transform of H ¦Ó ( s, t, x ), the expected total length of the intervals accepted during [0, t ], t ¡Ý s, by the threshold packing policy with parameters s, ¦Ó. Note that threshold packing reduces to simple greedy packing if ¦Ó = 0 or if s = 0. The formulas in the next section will verify that K ( t,x ) = H 0 ( t,t,x ). As noted in [3], there are many potential applications covered by models like ours. However, relatively new applications in existing and proposed communication systems, e.g., teleconferencing and video-on-demand systems, have given a fresh impetus to research on reservation systems. Previous work in the communications field is quite recent and focuses more on engineering problems than mathematical foundations; past research has dealt with the implementation issues of incorporating distributed advance-notice reservation protocols in current networks, and with the algorithmic issues concerned with well utilized resources in reservation systems (see [3, 4, 5] for many references). For the analysis of mathematical models different from our own, see the work of Virtamo [5] and Greenberg, Srikant, and Whitt [4].