Threshold policies for single-resource reservation systems. - Coffman, Jr., E. G. (Sigmetrics :performance evaluation review.)
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) = H0(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].