Kelly's LAN model revisited

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For a given k ≥ 1, subintervals of a given interval [0, X ] arrive at random and are accepted (allocated) so long as they overlap fewer than k subintervals already accepted. Subintervals not accepted are cleared, while accepted subintervals remain allocated for random retention times before they are released and made available to subsequent arrivals. Thus, the system operates as a generalized many-server queue under a loss protocol. We study a discretized version of this model that appears in reference theories for a number of applications; the one of most interest here is linear communication networks, a model originated by Kelly [2]. Other applications include surface adsorption/desorption processes and reservation systems [3, 1]. The interval [0, X ], X an integer, is subdivided by the integers into slots of length 1. An interval is always composed of consecutive slots, and a configuration C of intervals is simply a finite set of intervals in [0, X ]. A configuration C is admissible if every non-integer point in [0, X ] is covered by at most k intervals in C. Denote the set of admissible configurations on the interval [0, X ] by C X . Assume that, for any integer point i, intervals of length l with left endpoint i arrive at rate λ l ; the arrivals of intervals at different points and of different lengths are independent. A newly arrived interval is included in the configuration if the resulting configuration is admissible; otherwise the interval is rejected. It is convenient to assume that the arrival rates λ l vanish for all but a finite number of lengths l, say λ l > 0, 1 ≤ l ≤ L, and λ l = 0 otherwise. The departure of intervals from configurations has a similar description: the flow of "killing" signals for intervals of length l arrive at each integer i at rate µ l . If at the time such a signal arrives, there is at least one interval of length l with its left endpoint at i in the configuration, then one of them leaves. Our primary interest is in steady-state estimates of the vacant space, i.e., the total length of available subintervals kX - ∑l i , where the l i are the lengths of the subintervals currently allocated. We obtain explicit results for k = 1 and for general k with all subinterval lengths equal to 2, the classical dimer case of chemical applications. Our analysis focuses on the asymptotic regime of large retention times, and brings out an apparently new, broadly useful technique for extracting asymptotic behavior from generating functions in two dimensions. Our model, as proposed by Kelly [2], arises in a study of one-dimensional communication networks (LAN's). In this application, intervals correspond to the circuits connecting communicating parties and [0, X ] represents the bus. Kelly's main results apply to the case k = 1 and to the case of general k with interval lengths governed by a geometric law. The focus here is on space utilization, so the results here add to the earlier theory in three principal ways. First, we give expected vacant space for k = 1, with special emphasis on small-µ asymptotics. Behavior in this regime is quite different from that seen in the "jamming" limit (absorbing state) of the pure filling model (all µ's are identically 0). Second, the important dimer case of chemical applications, where all intervals have length 2, is covered. Finally, the approach of the analysis itself appears to be new and to hold promise for the analysis of similar Markov chains. In very broad terms, expected vacant space is expressed in terms of the geometric properties of a certain plane curve defined by a bivariate generating function.