act because of air traffic control separation requirements. For example, in the case of a pair of intersecting runways, the location of the intersection relative to the points where takeoffs are initiated or where landing aircraft touch down greatly affects the combined capacity of the two runways. Similarly, in the case of two parallel runways, the capacity depends on the distance between the centerlines of the runways. Approximate capacity analyses for airports with two parallel or intersecting runways are quite straightforward. Multirunway analytical capacity models also provide good approximate estimates of true capacity in cases involving three or more active runways, as long as the runway configurations can be “decomposed” into semi-independent parts, each consistin of one or two runways. Such models have proved extremely valuable in airport planning, as well as in assessing the impacts of proposed procedural or technological changes on airport capacity. Another topic of intensive study has been the estimation, through the use of queueing models, of the delays caused by the lack of sufficient runway capacity. This is a problem that poses a serious challenge to operations researchers: The closed-form results developed in the voluminous literature of classical steadystate queueing theory are largely nonapplicable—at least when it comes to the really interesting cases. The reason is that airport queues are, in general, strongly nonstationary. The demand rates and, in changing weather conditions, the service rates at most major airports vary strongly over the course of a typical day. Moreover, the demand rates may exceed capacity ( > 1), possibly for extended periods of time, most often when weather conditions are less than optimal. This has motivated the development of numerical approaches to the problem of computing airport delays analytically. In another landmark paper, Koopman (1972) argued—and showed through examples drawn from New York’s Kennedy and LaGuardia Airports, at the time among the world’s busiest—that the queueing behavior of an airport with k “runway equivalents” (i.e., k nearly independent servers) can be bounded by the characteristics of the M(t)/M(t)/k and the M(t)/D(t)/k queueing models, each providing “worst-case” and “best-case” estimates, respectively. Note that this allows for dynamic changes in the service rates, as well as in the demand rates. Extending the work of Koopman (1972), the M t/Ek t/k system was proposed by Kivestu (1976) as a model that could be used to directly compute approximate queueing statistics for airports— rather than separately solving the M(t)/M(t)/k and M(t)/D(t)/k models and then somehow interpolating their results. (Note that negative exponential service times (M and constant service times (D) are simply special cases of the Erlang (Ek) family, with k = 1 and k = , respectively.) Kivestu (1976) noted that k should be determined from the relationship E S /S = √ k, where E S and S denote the expected value and S denote the expected value and the standard deviation of the service times and can be estimated from field data. He also developed a powerful numerical approximation scheme that computes the (time varying) state probabilities for the M t/Ek t/k system efficiently. Malone (1995) has demonstrated the accuracy and practicality of Kivestu’s (1976) approach and developed additional efficient approximation methods, well suited to the analysis of dynamic airfield queues. Fan and Odoni (2002) provide a description of the application of Kivestu’s (1976) model to a study of the gridlock conditions that prevailed at LaGuardia Airport in 2000 and early 2001. Additional (numerical) analytical models for computing airport delays have been developed over the last few years. Peterson et al. (1995) and Daniel (1995) describe two different models for computing delays at hub airports, which are characterized by sharp “banks” or “waves” of arrivals and departures. Hansen (2002) has used a deterministic model, based on the notion of cumulative diagrams, to compute delay externalities at Los Angeles International Airport. Finally, Long et al. (1999) and Malone (1995) present two dynamic queueing network models and their application to the study of congestion in the National Airspace System. Ingolfsson et al. (2002) offer a comprehensive survey and comparison of several alternative approaches to the analysis of nonstationary queueing systems. Many of the best features of some of the analytical capacity and delay models just described have been integrated recently in a number of new software
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