Chapter+2

  ** Introduction ** In the previous chapter, I have explained the traffic from two directions shares the same infrastructure. Well-known examples are roadblocks caused by traffic accidents or road maintenance (Son, 1999). We encountered a similar setting in a project on the design of an underground transportation system near Schiphol Airport in the Netherlands (van der Heijden et al., 2002a). Here Automatic Guided Vehicles (AGVs) carry freight through underground tubes between various terminals. To reduce infrastructure investment, some terminals are connected by a single tube for traffic in both directions. Intelligent control of the driving direction is required to guarantee acceptable order throughput times. Although this problem is related to traditional traffic systems, an essential difference is that the behaviour of AGVs can be directly influenced. Moreover, quite precise information is available on expected arrival times of AGVs at the two-way track entrance, because the driving behaviour of AGVs is more predictable than that of passenger cars and because routes are known in advance. To exploit this information, we develop new control rules, focusing on delay reduction.  ** In the study of traffic, two-way track control is related to junction control supported by traffic signals, see Newell (1988) for an overview. In the early days traffic signals were usually scheduled according to a predetermined scheme (cf. Bell, 1992). A periodic control rule is a good example of such a scheme. Newell (1988) gave a relation between the mean waiting times in a deterministic and a stochastic setting, assuming that the clearance time is deterministic. He showed that a deterministic approach suffices if the clearance time goes to infinity, but also that this is not applicable for most realistic settings. Despite the long clearance times in our application, the difference in delay between the stochastic and deterministic case is still around 50% (Newell, 1988, Eq. 2.3.12), apart from the fact that we do not face deterministic clearance times. Note that in the analysis of intersections the vehicles mostly are regarded as a continuous fluid (cf. Newell, 2002). In our dynamic control rules we exploit the fact that in reality it is a discrete arrival process. In case arrival rates are not too high, improvements can be expected from dynamic control rules.  Stochastic models for periodic control have been addressed by several authors. Mung et al. (1996) derived distributions of queue lengths at fixed time traffic signals. Heidemann (1994) derived analytical results on statistical distributions of queue lengths and delays at traffic signals, given Poisson arrivals. He compared these results with several approximations (Webster, 1958; Miller, 1968). Hu et al. (1997) extended Heidemann_s model to the multi-lane case, where multiple vehicles may enter the traffic intersection simultaneously. As is argued in van der Heijden et al. (2002b), these models are not applicable to the two-way tube problem, because the effective green period and the effective red period are not constant and known. Therefore, van der Heijden et al. (2002b) developed an approximation for the mean waiting time in periodic systems with random clearance times. Still, none of the papers above exploit the information on the specific system state (AGVs in queue and en route to the two-way tube) to reduce average waiting times. This is not surprising, because the availability of such information is usually limited. As explained in theintroduction, the AGV systems that we have in mind can provide this additional information.  Situations analogous to our two-way tube arise in traffic control, when one lane of a two-lane highway is closed because of maintenance or construction activities. Ceder and Regueros (1990)and Ceder (2000) used a periodic model and delay formula based on Webster (1958) as input for an optimization model determining the lane closure policy and the lane closure length. Cassidy et al. (1994) expanded a deterministic equation for the delay using a statistical analysis of empirical data. An explicit control policy has not been defined. Shibuya et al. (1996) optimised green intervals using a regression equation for the delay. They observed discrepancies with simulated green intervals and concluded that effective control is very difficult under heavy traffic flows. In Son (1999), expressions for the mean delay have been derived, based on the assumption that vehicles arriving within a period H after a queue has entered the two-way track are allowed to enter the two-way track as well (green time extension). Here the value of H has been estimated empirically using discrete choice techniques. All these papers do not focus on waiting time reduction using an adaptive, state dependent control policy.  Finally, we note that problems rather similar to the two-way tube problem have been studied inthe machine scheduling literature, particularly in choosing batch sizes for ovens, see e.g. Uzsoy et al. (1992, 1994), Webster and Baker (1995), van der Zee et al. (1997) and Potts and Kovalyov(2000). Similarities with the two-way tube system can be found in the fact that goods (AGVs) have to be ‘‘batched’’, with the processing time of a batch equal to the driving time on the two-way track. An essential difference is the fact that processing times are fixed for oven systems, because they are related to static product and process characteristics. However, the processing time of a convoy in a two-way tube depends on the time between the first and the last AGV in a convoy. Furthermore, oven systems set restrictions on the number of products in a batch, but there is no a priori limit on the number of AGVs in a convoy.
 * 2.0 Literature Review **
 * Literature review 

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