## 1 Introduction  
The study of vehicles' interactions is a key topic in fatigue assessment of road and railway bridges. Strictly speaking, two different types of vehicles' interactions can be identified, depending on two vehicles or more traveling simultaneously on the bridges or not. However, as fatigue assessment is not influenced by cars and light vehicles, for road bridges only lorries need to be considered in practice. Interactions due to simultaneity originate from vehicles traveling at the same time on the bridge, on the same lane and/or on different lanes or tracks, causing a stress history, i.e., the stress-time diagram that cannot be obtained by the trivial assembly in sequence of the stress histories produced individually by each vehicle. In fact, in this case the stress history depends also on the relative positions of the vehicles instant by instant. Moreover, vehicles' simultaneity is a relevant problem not only for fatigue assessment, but also for the study of traffic loadings, as well as for more general traffic problems (see, for example [1; 2; 3; 4; 5; 6; 7; 8; 9]). Another relevant kind of interactions occurs when the union of two individual, and non-simultaneous, stress histories induces a fatigue damage bigger than the sum of fatigue damages induced by the individual stress histories [10; 11]. This ensues as a typical consequence of the application of most common cycle counting methods, such as, for instance, the range pair method; the level crossing method; the range mean method; the reservoir method; and the rainflow method, described in detail in [12; 13; 14; 15; 16; 17; 18]. The phenomenon is particularly emblematic for bridges [19; 20], where stress cycles are typically identified by means of the reservoir or the rainflow method. The study is extremely topical as conventional fatigue load models provided in modern bridge codes, such as EN1991-2 [21] (2003), disregard or include only partially effects of interactions. For example, in EN1991-2 (note 2 of Section 4.6.1(2), items c and e) is explicitly stated that the use of road bridges' fatigue load models no. 2 and no. 4, which are illustrated hereafter, is allowed only when \"the simultaneous presence of several lorries on the bridge can be neglected. If that is not the case, they should be used only if they are supplemented by additional data.\"  
Obviously, the development of appropriate theoretical models taking into account vehicle interactions and stress history combinations could improve fatigue assessment procedures, extending considerably the field of application of such notional fatigue load models, also providing a means for future improvements of bridge codes. At the same time, these theoretical models may well simplify the generation of artificial traffic samples, better addressing advanced Monte Carlo sampling techniques, such as the stratified sampling or the Latin hypercube sampling [22]. Actually, several different theoretical and numerical approaches have been proposed by different authors to take into account this kind of interaction problems, but they cannot be generalized, since they are specifically addressed to assess some particularly relevant bridge (see, for example [8; 9; 23; 24]). Starting from the aforementioned considerations, an original procedure has been developed to analytically solve the problem. The basic idea of the proposed method is that the problem can be considerably simplified identifying and solving in sequence three different cases, each one corresponding to one step of the proposed procedure. In the first two steps, the simultaneity is tackled in the framework of the queuing theory. The queuing theory, initially developed by [PERSON] [25; 26; 27; 28], is the subject of many papers and books [29; 30; 31], and it is commonly used to approach a wide range of engineering (see, for example [32; 33; 34]) or social problems [35].