## 1 Introduction
Most recently, it has been also applied to model the effects of breakdown of large network systems [36; 37; 38; 39; 40; 41], also in view of optimal planning of maintenance interventions. In the present study, innovative approaches and solutions are proposed to tackle interactions due to simultaneity. The innovative study stems from the classical queuing theory, hypothesizing that the stochastic process of vehicle arrivals is a Markov process and modeling the bridge as a service system. In addition, the problem has been radically simplified solving in succession two exhaustive subcases:* Subcase 1 refers to vehicles simultaneously traveling in one lane. This subcase evidently mainly pertains to road bridges, but it could be relevant also for bridges hosting light railway or tramway lanes. In this circumstance, once assigned the vehicle flow in a given time interval, the outcome of the proposed procedure is an adapted vehicle flow on the lane, composed by single vehicles and by vehicle convoys, if any, running individually on the considered lane. * Subcase 2 refers to simultaneity on different lanes, and it can be extremely relevant also for railway bridges. Starting from the adapted vehicle flows on each lane resulting from subcase 1, the outcome of the method is a suitably modified flow, \"non-interacting traffic flow\", composed by single vehicles; convoys; clusters of vehicles and convoys, separately traveling the bridge. Since the so modified traffic flow is composed by elements which cannot be simultaneously on the bridge, the global stress history can be seen as a random assembly of the individual stress histories induced by each element of the non-interacting traffic flow itself. These stress histories are not influenced by simultaneity, but can combine, as said before. Combination of stress histories is a complex issue, especially when, as in the Eurocodes, fatigue load models are composed by a set of standardized lorries. In fact, to apply the model it is not sufficient to consider the individual stress histories, but it is necessary to consider properly also the combined stress histories. With this aim, it is necessary to answer the following questions:  
* What are the conditions for the combination? * In which order stress histories combine each other? * What is the expected number of occurrences of each combined stress history? * What is the expected remaining number of occurrences of each individual stress history? Which were not previously solved, at least in a practicable form. In reality, these complicated aspects could be in any case faced resorting to sophisticated and time-consuming simulations based on Monte Carlo methods, but undeniably elementary and clear solutions could be extremely helpful. The innovative methodology proposed in the paper is a powerful tool to deal with the problem, also fixing the conditions that govern the combination. The main outcomes of the method are recursive formulae allowing to evaluate the expected number of occurrences of the combined stress histories, as well as the expected residual number of occurrences of each individual stress history. Moreover, the illustration of the practical application of the method in a relevant case study, concerning the evaluation of the stress spectrum at the bottom flange of an orthotropic steel deck, underlines the usefulness and the necessity of such a tool. Finally, the proposed methodologies have been implemented referring to two important examples regarding influence of the reference length \(L\) and of the traffic flow \(N\) on simultaneity effects. In the first example, the discussion focuses on simultaneity effects on one lane, while in the second example it focuses on simultaneity effect on several lanes. In the second example, it is also discussed some improvement of the values of the damage equivalent factors for multilane traffic effect provided in Eurocodes.