## 3 Improvement of the Effectiveness of the SOCP Method--Use Cases  
From the presented description of the SOCP method, it can be seen that the deadlock-free zones are exclusively unidirectional (Figure 1a), i.e., zones with a single predecessor and successor, where the predecessor and successor are not the same zone. It should be noted, however, that in the source work [14] another deadlock-free zone is also introduced, the analogy of which is presented in the transportation system in Figure 1b. This zone has a single predecessor and successor, but both the predecessor and the successor are the same zone. This means that this deadlock-free zone is always adjacent to a deadlock-risk zone and as a consequence will always form a deadlock-free segment with a capacity equal to one. This situation is generally unfavorable as shown by several examples in [2]. It should be noted, however, that for a certain group of use cases it is possible to relax the strict rules of the method, and the use of the deadlock-free zone shown in Figure 1b will make it possible to obtain significantly better results than with the general method as will be shown later in the paper. Let us take a closer look at the first rule of the method concerning the necessity of process reservations of the capacity unit of all deadlock-free segments located on the AGV guided path. The rationale for the necessity of process reservation in this way instead of staged reservations in consecutive segments is the possibility of restricted deadlock, a concept introduced by [PERSON] [13]. This is a type of deadlock that does not concern the AGVs moving in the transportation system but concerns the method of deadlock handling itself. To show the essence of restricted deadlock in the example of the SOCP method, let us consider a simple transportation system shown in Figure 2.  
In this system, there are two transport tasks \(t_{1}\) and \(t_{2}\). It is easy to see that in the presented example deadlock cannot occur, so there is no need to use the SOCP method. It is demonstrated here because of its simplicity and the possibility of presenting the restricted deadlock issue clearly. According to the SOCP method, to be able to start a transport task, it is necessary to reserve one unit of capacity of the deadlock-free logical segment inside all such segments accessed by an AGV moving along the guide path. This means that both task \(t_{1}\) and task \(t_{2}\) must have \(z_{2}\) and \(z_{7}\) zones reserved to proceed to the execution stage. Let us now consider what would be the result of a staged execution of the process reservation, meaning in this case reservation of the nearest deadlock-free zone first and then, after the AGV reaches it, reservation of the next one. In this case, assuming that the start of both transport tasks is close in time, a situation might arise in which vehicle \(v_{1}\), before the start, makes a process reservation of zone \(z_{2}\) and after reaching it wants to make a process reservation of zone \(z_{7}\), and this zone is reserved for vehicle \(v_{2}\), which had to reserve it before the start of the task. Once \(v_{2}\) reaches the zone, however, it must make a process reservation for zone \(z_{2}\) to be able to continue its movement. In this case, we are dealing with a stop in the functioning of the transportation system due to restricted deadlock i.e., the impossibility of meeting the conditions set by the method of deadlock handling. The presented example shows that there are situations in which it is necessary to make process reservations in all deadlock-free segments of the AGV path, as presented in the first rule of the SOCP method. This means, therefore, that in a general-purpose method applicable to any structure of transportation systems, it is necessary to fulfill the conditions of this rule in order not to lead to a restricted deadlock.