Graph-Based Deadlock Analysis and Prevention for Robust Intelligent Intersection Management
Authors: 
Kai-En Lin, Kuan-Chun Wang, Yu-Heng Chen, Li-Heng Lin, Ying-Hua Lee, Chung-Wei Lin, Iris Hui-Ru JiangAuthors Info & Claims
Kai-En Lin, Kuan-Chun Wang, Yu-Heng Chen, Li-Heng Lin, Ying-Hua Lee, Chung-Wei Lin, Iris Hui-Ru JiangAuthors Info & ClaimsArticle No.: 33, Pages 1 - 22
Abstract
Intersection management systems, with the assistance of vehicular networks and autonomous vehicles, have the potential to perform traffic control more precisely than contemporary signalized intersections. However, as infrastructural intersection management controllers do not directly activate motions of vehicles, it is possible that the vehicles fail to follow the instructions from controllers, undermining system properties such as deadlock-freeness and traffic performance. In this article, we consider a class of robustness issues, the time violations, which stem from possible discrepancies between scheduled orders and real executions. We refine a graph-based intersection model to build our theoretical foundations and analyze potential deadlocks and their resolvability. We develop solutions that mitigate negative effects of time violations. In particular, we propose a Robustness-Aware Greedy Scheduling algorithm for robust scheduling and evaluate the deadlock-free robustness of different intersection models and scheduling algorithms. Experimental results show that the Robustness-Aware Greedy Scheduling algorithm is able to significantly improve robustness and keep a good balance with traffic performance.
Appendix
To prove Theorem 1, we need some intermediate results regarding the entanglement property. We first extend the definition of an execution sequence as follows.
An execution sequence S of G is a list of \(k\le |V|\) vertices in V where each vertex appears at most once:
—
S is incomplete if \(k\lt |V|\) (i.e., not all vertices in V are in S).
—
S is valid if there is no pair of vertices \(v_{i,j}, v_{i^{\prime },j^{\prime }}\) in S satisfying both (1) there exists a path from \(v_{i,j}\) to \(v_{i^{\prime },j^{\prime }}\) in the extended timing conflict graph \(\tilde{G}\) , and (2) \(v_{i,j}\) has a larger index than \(v_{i^{\prime },j^{\prime }}\) in S.
Given a valid and incomplete execution sequence S, the corresponding execution state \(V_S\) is a subset of vertices in S, where \(v_{i,j}\in V_S\) if \(v_{i,j}\) is the last vertex of the execution subsequence \(S_i\) of S collecting all vertices associated with vehicle \(\delta _i\) .
The intuition of \(V_s\) is that each \(v_{i,j}\in V_S\) represents that conflict zone \(\xi _j\) is occupied by vehicle \(\delta _i\) after S is executed. The preceding definition of an execution sequences with length \(|V|\) is equivalent to the definition in Definition 3. The main idea here is to define an execution sequence with a smaller length, which can be considered as a prefix of a complete execution sequence. In the following, we introduce a class of execution states that are related to non-resolvable deadlocks.
Consider a valid and incomplete execution sequence S and its corresponding execution state \(V_S\) . \(V_S\) is an n-trap if the following apply:
—
There is no valid and complete execution sequence with prefix S (i.e., a deadlock will eventually occur).
—
The longest valid execution sequence with prefix S has length \(|S|+n\) .
Accordingly, a 0-trap indicates that there exists a fully occupied dependency cycle, where an n-trap indicates that there will be a fully occupied dependency cycle within n steps, causing a deadlock.
Among all trap states, 1-traps are of our main interest. For example, the execution state depicted in Figure 3 is a 1-trap, as the only possible moves are (1) vehicle 1 proceeds to conflict zone 2 and (2) vehicle 2 proceeds to conflict zone 2, both of which result in circular blocking. Note that before a dependency cycle is fully occupied, there must be a state being a 1-trap. Next we provide some properties of 1-traps.
Given an intersection I, there exists a 1-trap if and only if \(G_I\) contains an entanglement.
Proof of Lemma 1:
(⇐)
Let \(C_1,\ldots ,C_n\) be the dependency cycles within the entanglement. Consider the state where all dependency cycles other than the singularity, \(\xi ^*\) , are occupied (by vehicles requesting the next conflict zones on their dependency cycles, respectively). Then, there are n unblocked vehicles in the system, requesting to occupy \(\xi ^*\) . However, moving any unblocked vehicle to \(\xi ^*\) results in the full occupation of one of \(C_1,\ldots ,C_n\) , making the resulting state a 0-trap. Thus, every possible sequence ends right after a single move, which concludes that it is a 1-trap.
( \(⇒\) )
Assume that I has a 1-trap \(V_S\) but does not contain an entanglement. By the assumption, there exists a vertex \(v_{i_1, j}\) such that \(S_1=S\ \Vert \ v_{i_1,j}\) is a valid execution sequence, with \(V_{S_1}\) being a 0-trap and \(C_1\) being a corresponding fully occupied dependency cycle.
Without loss of generality, consider the vehicle \(\delta _1\) in the dependency cycle \(C_1\) at \(V_{S_1}\) that is blocked by \(\delta _{i_1}\) . Since \(S_2=S\ \Vert \ v_{i_1,j}\) is also a valid execution sequence, \(V_{S_2}\) is a 0-trap with some corresponding dependency cycle \(C_2\) (which is different from \(C_1\) ).
Similarly, without loss of generality, consider vehicle \(\delta _2\) in the dependency cycle \(C_2\) at \(V_{S_2}\) that is blocked by \(\delta _{i_2}\) . Since I is entanglement-free, \(\delta _2\) must not have its trajectory entering \(C_1\) or \(C_2\) . Therefore, we get \(S_3=S+v_{i_2,j}\) and \(V_{S_3}\) is a 0-trap with some corresponding dependency cycle \(C_3\) , where \(C_3\ne C_1\) and \(C_3\ne C_2\) .
By continuing this argument, we can find an infinite sequence \(\langle C_i\rangle _{i\in \mathbb {N}}\) of distinct dependency cycles. However, this is clearly impossible since the intersection graph \(G_I\) is finite. Therefore, there must be some \(\delta _m\) having trajectory entering one of \(C_1,\ldots , C_{m}\) , and thus the dependency cycles \(C_1,\ldots ,C_m\) form an entanglement with singularity \(\xi _j\) .
Lemma 1 indicates an important connection between 1-traps and entanglements. Since any intersection I that grants the existence of n-traps with \(n\gt 1\) also grants the existence of 1-traps, if I is guaranteed to be 1-trap-free, then we can also rule out the existence of more complicated trap states. We can then derive Theorem 1 from Lemma 1 with the following proof of Theorem 1:
(⇐)
It is straightforward since it implies that the cycle is non-resolvable.
( \(⇒\) )
Consider an entanglement-free intersection I, where \(v_{i, j}\) induces a time violation at execution step k. Let \(S=\langle s_1,\ldots ,s_{|V|}\rangle\) be the original execution sequence and consider the execution state of \(S^{\prime }=\langle s_1,\ldots ,s_k, v_{i,j}\rangle\) . The existence of some reversible Type-3 edges in all cycles implies that there is no fully occupied dependency cycle, then implying that \(V_{S^{\prime }}\) is not a 0-trap. From Lemma 1, as I is entanglement-free and \(V_{S^{\prime }}\) is not a 0-trap, we conclude that \(V_{S^{\prime }}\) is a non-trap state. Thus, reschedulability of \(G^{\prime }\) is guaranteed by \(V_{S^{\prime }}\) being a non-trap state, indicating the deadlock is resolvable. The proof is completed by the contraposition.
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- Graph-Based Deadlock Analysis and Prevention for Robust Intelligent Intersection Management
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Publication History
Published: 13 July 2024
Online AM: 08 November 2023
Accepted: 25 October 2023
Revised: 22 October 2023
Received: 25 February 2023
Published in TCPS Volume 8, Issue 3
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