Qualitative Analysis of Commercial Services in MEC as Phased-Mission Systems

Currently, mobile edge computing (MEC) is one of the most popular techniques used to respond to real-time services from a wide range of mobile terminals. Compared with single-phase systems, commercial services in MEC can be modeled as phased-mission systems (PMS) and are much more complex, because of the dependencies across the phases. Over the past decade, researchers have proposed a set of new algorithms based on BDD for fault tree analysis of a wide range of PMS with various mission requirements and failure behaviors. (e analysis to be performed on a fault tree can be either qualitative or quantitative. For the quantitative fault tree analysis of PMS by means of BDD, much work has been conducted. However, for the qualitative fault tree analysis of PMS bymeans of BDD, nomuch related work can be found. In this paper, we have presented some efficient methods to calculate the MCS encoding by a PMS BDD. Firstly, three kinds of redundancy relations-inclusive relation, internal-implication relation, and external-implication relation-within the cut set are identified, which prevent the cut set from being minimal cut set. (en, three BDD operations, IncRed, InImpRed, and ExImpRed, are developed, respectively, for the elimination of these redundancy relations. Using some proper combinations of these operations, MCS can be calculated correctly. As an illustration, some experimental results on a benchmark MEC system are given.


Introduction
Currently, mobile edge computing (MEC) is one of the most popular techniques used to respond to real-time services from a wide range of mobile terminals [1][2][3][4]. Edge computing is a distributed computing topology where information processing is placed closer to the things or people that produce and/or consume that information. It means that MEC provides cloud-computing capabilities at the edge of the mobile network in close proximity to mobile subscribers; it is also considered as one of the key pillars for meeting the demanding KPIs of 5G.
Mobile edge computing integrates cloud computing (CC) into mobile networks, prolonging the battery life of mobile users (MUs). However, this mode may cause significant mission complexity. Compared with single-phase systems, commercial services in MEC can be modeled as phased-mission systems (PMS). A MEC PMS is defined as a commercial service, which is subject to multiple, consecutive, nonoverlapping phases of operation. During each phase, it has to accomplish a specified task. us, the MEC configuration, failure criterion, and/or failure behavior can change from phase to phase [5][6][7][8].
Much early work has been conducted on the fault tree analysis of PMS. Qualitative analysis of commercial services in MEC as phased-mission systems is much more complex, because of the dependencies across the phases. For instance, the state of a MEC node at the beginning of a new phase is identical to its state at the end of the previous phase [9][10][11]. Over the past decade, researchers have proposed a set of new algorithms based on the binary decision diagram (BDD) for fault tree analysis of a wide range of PMS with various mission requirements and failure behaviors [12][13][14][15]. Due to the nature of the BDD, cancellation of common components among the phases can be combined with the BDD generation, without additional operations, and the sum of disjoint products (SDP) can be implicitly represented by the final PMS BDD. Several experiments show that BDD-based algorithm is more efficient than the algorithm based on SDP, in both computation time and storage space; this efficiency allows the study of some practical, large MEC PMS [16][17][18]. e analysis to be performed on a fault tree can be either qualitative or quantitative. Qualitative analysis involves calculating the minimal cut set (MCS), that is, listing all possible smallest combinations of basic events, which cause the top event. Quantitative analysis, on the other hand, involves calculating the probability of the top event occurring from the probabilities of the basic events. For the qualitative fault tree analysis of MEC PMS by means of BDD, no much related work can be found. In this paper, we focus on this line of research, that is, calculating the MCS encoding by a PMS BDD. e major contributions of our work are the following ones: (1) Different kinds of redundancy relations within the cut set encoding by a PMS BDD are identified, and it is these relations that prevent the cut set from being minimal cut set. (2) Two BDD operations are proposed to eliminate these redundancy relations from a PMS BDD. One operation can only eliminate the external-implication relations relating to one component, and the other operation eliminates all the inclusive relations from the cut set. (3) MCS calculation method is developed by combining the proposed two BDD operations for a PMS BDD with forward ordering. e remainder of the paper is organized as follows. Section 2 introduces the fundamentals of fault tree and BDD for the qualitative analysis of MEC PMS. Section 3 proposes three kinds of redundancy relations within a PMS BDD, several BDD operations eliminating these redundancy relations, and MCS calculation methods based on these BDD operations. Section 4 gives some experimental results on a benchmark MEC PMS. Last, conclusions are given in Section 5.

Fault Tree.
Fault tree analysis is an important technique for reliability and safety analysis. Bell Telephone Laboratories developed the concept in 1962 for the US Air Force for use with the Minuteman system. It was later adopted and extensively applied by the Boeing Company.
Fault tree diagrams are logic block diagrams that display the state of a system (top event) in terms of the states of its components (basic events). It uses a graphic "model" of the pathways within a system that can lead to a foreseeable, undesirable loss event (or a failure). Fault trees are built using gates and events. e two most commonly used gates in a fault tree are the AND and OR gates. If both events need to occur to cause the top event to occur, they are connected by an AND gate. Alternatively, if the occurrence of either event causes the top event to occur, then these events are connected using an OR gate. Notice that NOT gate is not considered in this paper. us, the fault trees analyzed in the following sections are coherent fault trees.
For MEC PMS, there are two kinds of fault tree models: phase-level fault tree, which represents the causal chain between component failure and phase failure, and systemlevel fault tree, which represents the causal chain between phase failure and service failure. As an illustration example, Figure 1 depicts a small MEC PMS fault tree.
For the purpose of this paper, MEC PMS fault trees are essentially considered as Boolean functions, that is, terms inductively built over the two constants 0 and 1, a set of variables X, and usual logical connectives ∧ (AND) ∨ (OR).
e Boolean function F associated with the fault tree in Figure 1 is

BDD.
BDD is a compact encoding of the truth tables of Boolean function [19]. e BDD representation is based on the Shannon decomposition. Let F be a Boolean function that depends on the variable x; then the following equality holds: BDD has two sink nodes, labeled 0 and 1, representing the two corresponding constants 0 and 1. Each nonsink node is labeled with a Boolean variable x and has two outgoing edges that represent the two corresponding expressions in the Shannon decomposition. ese two edges are called E-edge (or 0-edge) and T-edge (or 1-edge), respectively. e node linked by the T-edge represents the Boolean expression F[x←1]; E-edge represents the Boolean expression F[x←0].
us, each nonsink node in a BDD encodes an ite format. An ordered BDD is a BDD with the constraint that the variables are ordered and every source-to-sink path in the ordered BDD visits the variables in ascending order. A reduced ordered BDD is an ordered BDD where each node represents a distinct Boolean expression.

Variable Ordering.
A variable in a PMS BDD indicates the component that the variable belongs to and the phase in which the component exists. For example, C i is the state indicator variable of component C at phase i. Variable ordering can be generated in the following steps: componentlevel ordering and ordering variables on phases.
(1) Component-level ordering: the components in PMS are s-independent from each other. erefore, the ordering heuristics for ordinary BDD can be applied to component-level ordering for PMS BDD. (2) Phase ordering: there are two classes of phase ordering: forward ordering and backward ordering. In forward ordering, the variable order is the same as the phase order. In backward ordering, the variable order is the reverse of the phase order.
To illustrate the above ordering method, the PMS in Figure 1 can be used. Applying DFLM to the fault tree in "do not care" form shown in Figure 2, where phase indexes are unconsidered, the component-level ordering is A < B < C. Extending this component-level ordering over different phase indexes, the forward DFLM ordering is

BDD Generation.
In order to compute the BDD associated with a Boolean function F, the following principle is applied: (1) If F is a constant, then one associates with F the corresponding sink node 0 or 1.
where <> is the binary connective ∧ or ∨ and G and H are functions, then one computes the BDDs associated with G and H and then performs the operation <> on these two BDDs.
Given two Boolean functions G and H encoded by the BDDs G � ite (x, G 1 , G 0 ) and H � ite (y, H 1 , H 0 ), it is possible to compute directly on G and H any logical operation between G and H by means of the following calculation: Compared with the single-phase system, BDD generation of PMS is much more complex because of the dependencies of component states across the phases. Use G and H to represent phase BDD of phase i and phase j, respectively (i < j). When expanded with regard to x i and x j , they, respectively, can be written as Following special BDD operation, to deal with the crossphase dependencies associated with the operation <> on G and H, With the help of BDD operations (2) and (3), PMS BDD can be generated from PMS fault tree. When different orderings are used, the sizes of PMS BDD are different. As shown in Figure 3, the size of BDD with forward DFLM ordering is 15.

Cut Set and Minimal Cut Set.
In order to introduce formally the notion of minimal cut set, we need the following definitions.  Figure 1: An example MEC PMS fault tree.

Security and Communication Networks
. , x n be a set of Boolean variables. An assignment of X is a mapping from X into {0, 1}. Assignment and subset of X are equivalent objects. A unique assignment σ corresponds to a subset of X, X(σ), where Definition 2. Let σ be an assignment of X and F be a monotonic Boolean function on X.
Definition 3. For any ρ ∈ CS[F], ρ is a minimal cut if for any assignment σ, e set of minimal cuts of F is denoted by MCS[F]. According to Definition 3, the following property holds.  Let F be the BDD associated with F. Each path from the root of F to sink node 1 corresponds to an assignment σ and defines a cut of F, ρ, as follows: x ∈ ρ iff the path goes through a node labeled by x and goes out of this node on the T-edge. e set of cuts encoded by F is denoted by CS[F]. Due to the fact that F compactly encodes the truth table of F in Shannon's form by means of subtree sharing, CS[F] is a cut set of F.
What is more, PMS BDD F is minimal if for any path from the root of F to sink node 1, the cut identified by it is minimal. e set of cuts encoded by minimal F, denoted by MCS[F], is still a minimal cut set of F.
One One One One One One One One One One According to forward ordering, the following property holds.
Property 2. Let F be a PMS BDD with forward ordering. en, where X � x 1 , x 2 , . . . , x n includes all variables relating to component X and n is the maximum of phase number.
Proof. Due to the fact that PMS considered in this paper is not maintained, a component remains down for the remaining phases if it is down in one phase; that is, x j will never appear at the T-edge of node x i . In the following section, we try to show how to calculate the MCS[F] from PMS BDD F or CS[F].
If ω is a cut of F 0 , then ω does not contain x and ω is also a cut of F. If δ is a cut of F 1 , δ augmented of x, ρ � δU x { }, is a cut of F. If ρ includes ω, we say that F has inclusive relation between ω and ρ. Notice that ω will never include ρ. More formally, we have the following definition.

Definition 4. Let F be a PMS BDD; F has inclusive relation across different cuts within CS
From the point of view of MCS, inclusive relation is one kind of redundancy relation. at is to say, in order to make CS[F] be MCS, ρ 11 should be dropped.
It is obvious that the cut sets encoded by PMS BDD shown in Figures 4 and 5 have inclusive relations across different cuts. For example, A 3 , C 1 is a cut, but its subset C 1 is still a cut. e following theorem gives an inductive principle to eliminate the inclusive relations across cuts.
where IncRed (F) is the cut set of F which has no inclusive relation across different cuts. Proof Using eorem 1, an operation (IncRed) can be derived to eliminate all the inclusive relations within CS[F]. Reference [20] has implemented a similar operation called "Minsol" under the context of fault tree analysis of singlephase system. us, the operation implementation would not be given here.
As an illustration, IncRed operation is used to transform BDD in Figure 3 into the one as shown in Figure 4. e cut set encoded by the PMS BDD shown in Figure 4 is

External-Implication Relation Elimination. Let
be a PMS BDD with forward ordering. If ρ 1 is a cut of G 1 , ρ 1 contains x j and ρ 1 U x j is also a cut of F. If ρ 2 is a cut of F 1 , ρ 2 U x i is a cut of F. If ρ 1 includes ρ 2 , we say that F has external-implication relation between cut ρ 1 U x j and cut ρ 2 U x i . More formally, we have the following definition.
where ρ < � y |(y ∈ ρ)∧(order(y) < order(x i )) . Notice that ρ 1 , ρ 2 will never contain variables within X − x i , x j according to Property 2. From the point of view of MCS, external-implication relation is one kind of redundancy relation. Notice that x i implies x j , and x i , x j and x i are equivalent objects. In order to make CS[F] be MCS, ρ 2 should be dropped. Consider the PMS BDD shown in Figure 5. e cut set encoded by this BDD is Due to the externalimplication relation, A 1 should be eliminated from the cut set, and so does A 2 , B 2 , C 3 . e following theorem gives an inductive principle to eliminate the external-implication relation across different cuts. (y j , G 1 , G 0 )), i < j, be a PMS BDD with forward ordering. en,  Figure 3.

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where ExImpRed (F) is the cut set of F which has no externalimplication relation across different cuts.
Proof. Two cases are possible: (1) x ≠ y. F 0 does not contain variables relating to component X.
If ρ is a cut of F, ρ is either a cut of F 0 (ρ does not contain x i ) or a cut of F 1 augmented of x i (ρ only contains one variable x i relating to component X).
us, there is no external-implication relation relating to x i within the cut set, and we obtain that (2) x � y. ere might be some cuts that have externalimplication relation relating to x i .
If ρ is a cut of F 0 and ρ contains x j , ρ is also a cut of F. If δ is a cut of F 1 , δ augmented of x i is a cut of F. If (δ⊇ρ/ x j ), ρ and δU x i have external-implication relation. at is to say, δ should be eliminated. According to the fact that ρ contains x j , (ρ/ x j ) is a cut of G 1 . us, we obtain that According to eorem 2, the following property holds. □ Property 4. Let F be a PMS BDD with forward ordering. en, where extend operation means adding variables with higher index to a cut. For example, extend As an illustration, the ExImpRed operation is used to transform BDD in Figure 4 into the BDD shown in Figure 5. e cut set encoded by this BDD is  A 2 , A 3 , B 3 , C 3 , B 1 , B 2 , C 2 , C 1 . is cut set has no external-implication relation across different cuts.

MCS Calculation
Theorem 3. Let F be a PMS BDD with forward ordering. en, Proof. We have known that CS[F] might have inclusive relations and external-implication relations across the included cuts. According to Property 2, we know that x i and x j will not be simultaneously included by a cut. us, there is no internal-implication relation. According to eorem 2, we know that ExImpRed operation can only eliminate the external-implication relations relating to one component. However, CS[F] might have external-implication relations relating to more than one component. For example, consider a CS[F] has two cuts (one is A 2 , B 2 and the other is A 1 , B 1 . Notice that x i implies x j and A 1 , B 1 , A 2 , B 2 and A 1 , B 1 are equivalent objects. us, A 1 , B 1 should be dropped. However, our ExImpRed operation cannot use A 2 , B 2 to eliminate A 1 , B 1 . Now, we show that the ExImpRed operation can do this elimination indirectly. If both A 2 , B 2 and A 1 , B 1 belong to CS[F], then F must have a structure as shown in Figure 6. ere are two points: (1) according to the BDD generation process, B 1 will appear at the T-edge of node A 2 under the condition that the T-edge of node A 1 has a B 1 and the T-subBDD of node A 2 has a B 2 ; (2) the T-edge of this B 1 is 1 due to the fact that A 2 , B 2 is a cut. us, A 2 , B 1 must be included by CS [F]. Now, the ExImpRed operation can use A 2 , B 1 to eliminate A 1 , B 1 and then use A 2 , B 2 to eliminate A 2 , B 1 .
us, our ExImpRed operation can eliminate all kinds of externalimplication relations within CS [F].
ExImpRed operation eliminates all the external-implication relations from the cut set CS[F]. However, it will disturb some inclusive relations. For example, consider the PMS BDD F shown in Figure 7, where cut set CS[F] � A 1 , A 2 , B 1 , C 1 , A 2 , B 2 , C 1 , B 1 , C 1 . After the ExImpRed operation, we get ExImpRed(F) � A 1 , A 2 , B 2 , C 1 , B 1 , C 1 }. Here, the inclusive relation between cut B 1 , C 1 and cut A 2 , B 1 , C 1 is destroyed. us, after another IncRed operation, we get IncRed (ExImpRed(F)) � On the other hand, IncRed operation eliminates all the inclusive relations from the cut set CS[F]. However, it will disturb some external-implication relations. For the PMS BDD F shown in Figure 7, after the IncRed operation, we get IncRed(F) � A 1 , A 2 , B 2 , C 1 , B 1 , C 1 . Here, the external-implication relation between cut A 2 , B 1 , C 1 and cut A 2 , B 2 , C 1 is destroyed.

Application
With edge, compute and storage systems reside at the edge as well, as close as possible to the component, device, application, or human that produces the data being processed. e purpose is to remove processing latency, because the data need not be sent from the edge of the network to a central processing system and then back to the edge. e applications for edge make sense: Internet of thingsconnected devices are a clear use for edge computing architecture. With remote sensors installed on a machine, component, or device, they generate massive amounts of data. If that data is sent back across a long network link to be analyzed, logged, and tracked, that takes much more time than if the data is processed at the edge, close to the source of the data.
ere are patterns of failure. (1) If you place too much at the edge, it is easy to overwhelm the smaller processor and storage platforms that exist there. In some cases, storage could be limited to a few gigabytes and processing using a single CPU. Power and size restrictions are really what set the limits. (2) Another pattern is failure to integrate security from concept to production. Security is systemic to edge computing architectures and centralized processing. Security needs to span both and use mechanisms such as identity and access management. Encryption is not a nice-to-have, but rather a requirement for device safety.
Consider a benchmark MEC PMS borrowed from [21]. is MEC PMS has the following: (1) It has 7 groups of components.
(3) Five MEC node configurations are shown in Figure 8. (4) Four MEC service configurations are listed in Table 1. Figure 6: e PMS BDD encoding A 2 , B 2 and A 1 , B 1 .

Zero One
One Zero Zero Zero

Conclusion
Based on the mobile edge computing techniques, commercial service providers, such as video content providers, can benefit from low-latency edge resources to provide their users with more efficient service acquisition, thereby improving the quality of experience [22][23][24]. Compared with single-phase systems, fault tree analysis of MEC PMS is much more complex, because of the dependencies across the phases. e analysis to be performed on a fault tree can be either qualitative or quantitative. For the qualitative fault tree analysis of PMS by means of BDD, no much related work can be found. In this paper, we have presented some efficient methods to calculate the MCS encoding by a PMS BDD. e basic idea is to eliminate different kinds of redundancy relations from a cut set encoding by a PMS BDD. ese methods are based on several BDD operations, such as IncRed and ExImpRed.
e IncRed can eliminate all inclusive relations. e ExImpRed can eliminate all externalimplication relations. Using some proper combinations of these operations, MCS can be calculated correctly.
Smart city is a fast-developing system enabled by the Internet of things (IoT) with massive collaborative services (e.g., intelligent transportation and collaborative diagnosis) [25,26]. erefore, one direction of our future work is to consider the qualitative analysis of commercial services in MEC smart city. As an extension of this research work, we will also improve the presented BDD-based analysis methodology for more generalized MEC service systems [27].

Data Availability
e MCS calculation data used to support the findings of this study may be released upon application to the Fujian Province University Key Laboratory of Computational Science, who can be contacted at CSlab@hqu.edu.cn.

Conflicts of Interest
e authors declare that they have no conflicts of interest.