System reliability analysis for a cloud-based network under edge server capacity and budget constraints

Abstract

In this paper, a modern computer network, cloud-based network, which comprises internet of things (IoT), edge servers, and cloud servers for data transmission, is investigated and evaluated. A cloud-based network is modeled as a graph having a set of nodes and a set of links. Each link represents a transmission route, and each node represents a device, such as an IoT device, edge server, and cloud server. In practical, a transmission route comprises several physical lines or virtual channels. Each physical line (virtual channel) may provide a capacity or may fail to imply several and stochastic states. Such a cloud-based network is called a stochastic flow cloud-based network (SCN) herein. System reliability for an SCN is then evaluated. It is defined as the probability of the data being successfully transmitted through the SCN under edge server capacity and budget constraints. The SCN is modeled firstly in order to elucidate the flow relationship among the whole system; capacity limitation of the edge servers and costs of data transmission/process are also considered. Subsequently, we conclude an algorithm to evaluate system reliability. Supervisors can manage the SCN based on system reliability which presents the system capability with capacity and budget consideration.

Appendix

1.1 A.1 The RSDP algorithm

Suppose that there are l (D, L, B)- LSV, termed X¹, X², …, X^l. Input all LSV to compute $\Pr \{ \cup_{r = 1,2, \ldots ,l} \{ X|X \ge X^{r} \}$ as follows.

1.2 A.2 Proof of Lemma 1

Suppose X satisfies (D, L, B), i.e., there exists an F∈ ω such that the amount of F is not less than both demand $D(\sum\nolimits_{r} {\left\{ {f_{r} \left| {P_{r} \in {\text{B}}\left( {G_{j} ,{\mathbf{T}}} \right)} \right.} \right\}} \ge d_{i} )$ and processed demand $\left( {\sum\nolimits_{r} {\left\{ {f_{r} \left| {P_{r} \in {\text{B}}\left( {{\mathbf{S}},G_{j} } \right)} \right.} \right\}} \ge d_{e}^{*} } \right)$. Without loss of generality for f_j and d_i, assume that P₁ ∈ B(I_i, E_e), f₁ > 0 and $\sum\limits_{r} {\left\{ {f_{r} \left| {P_{r} \in {\text{B}}\left( {G_{j} ,{\mathbf{T}}} \right)} \right.} \right\}} = d_{1} + 1$. Let $F^{{\prime }} = (f_{1}^{\prime } ,f_{2}^{\prime } , \ldots ,f_{{m_{1} }}^{{\prime }} ,f_{{m_{1} + 1}}^{{\prime }} , \ldots ,f_{{m_{1} + m_{2} }}^{{\prime }} ) = (f_{1}^{\prime } - 1,f_{2} , \ldots ,f_{{m_{1} }} ,f_{{m_{1} + 1}} , \ldots ,f_{{m_{1} + m_{2} }} )$. Then, F′ < F and $\sum\nolimits_{r} {\left\{ {f_{r} \left| {P_{r} \in {\text{B}}\left( {G_{j} ,{\mathbf{T}}} \right)} \right.} \right\}} = \sum\nolimits_{r} {\left\{ {f_{r} \left| {P_{r} \in {\text{B}}\left( {G_{j} ,{\mathbf{T}}} \right)} \right.} \right\}} - 1 = d_{1}$. This implies that we can find an F∈ ω with $\sum\nolimits_{r} {\left\{ {f_{r} \left| {P_{r} \in {\text{B}}\left( {G_{j} ,{\mathbf{T}}} \right)} \right.} \right\}} = d_{1}$. Conversely, if there exists an F∈ ω which satisfies constraints (8-9), then X ∈ ω by Definition 1.□

1.3 A.3 Proof of Lemma 2

If there exists a link E_a such that $x_{a} > b_{ar} \ge \left\lceil {\sum\nolimits_{j} {\left\{ {f_{j} \left| {E_{a} \in P_{j} } \right.} \right\}} } \right\rceil > b_{ar - 1}$ and $x_{\alpha } = b_{\alpha r} \ge \left\lceil {\sum\nolimits_{j} {\left\{ {f_{j} \left| {E_{a} \in P_{j} } \right.} \right\}} } \right\rceil > b_{\alpha r - 1}$ for a ≠ α. According to the above hypothesis, F ∈ ω with Y = (y₁, y₂, …, y_n) where y_α = b_α and y_a = x_a for a ≠ α. Particularly, Y < X which conflicts that X is a (D, L, B)-LSV because $y_{a} = b_{a} \ge \left\lceil {\sum\nolimits_{j} {\left\{ {f_{j} \left| {E_{a} \in P_{j} } \right.} \right\}} } \right\rceil \quad \forall i$. Thus, $x_{a} = b_{a} \ge \left\lceil {\sum\nolimits_{j} {\left\{ {f_{j} \left| {E_{a} \in P_{j} } \right.} \right\}} } \right\rceil$. □

1.4 A.4 Proof of Lemma 3

Let X be not a (D, L, B)-LSV, but X ∈ γ_min that implies X ∈ γ. If there exists a (D, L, B)-LSV Y such that Y < X, it also implies Y ∈ γ which contradicts X ∈ γ_min. In turn, let X be a (D, L, B)-LSV, but X ∉ γ_min. X ∈ γ is known. Hence a Y ∈ γ is existed such that Y < X. Y is given by Y ∈ F_Y which contradicts that X is a (D, L, B)-LSV. Hence γ_min is the set of (D, L, B)-LSV. □