Quantitative understanding serial-parallel hybrid sfc services: a dependability perspective

Abstract

Network function virtualization (NFV) has been explored to be integrated with multi-access edge computing (MEC) to facilitate the development of 5G (fifth-generation) network. Latency-sensitive applications can be deployed as serial-parallel hybrid service function chains (SP-SFCs) in the MEC-NFV environment. SP-SFCs are deployed on resource-limited devices in the edge and therefore are vulnerable to software aging, which can reduce the SFC service dependability. Rejuvenation technique can mitigate the impact of software aging but its effectiveness is influenced by the rejuvenation trigger interval. This paper explores a semi-Markov model approach to quantitatively evaluate the impact of different rejuvenation trigger intervals on the SFC service dependability in terms of reliability and availability. In contrast to the existing studies, our model captures the behaviors of an SP-SFC system consisting of any number of SFs from suffering from software aging until recovery under the condition that time intervals of failure and recovery events follow general distributions. Our model and formulas are verified by developing a simulator. The results of numerical experiments show the optimal rejuvenation trigger intervals, which can help service providers to maximize the benefits.

Appendices

Appendix A

In this section, we give the detailed process for calculating SFC service availability mentioned in Sect. 3.4 of the main paper. The notations used in this appendix are defined in Table 3 of the main paper.

The non-null elements of the TPM ${\text{P}}_{PS}$ are given by Eqs. (8)–(24),

$$p_{{PS_{0} PS_{2i + 1} }} = \int_{0}^{\infty } {\prod\nolimits_{{r \in B_{i}^{^{\prime}} }} {{(}1 - F_{{{\text{as}}r}} {(}t{))}} \prod\nolimits_{j \in C} {{(}1 - F_{{{\text{ap}}j}} {(}t{))}} dF_{{{\text{as}}i}} } {(}t{)}$$

(8)

$$p_{{PS_{0} PS_{2n + 3j} }} = \int_{0}^{\infty } {\prod\nolimits_{i \in B} {{(}1 - F_{{{\text{as}}i}} {(}t{))}} \prod\nolimits_{{r^{\prime} \in C_{j}^{^{\prime}} }} {{(}1 - F_{{{\text{ap}}r^{\prime}}} {(}t{))}} dF_{{{\text{ap}}j}} } {(}t{)}$$

(9)

$$p_{{PS_{2n + 3j + 1} PS_{2} }} = \int_{0}^{\infty } {{(}1 - F_{{{\text{fdp}}j}} {(}t{))(}1 - F_{{{\text{mp}}j}} {(}t{))}dF_{{{\text{dp}}j}} } {(}t{)}$$

(10)

$$p_{{PS_{2n + 3j + 1} PS_{2n + 3j + 2} }} = \int_{0}^{\infty } {{(}1 - F_{{{\text{mp}}j}} {(}t{))(}1 - F_{{{\text{dp}}j}} {(}t{))}dF_{{{\text{fdp}}j}} } {(}t{)}$$

(11)

$$p_{{PS_{2n + 3j + 1} PS_{0} }} = \int_{0}^{\infty } {{(}1 - F_{{{\text{fdp}}j}} {(}t{))(}1 - F_{{{\text{dp}}j}} {(}t{))}dF_{{{\text{mp}}j}} } {(}t{)}$$

(12)

$$p_{{PS_{2i + 2} PS_{{2}} }} = \int_{0}^{\infty } {{(}1 - F_{{{\text{ms}}i}} {(}t{))(}1 - F_{{{\text{fds}}i}} {(}t{))}dF_{{{\text{ds}}i}} } {(}t{)}$$

(13)

$$p_{{PS_{2i + 2} PS_{1} }} = \int_{0}^{\infty } {{(}1 - F_{{{\text{ms}}i}} {(}t{))(}1 - F_{{{\text{ds}}i}} {(}t{))}dF_{{{\text{fds}}i}} } {(}t{)}$$

(14)

$$p_{{PS_{2i + 2} PS_{0} }} = \int_{0}^{\infty } {{(}1 - F_{{{\text{fds}}i}} {(}t{))(}1 - F_{{{\text{ds}}i}} {(}t{))}dF_{{{\text{ms}}i}} } {(}t{)}$$

(15)

$$p_{{PS_{2n + 3j} PS_{2} }} = \int_{0}^{\infty } {{(}1 - F_{{{\text{p}}j}} {(}t{))(}1 - F_{{{\text{fp}}j}} {(}t{))}dF_{{{\text{dp}}j}} } {(}t{)}$$

(16)

$$p_{{PS_{2n + 3j} PS_{2n + 3j + 2} }} = \int_{0}^{\infty } {{(}1 - F_{{{\text{dp}}j}} {(}t{))(}1 - F_{{{\text{p}}j}} {(}t{))}dF_{{{\text{fp}}j}} } {(}t{)}$$

(17)

$$p_{{PS_{2i + 1} PS_{2} }} = \int_{0}^{\infty } {{(}1 - F_{{{\text{fs}}i}} {(}t{))(}1 - F_{{{\text{s}}i}} {(}t{))}dF_{{{\text{ds}}i}} } {(}t{)}$$

(18)

$$p_{{PS_{2i + 1} PS_{1} }} = \int_{0}^{\infty } {{(}1 - F_{{{\text{ds}}i}} {(}t{))(}1 - F_{{{\text{s}}i}} {(}t{))}dF_{{{\text{fs}}i}} } {(}t{)}$$

(19)

$$p_{{PS_{2n + 3j} PS_{2n + 3j + 1} }} = 1 - p_{{PS_{2n + 3j} PS_{2n + 3j + 2} }} - p_{{PS_{2n + 3j} PS_{2} }}$$

(20)

$$p_{{PS_{2i + 1} PS_{2i + 2} }} = 1 - p_{{PS_{2i + 1} PS_{1} }} - p_{{PS_{2i + 1} PS_{2} }}$$

(21)

$$p_{{PS_{2n + 3j + 2} PS_{0} }} = 1$$

(22)

$$p_{{PS_{2} PS_{0} }} = 1$$

(23)

$$p_{{PS_{1} PS_{0} }} = 1$$

(24)

where $B_{i}^{{\prime}} = {{\{ }}\left. r \right|0 < r \le n{,}r \ne i{{\} ,}}$$C_{j}^{{\prime}} = {{\{ }}\left. {r{^{\prime}}} \right|0 < r{^{\prime}} \le m,r{^{\prime}} \ne j{{\} }}$$B = {{\{ }}\left. i \right|0 < i \le n{{\} ,}}$ and $C = {{\{ }}\left. j \right|0 < j \le m{{\} }}$. The equations of calculating the steady-state probability $V_{{PS_{i} }}$ of the EDTMC at system state $PS_{i}$ are given by Eqs. (25)–(32),

$$\begin{aligned} V_{{PS_{2} }} & = {(}\sum\nolimits_{i \in B} {p_{{PS_{0} PS_{2i + 1} }} p_{{PS_{2i + 1} PS_{2} }} }\\ & + \sum\nolimits_{j \in C} {p_{{PS_{0} PS_{2n + 3j} }} p_{{PS_{2n + 3j} PS_{2} }} } \\ & + \sum\nolimits_{j \in C} {p_{{PS_{0} PS_{2n + 3j} }} p_{{PS_{2n + 3j} PS_{2n + 3j + 1} }} p_{{PS_{2n + 3j + 1} PS_{2} }} } {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \\ & + \sum\nolimits_{i \in B} {p_{{PS_{0} PS_{2i + 1} }} p_{{PS_{2i + 1} PS_{2i + 2} }} p_{{PS_{2i + 2} PS_{2} }} } {)}/W \end{aligned}$$

(25)

$$\begin{aligned} V_{{PS_{2n + 3j + 2} }} & = {(}p_{{PS_{0} PS_{2n + 3j} }} p_{{PS_{2n + 3j} PS_{2n + 3j + 2} }} \\ & + p_{{PS_{0} PS_{2n + 3j} }} p_{{PS_{2n + 3j} PS_{2n + 3j + 1} }} p_{{PS_{2n + 3j + 1} PS_{2n + 3j + 2} }} {)}/W \end{aligned}$$

(26)

$$\begin{aligned} V_{{PS_{1} }} & = {(}\sum\nolimits_{i \in B} {p_{{PS_{0} PS_{2i + 1} }} p_{{PS_{2i + 1} PS_{1} }} } \\ & + \sum\nolimits_{i \in B} {p_{{PS_{0} PS_{2i + 1} }} p_{{PS_{2i + 1} PS_{2i + 2} }} p_{{PS_{2i + 2} PS_{1} }} } {)}/W \end{aligned}$$

(27)

$$V_{{PS_{2n + 3j + 1} }} = p_{{PS_{0} PS_{2n + 3j} }} p_{{PS_{2n + 3j} PS_{2n + 3j + 1} }} /W$$

(28)

$$V_{{PS_{2i + 2} }} = {(}p_{{PS_{0} PS_{2i + 1} }} p_{{PS_{2i + 1} PS_{2i + 2} }} {)}/W$$

(29)

$$V_{{PS_{2n + 3j} }} = p_{{PS_{0} PS_{2n + 3j} }} /W$$

(30)

$$V_{{PS_{2i + 1} }} = p_{{PS_{0} PS_{2i + 1} }} /W$$

(31)

$$V_{{PS_{0} }} = 1/W$$

(32)

where $W = \sum_{i \in B} {p_{{PS_{0} PS_{2i + 1} }} } + \sum_{i \in B} {p_{{PS_{0} PS_{2i + 1} }} p_{{PS_{2i + 1} PS_{2i + 2} }} p_{{PS_{2i + 2} PS_{2} }} } {\kern 1pt} + 1$

$$+ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sum\nolimits_{j \in C} {p_{{PS_{0} PS_{2n + 3j} }} p_{{PS_{2n + 3j} PS_{2n + 3j + 1} }} p_{{PS_{2n + 3j + 1} PS_{2n + 3j + 2} }} + {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sum\nolimits_{i \in B} {p_{{PS_{0} PS_{2i + 1} }} } } {\kern 1pt}$$

$$p_{{PS_{2i + 2} PS_{1} }} p_{{PS_{2i + 1} PS_{2i + 2} }} + {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sum\nolimits_{j \in C} {p_{{PS_{0} PS_{2n + 3j} }} p_{{PS_{2n + 3j} PS_{2n + 3j + 1} }} p_{{PS_{2n + 3j + 1} PS_{2} }} } {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + {\kern 1pt} {\kern 1pt}$$

$$\sum\nolimits_{i \in B} {p_{{PS_{0} PS_{2i + 1} }} } p_{{PS_{2i + 1} PS_{2} }} + \sum\nolimits_{i \in B} {p_{{PS_{0} PS_{2i + 1} }} p_{{PS_{2i + 1} PS_{2i + 2} }} } + \sum\nolimits_{j \in C} {p_{{PS_{0} PS_{2n + 3j} }} }$$

$$p_{{PS_{2n + 3j} PS_{2} }} + \sum\nolimits_{j \in C} {p_{{PS_{0} PS_{2n + 3j} }} p_{{PS_{2n + 3j} PS_{2n + 3j + 1} }} + {\kern 1pt} \sum\nolimits_{i \in B} {p_{{PS_{0} PS_{2i + 1} }} p_{{PS_{2i + 1} PS_{1} }} } }$$

$$+ \sum\nolimits_{j \in C} {p_{{PS_{0} PS_{2n + 3j} }} } p_{{PS_{2n + 3j} PS_{2n + 3j + 2} }} {.}$$

The general equations which can be used to calculate the mean sojourn time $h_{{PS_{i} }}$ at system state $PS_{i}$ are give by Eqs. (33)–(40).

$$h_{{PS_{2n + 3j + 1} }} = \int_{0}^{\infty } {{(}1 - F_{{{\text{fdp}}j}} {(}t{))(}1 - F_{{{\text{dp}}j}} {(}t{))(}1 - F_{{{\text{mp}}j}} {(}t{))}d} t$$

(33)

$$h_{{PS_{2n + 3j} }} = \int_{0}^{\infty } {{(}1 - F_{{{\text{dp}}j}} {(}t{))(}1 - F_{{{\text{p}}j}} {(}t{))(}1 - F_{{{\text{fp}}j}} {(}t{))}d} t$$

(34)

$$h_{{PS_{2i + 2} }} = \int_{0}^{\infty } {{(}1 - F_{{{\text{fds}}i}} {(}t{))(}1 - F_{{{\text{ds}}i}} {(}t{))(}1 - F_{{{\text{ms}}i}} {(}t{))}d} t$$

(35)

$$h_{{PS_{2i + 1} }} = \int_{0}^{\infty } {{(}1 - F_{{{\text{ds}}i}} {(}t{))(}1 - F_{{{\text{fs}}i}} {(}t{))(}1 - F_{{{\text{s}}i}} {(}t{))}d} t$$

(36)

$$h_{{PS_{0} }} = \int_{0}^{\infty } {\prod\nolimits_{i \in B} {{(}1 - F_{{{\text{as}}i}} {(}t{))}} \prod\nolimits_{j \in C} {{(}1 - F_{{{\text{ap}}j}} {(}t{))}} d} t$$

(37)

$$h_{{PS_{2n + 3j + 2} }} = \int_{0}^{\infty } {(1 - F_{{{\text{rp}}j}} {(}t{))}dt}$$

(38)

$$h_{{PS_{2} }} = \int_{0}^{\infty } {(1 - F_{{{\text{SR}}}} {(}t{))}dt}$$

(39)

$$h_{{PS_{1} }} = \int_{0}^{\infty } {(1 - F_{{\text{R}}} {(}t{))}dt}$$

(40)

Appendix B

In this section, we give the detailed process for calculating the MTTF of SFC service mentioned in Sect. 3.5 of the main paper. The notations used in this appendix are defined in Table 3 of the main paper.

The equations of calculating the expected number of visits $V_{{PS_{i*} }}^{*}$ to system state $PS_{i*}$ until absorption are given by Eqs. (41)–(46),

$$\begin{aligned} V_{{S_{2n + 3j} }}^{*} & = - {(}p_{{PS_{0} PS_{2n + 3j - 2} }} p_{{PS_{2n + 3j - 2} PS_{2n + 3j - 1} }} p_{{PS_{2n + 3j - 1} PS_{2n + 3j} }} \\ & {{ + p_{{PS_{0} PS_{2n + 3j - 2} }} p_{{PS_{2n + 3j - 2} PS_{2n + 3j} }} {)}} \mathord{\left/ {\vphantom {{ + p_{{PS_{0} PS_{2n + 3j - 2} }} p_{{PS_{2n + 3j - 2} PS_{2n + 3j} }} {)}} M}} \right. \kern-\nulldelimiterspace} M} \end{aligned}$$

(41)

$$V_{{S_{2n + 3j - 1} }}^{*} = - {{p_{{PS_{0} PS_{2n + 3j - 2} }} p_{{PS_{2n + 3j - 2} PS_{2n + 3j - 1} }} } \mathord{\left/ {\vphantom {{p_{{PS_{0} PS_{2n + 3j - 2} }} p_{{PS_{2n + 3j - 2} PS_{2n + 3j - 1} }} } M}} \right. \kern-\nulldelimiterspace} M}$$

(42)

$$V_{{S_{2i} }}^{*} = {{ - p_{{PS_{0} PS_{2i - 1} }} p_{{PS_{2i - 1} PS_{2i} }} } \mathord{\left/ {\vphantom {{ - p_{{PS_{0} PS_{2i - 1} }} p_{{PS_{2i - 1} PS_{2i} }} } M}} \right. \kern-\nulldelimiterspace} M}$$

(43)

$$V_{{S_{2n + 3j - 2} }}^{*} = {{ - p_{{PS_{0} PS_{2n + 3j - 2} }} } \mathord{\left/ {\vphantom {{ - p_{{PS_{0} PS_{2n + 3j - 2} }} } M}} \right. \kern-\nulldelimiterspace} M}$$

(44)

$$V_{{S_{2i - 1} }}^{*} = - {{p_{{PS_{0} PS_{2i - 1} }} } \mathord{\left/ {\vphantom {{p_{{PS_{0} PS_{2i - 1} }} } M}} \right. \kern-\nulldelimiterspace} M}$$

(45)

$$V_{{S_{0} }}^{*} = {{ - 1} \mathord{\left/ {\vphantom {{ - 1} M}} \right. \kern-\nulldelimiterspace} M}$$

(46)

where $M = \sum\nolimits_{j \in C} {p_{{PS_{0} PS_{2n + 3j - 2} }} p_{{PS_{2n + 3j - 2} PS_{2n + 3j - 1} }} p_{{PS_{2n + 3j - 1} PS_{0} }} } - 1$$+ \sum\nolimits_{j \in C} {p_{{PS_{0} PS_{2n + 3j - 2} }} p_{{PS_{2n + 3j - 2} PS_{2n + 3j} }} p_{{PS_{2n + 3j} PS_{0} }} } { + }\sum\nolimits_{i \in B} {p_{{PS_{0} PS_{2i - 1} }} } p_{{PS_{2i - 1} PS_{2i} }}$$p_{{PS_{2i} PS_{0} }} + \sum\nolimits_{j \in C} {p_{{PS_{0} PS_{2n + 3j - 2} }} p_{{PS_{2n + 3j - 2} PS_{2n + 3j - 1} }} p_{{PS_{2n + 3j - 1} PS_{2n + 3j} }} } p_{{PS_{2n + 3j} PS_{0} }}$.