# Reliability estimation for a stochastic production system with finite buffer storage by a simulation approach

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## Abstract

This study develops a novel Monte Carlo simulation (MCS) approach to estimate system reliability for a stochastic production system with finite buffer storage. System reliability indicates the probability of all workstations providing sufficient capacities to satisfy a specified demand, as well as that all buffer stations are not running out of storage. First, buffer stations are modeled in a stochastic production network (SPN) model and their storage usage is analyzed based on the network-structured SPN. Second, an MCS is developed to generate the system state and to check the storage usage of buffer stations to determine whether the demand can be satisfied. After repeated simulations, the system reliability of the SPN can be estimated. Experimental results show that the proposed MCS approach is effective and efficient in estimating system reliability with reasonable quality for an SPN within a reasonable time. More importantly, system reliability will be overestimated with infinite buffer storage, and thus, it is worth studying finite buffer storage.

## Keywords

Mote Carlo simulation (MCS) Stochastic production network (SPN) Finite buffer storage System reliability## Abbreviation

- AOA
Activity-on-arc

- MCS
Monte Carlo simulation

- SPN
Stochastic production network

- WIP
Work-in-process

## List of symbols

**N**Set of nodes (inspection stations)

*n*Number of arcs (workstations)

- \(a_{i}\)
*i*th workstation**A**\(\{a_{i}|i = 1, 2, {\ldots }, n\}\): the set of workstations

- \(b_{i,i+1}\)
Buffer station installed after \(a_{i}\)

**B**{\(b_{i,i+1}|i\): a buffer station is installed after \(a_{i}\)}: the set of buffer stations

*r*,*k*Defective WIP output from the

*r*th workstation is reworked starting from the previous*k*workstations- \(\varGamma ^{(\mathrm{G})}\)
General processing route

- \(\varGamma ^{(\mathrm{R}|r,r-k)}\)
Rework route

- \(f_i^{({\mathrm{G}})}\)
Input flow for \(a_{i} \in \varGamma ^{({\mathrm{G}})}\)

- \(f_i^{({\mathrm{R}}|r,r-k)}\)
Input flow for \(a_{i} \in \varGamma ^{(\mathrm{R}|r,r-k)}\)

- \(l_{i}\)
Loading of \(a_{i}\) and \(l_{i}=f_i^{({\mathrm{G}})} +f_i^{({\mathrm{R}}|r,r-k)}\)

- \(M_{i}\)
Maximal capacity of \(a_{i}\)

- \(x_{i}\)
Capacity of each workstation \(a_{i}\)

- \(x_{i(\alpha )}\)
\(\alpha \)th possible capacity of \(a_{i}\), where \(\alpha = 1, 2, {\ldots }, c_{i}\)

- \(\pi _{i(\alpha )}\)
Pr(\(x_{i}=x_{i(\alpha )}\))

- \(p_{i}\)
Expected success rate of \(a_{i}\)

- \({{\varvec{\Delta }}}_{c_i \times c_{i+1} }\)
Matrix for different amount under all possible values of \(x_{i}\) and \(x_{i+1}\)

- \({{\varvec{\Lambda }}}_{c_i \times c_{i+1} }\)
Corresponding probability distribution matrix for \({{\varvec{\Delta }}}_{c_i \times c_{i+1} }\)

*D*Demand

- \(R_D^b\)
Reliability for finite buffer storage for

*D*

## Notes

### Funding

This work was supported by the Ministry of Science and Technology, Taiwan, Republic of China (Grant Number MOST 103-2218-E-507-001-MY3).

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