This paper presents an efficient exact analytical method for evaluating the performance of a twomachine manufacturing system with a finite buffer. Unlike existing work, it is assumed that the buffer is prone to timedependent failure, that is, failure that can occur even when the buffer is not working. First, Markov model is established for the system. Then transition equations are derived based on the system state analysis. After that, a solution technique is provided to obtain the results. Finally, numerical cases are carried out to explore the internal laws of the system. The relationships between system parameters and system performance are investigated. Furthermore, the difference between buffer subject to timedependent failure and buffer subject to operationdependent failure is discussed. The proposed method is the building block of approximate analytical methods which can greatly improve the accuracy when analyzing long systems.
In an ideal manufacturing system, the machines have the same cycle time and never break down. Parts flow through the system fluently and steadily. However, in real systems this ideal manufacturing process could be disrupted by the fluctuation of cycle times or machine failures. For instance, machines may vary a lot in their types and thus have different cycle times; the cycle time of a manual operation is not deterministic and may obey some random distribution; machines may be prone to failure and need to stop for maintenance. All the situations above can penalize the productivity of the system and increase losses in availability for the whole plant [
In order to cope with these situations, buffers are placed between machines which can highly improve the efficiency of the system. As a typical type of buffer, accumulating conveyor has been widely used in actual manufacturing systems especially in automated production lines. It provides storage and buffering between two consecutive machines. In practice, accumulating conveyors have finite buffer capacities. Since accumulating conveyors are mechanical devices, they are prone to failure. There are two types of failures, referred to as operationdependent failure and timedependent failure [
When configuring a system with buffers like accumulating conveyors, how to determine the parameters of those buffers is a complex problem. To solve the problem, performance evaluation methods which can identify the relationships between system parameters and system performance have been developed as powerful decision support tools when designing or optimizing systems.
Lots of work has been devoted to the research of performance evaluation technologies. Generally, it can be mainly classified into simulation methods and analytical methods. Simulation methods can model a system at any required level of detail and provide accurate results. But they are too time consuming to be used under some circumstances, for instance, the early design stage where lots of alternative configurations need to be evaluated. On the contrary, analytical methods are much more efficient. They can evaluate systems in a short time. Although the results are approximate, they are accurate enough at the early design stage of manufacturing systems because there is often significant inaccuracy in the input data [
Even though extensive research has been done on exact analytical methods, most of it has been focused on studying machine behaviour. By using an exact method based on discrete Markov model, Buzacott [
Buffer failures affect system performance significantly. Lipset et al. [
The purpose of this paper is to develop an exact analytical method for twomachine systems with buffers like accumulating conveyors by using Markov model. Based on this method, large systems with buffers subject to timedependent failure can be evaluated more accurately.
The remaining part of the paper is organized as follows. A detailed formulation of the problem is shown in Section
The twomachine manufacturing system studied in this paper consists of machine
The following assumptions are made:
The system is homogeneous which means the cycle times of machine
Both of the machines are unreliable. The failures are operationdependent. The times between failures and the times to repair of machine
Buffer
Machine
Machine
Let
To be noticed, the fact that machine
Let
According to the assumptions, if machine
Similarly, we have
For a buffer subject to operationdependent failure, it cannot fail when there is no part in it. On the contrary, as a buffer subject to timedependent failure, buffer
As the machines are prone to operationdependent failure, they can only fail when they are working. According to that, we have
As the machines cannot fail when the buffer breaks down, we have
Let
Consequently, the buffer level at time
The most important performance measures of a manufacturing system are the production rate and the average buffer level. The production rate
Because flow in a steady manufacturing system is conserved [
Let
A transient state in Markov process has no possible predecessor except itself or another transient state. The steady probability of a transient state is zero. Removing all transient states before analyzing the system can simplify the work.
The state
The state
The states
The state
The state
Similarly, it can be deduced that
After eliminating the transient states, the relationships between the steady states are analyzed. According to the number of parts in the buffer, these steady states are classified into 3 categories:
First, the internal state transitions are shown in Figure
Internal state transitions.
Then, the lower boundary state transitions are shown in Figure
Lower boundary state transitions.
Finally, Figure
Upper boundary state transitions.
First, the internal equations are solved. Markov process is a birthdeath process where the state transitions are of only two types: “births,” which increase the state variable by 1, and “deaths,” which decrease the state by 1. When deriving the transition equations, it can be observed that the buffer level can increase or decrease only by 1 between two consecutive time units. From Figure
By substituting
Equations (
For the equation set with buffer level
By substituting (
Let
From this equation, it should be noticed that
Since
Now we show the solution of the boundary equations. As shown in Figure
According to (
Similarly, for the upper boundary states,
The probabilities of other upper boundary states can be given according to (
Finally, these equations have to satisfy the normalization equation
The unknown constant
The developed method is validated in this section. In Section
Three representative cases in [
Configurations of cases.
Case 
Case 
Case 



0.005  0.001  0.005 

0.05  0.05  0.05 

0.001  0.005  0.005 

0.05  0.05  0.05 

Varied  Varied  Varied 

0.008  0.008  0.008 
Figures
Production rate for different combinations of buffer capacity and isolated efficiency (Case
Production rate for different combinations of buffer capacity and isolated efficiency (Case
Production rate for different combinations of buffer capacity and isolated efficiency (Case
Figures
Average buffer level for different combinations of buffer capacity and isolated efficiency (Case
Average buffer level for different combinations of buffer capacity and isolated efficiency (Case
Average buffer level for different combinations of buffer capacity and isolated efficiency (Case
The results above also prove the reversibility of manufacturing systems, a phenomenon discovered by Ammar and Gershwin [
A case is used to compare the two different types of buffers. The failure rates of machine
Comparison of the production rates generated by two different types of buffer.
Comparison of the average buffer levels generated by two different types of buffer.
In this paper, a twomachine manufacturing system with a finite buffer subject to timedependent failure is analyzed. The representative of this type of buffers is accumulating conveyor which is widely used in reality. As far as we know, only one method has been proposed to study the behaviour of this type of buffers. However, it is an approximate one. The main contribution of this paper is the development of an exact analytical method. A Markov model is established for the system. Then transition equations are derived and solved. Numerical experiments are implemented to investigate the relationships between buffer parameters and system performance. Also, the difference between buffer subject to timedependent failure and buffer subject to operationdependent failure is discussed. As the building block of approximate analytical methods for the analysis of large systems, the proposed method is of great significance.
Future work would be devoted to proposing a decomposition method for evaluating large systems and developing an exact analytical method for the analysis of inhomogeneous twomachine systems based on continuous Markov modelling techniques.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This project is supported by National Natural Science Foundation of China (Grant nos. 71401098 and 51205242).