### Analysis of Bernoulli Lines [chapter]

Jingshan Li, Semyon M. Meerkov
2008 Production Systems Engineering
Motivation: The problem of performance analysis of production systems consists of investigating their performance measures, e.g., production rate (P R), work-in-process (W IP i ), blockages (BL i ) and starvations (ST i ), as functions of machine and buffer parameters. In principle, this investigation can be carried out using computer simulations. However, the simulation approach has two drawbacks. First, it is not conducive to analysis of fundamental properties of production systems, e.g.,
more » ... tionships between system parameters and performance measures. Second, simulations require a relatively lengthy and costly process of developing a computer model and its multiple runs for statistical evaluation of the performance measures. In some cases, especially when numerous "what if" scenarios must be analyzed, this approach may become prohibitively expensive and slow. This problem is exacerbated by the exponential explosion of the dimensionality of the system as a function of buffer capacity. Indeed, even in the Bernoulli reliability case (i.e., when the machines are memoryless), a serial line with, say, 11 machines and buffers of capacity 9, has 10 10 states, which is overwhelming for simulations. Therefore, a quick, easy and revealing method for production systems analysis, based on formulas, rather than on simulations, is of importance. The purpose of this chapter is to present such a method for serial production lines with Bernoulli machines along with describing system-theoretic properties of these systems. Overview: The analytical approach to calculating P R, W IP i , BL i and ST i is based on the mathematical models of production systems discussed in Chapter 3. Due to the complex nature of interactions among the machines, closed-form expressions for their performance measures are all but impossible to derive, except for the case of systems with two machines. Therefore, the approach, developed here, is based on a two-stage procedure: First, analytical formulas for performance analysis of two-machine lines are derived and, second, an aggregation procedure is developed, which reduces longer systems to a set of coupled two-machine lines and recursively evaluates their performance characteristics. This approach, illustrated in Figure 4.1, leads to sufficiently accurate estimates 123 124 CHAPTER 4. ANALYSIS OF BERNOULLI LINES of performance measures P R, W IP i , BL i and ST i . PR WIP i