Process Analysis (Exponential Models)

Previously, we looked at how a production process with general inter-arrival and process times can be modelled based on inter-arrival and process time. MS Excel can be used to generate random numbers that are uniformly distributed between 0 and 1. In this section, we discuss how these random numbers can be transformed to follow other distributions and be used in conjunction with the model presented last week to simulate general production processes.

If inter-arrival time is exponentially distributed, the performance of the production process can be analyzed analytically (i.e., computed using formulas rather than via simulation). Here, we present these formulas and verify their validity through simulation.

At the end of this exercise, the student should be able to:

Generate general random numbers by sampling from a uniform distribution
Determine throughput and utilization of M/M/1 queues using formulas.
Determine the average production lead time, waiting time and WIP of M/M/1 queues using formulas.
Determine throughput and utilization of M/G/1 queues using formulas.
Determine the average production lead time, waiting time and WIP of M/G/1 queues using formulas.

Generating exponential random variables (8 questions)

Formulas for M/M/1 queues (10 questions)

Formulas for M/G/1 queues (10 questions)

Generating exponential random variables (8 questions)

Consider a random variable X such that: P(X=1) = 0.2, P(X=2) = 0.5 and P(X=3) = 0.3. Suppose Y is a random variable uniformly distributed between 0 and 1. A simple strategy to obtain X from Y is to sample from Y and then use the following conversion rule:

X=1 if 0 ≤ y < 0.2
X=2 if 0.2 ≤ y < 0.7
X=3 if 0.7 ≤ y ≤ 1

Mathematically, we can express this conversion rule as a function FX-1 (y), which is commonly referred to as the quantile function since it provides the value associated with the quantiles of the distribution as illustrated on the right.

The quantile function is particular useful because it allows us to generate the desired random numbers by sampling from a uniform distribution between 0 and 1, which is easily generated in commonly used software (e.g., MS Excel). For continuous probability distributions, the quantile function is simply the inverse of its cumulative distribution function (CDF). Hence, the quantile function is also referred to as the inverse cumulative distribution function.

Formulas for M/M/1 queues (10 questions)

Here, we consider a single workstation that receives and processes jobs. The inter-arrival time between jobs follows an exponential distribution with parameter λa. The process time of each job follows an exponential distribution with parameter λp.

This is commonly referred to as a M/M/1 queue. The “M” stands for Markov because the exponential distribution possesses the memoryless property. The “1” refers to the fact there is only one server.

The formulas for a M/M/1 queue where λa < λp are provided on the right.

For the questions in this section, suppose that an average of 5 jobs arrive each hour and it takes an average of 6 minutes to process each job.

Formulas for M/G/1 queues (10 questions)

In practice, the processing time of the workstation may not follow the exponential distribution. In this section, we consider the M/G/1 queue, where “G” implies that processing time follows a general distribution rather than the exponential distribution. As in the previous section, we assume that there is a single server and inter-arrival time between jobs follows an exponential distribution with parameter λa. In addition, we assume that the workstation has capacity λp and the variance of processing time is σ2

The formulas for a M/G/1 queue where λa < λp are provided on the right.

For the questions in this section, suppose that an average of 5 jobs arrive each hour and it takes an average of 6 minutes to process each job. Furthermore, the standard deviation of processing time is 3.464 minutes.

Process Analysis (Exponential Models)

Table of Contents

Generating exponential random variables (8 questions)

Formulas for M/M/1 queues (10 questions)

Formulas for M/G/1 queues (10 questions)