Software for Queueing Analysis

Chapter 8 gives a brief discussion of computer simulation for discrete events. The chapter lists software programs in the technical literature that outline programs for the simulation of discrete events, both of commercial origin and free programs. In addition to the lists submitted, the authors present specialized packages for analysis and simulation of waiting lines in the R language. Statistical considerations are presented, which must be taken into account when obtaining data from simulations in situations of waiting lines. Chapter 8 presents three packages of the statistical program R: the “queueing” analysis package provides versatile tools for analysis of birth- and death-based Markovian queueing models and single and multiclass product-form queueing networks; “simmer” package is a process-oriented and trajectory-based discrete-event simulation (DES) package for R; and, the purpose of the “queuecomputer” package is to calculate, deterministically, the outputs of a queueing network, given the arrival and service times of all the customers. It also uses simulation for the implementation of a method for the calculation of queues with arbitrary arrival and service times. For each theme, the authors show the use of the packages in R.

Special Topics in Queueing Theory

Topics covered in Chapter 7 are priority systems with preemptive or non-preemptive system, systems with N classes of customers, customers in groups: bulk arrivals, batch service, balking and reneging, and finite population. In a priority system, it is assumed that there are 1, 2, 3, …, N different classes or types of customers, where Type 1 customers are the most important while class N ones are the least important. When a server is available to serve a customer from the queue, the one with the highest priority level will go to the server to start their service process. In batch service, before starting the service process, a group or batch needs to be formed with a certain number of customers.

Systems With Limited Capacity

In factories it is common that there are a limited number of spaces for the pieces; similarly, waiting rooms in a hospital can accommodate a limited number of people. Chapter 6 is dedicated to multi-stage systems where the stations have limited space for the queue; the chapter begins with the M/M/1/K systems and the calculation of its performance measures; the following is the bowl phenomenon and its implications in the efficiency of a system; then the M/G/1/K systems and the approaches developed to estimate the performance of this class of systems are presented. The chapter ends with the presentation of some optimization problems related to the M/M/c/K and M/G/c/K systems. Several codes are proposed in Scilab Language to perform calculations automatically.

Open Queueing Networks

Systems often have two or more stages and a customer must go several stages before finalizing their service. The systems can be arranged in series or in network. Chapter 5 is dedicated specifically to the performance analysis of systems that have several stages both in series and those that have network arrangement; the theorems of Burke and Jackson are presented; the calculations of the flow and variability in a network and of the measures of performance (cycle time and work in process) are also exposed. Several codes are proposed in Scilab Language to perform calculations automatically. The chapter ends with a section devoted to identifying and analyzing the bottleneck in a system from a cost approach.

Single Stage Systems Markovian Case

In Chapter 3, the authors show the expressions of queueing theory for Markovian systems with a single stage. The chapter begins with definitions of stochastic processes and Markov chains; then; they present the models for calculating the work in process and the cycle time of systems with a single server, multiple servers, and systems with restriction on queue size. Later, the chapter explores heuristic rules to estimate the capacity of a system. The chapter ends with the monetary analysis of the system and the optimum selection of capacity.

Review of Probability Distributions

In Chapter 2, probability distributions are presented; the distributions exposed are those with more relation to the analysis and study of waiting lines; discrete distributions: binomial, geometric, Poisson; continuous distributions: uniform, exponential, erlang, and normal. Confidence intervals are calculated for some of the parameters of the distributions. A brief example of the generation of pseudorandom exponential times using a spreadsheet is presented. The chapter closes with the goodness-of-fit tests of probability distributions, especially the Anderson-Darling test. The statistical language of programming R is used in the exercises performed. Several codes are proposed in R Language to perform calculations automatically.

Single-Stage Non-Markovian Systems and Variability

The variability of a queueing system is the main theme of Chapter 4; some of the approaches that have been developed to obtain the performance measures are presented. Part of the material is devoted to variability due to equipment failures and their effect on performance measures, as well as in estimating the required capacity. The chapter ends with the topic on propagation of variability in a queue and the respective approximations to calculate the variability at the exit.

Quantitative Analysis

The first chapter presents concepts about quantitative models for decision making; the process that follows is used when constructing a model and definitions of queueing theory: work in process, cycle time, and congestion. Also presented and explained is the Kendall notation that will be used throughout the book. The chapter shows some examples of waiting line systems at the end.