Probability, stochastic processes, and queueing theory: the mathe-matics of computer performance modeling
Nelson (2013) published a book with the sole aim of it serving as both a reference ma-terial and a teaching tool to better understand mathematics and especially the topic on probability. The book is organized into texts that are well-structured and flows freely and has an easy access to tables both in the main body and appendices. The researcher and student are provided with a notation list throughout the book in a bid to easy navigation throughout the chapters. The book is fundamentally covered in two parts with part 1 covering probability in chapters 2 to 5 while stochastic processes are covered by chapter 6 to 10 that can be found in part 2 of the book. Both the reference and review materials and an in-depth analysis outside the main text scope are well covered.
The principles of probability and notions of randomness have been adversely men-tioned throughout Chapter 2 of the text citing the service times of customers and advents to computer systems. It commences with some simple question: “What does probability of a random event mean?” Nelson answers this question in the preceding appendix A in the chapter with probability being at the center stage via two approaches. Commonsense observations form the basis of the first approach; whereby the statistical average is used to calculate the probability of an event through an estimation course. This approach is referred to as ‘Frequen-cy-based’ of probability. The second approach is the one referred to as the ‘axiomatic ap-proach’ that is considered a modern formula to solving probability queries. The greatest ele-mentary formula of probability ascends once all measures are equiprobable and the main aim being to ascertain the probable occurrence of some events.
Random variables (variables that require randomized experiments to determine their values) are well illustrated and explained in chapter 4 of the book. They are called variables because of the way they are expressed. For instance, 3X and X2 are expressions referred to as variables because they represent unknown objects. The known values of these random vari-ables can only be ascertained probabilistically.
The theory of Queueing as defined by Nelson is the analysis of systems in a waiting line. ‘A waiting line of consumers seeking services from a server makes up a queue.’ It forms whenever the demand surpasses the supply owing to an engaged server. The queuing system has three main characteristics; in plus/arrival to the system, the waiting line or the queue and the server facility. The mathematical queuing theory approach examines how waiting lines are structured with a view of fixing the same. The theory was developed as a model by Agner Krarup in his quest to describe the telephone exchange conversation in Copenhagen. Business decisions are always made based on the results from operational research emanating from queueing theories. This idea is applicable in traffic engineering, computing, design factories and telecommunications.
A classified bibliography of research on optimal design and control of queues
There are unprecedented circumstances of uncertainty whereby decision making becomes a hitch to service care providers due to pending ques for service. Many firms are con-fronted with critical choices of choosing whom to serve first and at what juncture. For this reason, Crabill et al. 2007 provides a solution of effective management of servers to attend to customers’ needs effectively and efficiently. In instances where a firm has only less than two servers, it becomes a tough decision making journey for queues to be developed to serve the arriving customers. Consequently, the book gives a clear insight on how firms can optimally assign servers and route consumers to queues. Clients arrive at unspecified time and in ran-dom numbers and hence the book has models that review how best to address this uncertain-ties.
The second section of the book focuses on the pricing issue for any product innovated by the firm. Customers may be reluctant to purchase the product owing to its uniqueness and uniqueness in the market. This forces the marketing of the specified product a headache for the marketing personnel’s. The book however, provides a shoulder to lean on as it analyzes marketing strategies involving product promotions and auctions in a bid to make valuations to consumers. Once adequate quantities of the new product are sold out, the firm may be chal-lenged decision wise on when it may be appropriate to halt these promotions. The firm’s stopping time is well calculated by the authors who forecast and advice the firms on when to update the data input upon receiving valuations from the customers by way of variables. There is a symmetric equilibrium attained at the end to the text where the authors provide a model waiting line that will make it possible or consumers to have an ample time devoid of delays while embarking of their shopping escapades.
A queuing network model for the management of berth crane operations
Canonaco et al. on this journal entry brings forth the operations undertaken at berthing points with optimal management of container loading/discharge inside a terminal in real time. The authors suggests that when berth cranes that are mounted on rail create instances where expensive resources require maximization productivity, the vessels should be matched as a requirement bid to minimizing waiting times with sufficient rate of service accomplishment. A queuing model network is proposed in a bid to address the applied problem. The model solution can be approached fittingly by way of distinct-event simulation owing to its com-plexity. The use of simulator design has made it possible to attain a systematic representation of real policies and constraints of activity scheduling and resource allocations by exploiting a methodology referred to as ‘Event graph (EG)-centered methodology.’
The operations manager in the respective plant/machine may issue alternate policies to be introduced in an appropriate queuing network model through a board-like view that is to be matched by way of simulation. Throughput and completion time would hence be easily evaluated for all berth cranes on average measures. Mathematical experiments for simulator authentication besides real data are promising. Canonaco et al. 2008 goes further to acclaim that some resolutions overlap berth cranes to carrier assignments and hold sequencing and as-signment onto the similar crane and this can be enhanced by use of the projected simulation tool that is manager-approachable.
Queueing theory in manufacturing systems analysis and design: A classification of models for production and transfer lines
The viability and prosperity of any given business depends on its market performance and a well-structured mathematical accounting option of their expenditure. Papadopoulos et al. 2006 in their journal on waiting lines and queuing theory that affect operational research draws a picture of how a business might be affected if proper mechanisms are not in place to address the impending issues. The book provides solutions on how well a business could im-prove its manufacturing and service delivery sector through proper decision making with the question of queuing (waiting line) having been properly addressed. There are various assump-tions that are to be used when developing the theory in addition to ways of applying caveats when and if adopting the theory of waiting line. The importance of variability accounting in waiting line is vital since queuing equations are basic and provide data that is of typical en-actment under conditions that are steady.
Considering the viability amount that exists for a given scenario in a waiting line is critical as it gives perceptions needed to lower the effects and cultivate solutions that are able to enhance service delivery and reduce the operational costs. The mathematical tone of the book can be generalized as one that is attentive to formulas on the application as opposed to the deviation part. There are exceptions on deviations (use of random number approximations and state diagrams of probability disseminations) that are rarely provided in any book but which are present in the book analysis of simulation models. As a way of easing the grasping of materials existing in the book, there are practically illustrated examples that are specified at suitable points with few simulation methodologies in the text by use of customized spread-sheet software. This makes it easy for the reader to follow and understand the concepts and models of queuing.
The application of waiting line theory to industrial problems
Everyone is familiar of what a waiting line (Queue) entails. Congestion is said to be a contributing phenomenon to developing queues making our lives substantially strenuous as we are forced to wait for long. Hillier, (2004) penned an industrial engineering journal ad-dressing industrial problems caused by waiting line theory. A total of about 909 bibliograph-ical references are available as contributions made by renowned scholars on the subject matter as part of the book dated back to about fifty five years ago. The overpopulated world essen-tially experiences the theory of queuing and the upcoming generations have to be arithmeti-cally familiarized with these undue delays that they are prone to encounter at one point or the other in life.
The paper grants a categorized catalogue structured into six subdivisions— dynamic controller models, statistical design models, prototypes encompassing queue discipline con-trol, survey papers and tutorials, miscellaneous models, and books having detailed portions fanatical to design and controller. Some of the exceeding classifications are supplementary subdivided when appropriate, and descriptions of the categories of models in each grouping and subsection herald the authentic gradient of references. Credentials that descend into sup-plementary categories are referenced into cross-sections. The book is informative as well as engaging to the reader matters as it provides insights on experimental designs with matters queuing. There are consulting dialogs presented by the authors in an enjoyable and entertain-ing manner with the technical aspects brilliantly presented. As the saying goes, “Design for the experiments, do not experiment for the designs.”
The book illustrates optimal designed that are tailor-made for mathematicians to em-ploy effectively in solving clientele real issues using computerized enabled software. This can also be applied in industries to address real issues on; ways of screening queuing options with a dozen factors that need investigations, what needs to be done in case one has more than three variables to be run in a day?, what analysis and experimental design can be undertaken in situations where a factor can only be changed ones in the course of the study? In case there is a mixture of ingredients, what criteria am I supposed to use to incorporate other elements? How best can I manage the time used in warming up the experiments prior to the study con-clusion without interfering with other tentative procedures? When presented with categorical factors, is it really possible for me to effectively account for the cost of the experiment? In my quest to resolve ambiguities in circumstances when my experiment is botched, what do I do to add runs to the procedure? The book answers these questions exhaustively and with precise examples that compare and evaluate queuing models. Researchers have unlimited op-tions of making trade-off that are sensible between the amount of information obtained and the cost of experimentation.
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