An Introduction to Probability Theory and its Applications, Volume I

Power grids, flexible manufacturing, cellular communications: interconnectedness has consequences. This remarkable book gives the tools and philosophy you need to build network models detailed enough to capture essential dynamics but simple enough to expose the structure of effective control solutions and to clarify analysis. Core chapters assume only exposure to stochastic processes and linear algebra at the undergraduate level; later chapters are for advanced graduate students and researchers/practitioners. This gradual development bridges classical theory with the state-of-the-art. The workload model that is the basis of traditional analysis of the single queue becomes a foundation for workload relaxations used in the treatment of complex networks. Lyapunov functions and dynamic programming equations lead to the celebrated MaxWeight policy along with many generalizations. Other topics include methods for synthesizing hedging and safety stocks, stability theory for networks, and techniques for accelerated simulation. Examples and figures throughout make ideas concrete. Solutions to end-of-chapter exercises available on a companion website.

http://www.meyn.ece.ufl.edu/archive/spm_files/Presentations/Atlanta05.pdf

Control Techniques for Complex Networks

Successful development of cloud computing paradigm necessitates accurate performance evaluation of cloud data centers. As exact modeling of cloud centers is not feasible due to the nature of cloud centers and diversity of user requests, we describe a novel approximate analytical model for performance evaluation of cloud server farms and solve it to obtain accurate estimation of the complete probability distribution of the request response time and other important performance indicators. The model allows cloud operators to determine the relationship between the number of servers and input buffer size, on one side, and the performance indicators such as mean number of tasks in the system, blocking probability, and probability that a task will obtain immediate service, on the other.

Performance Analysis of Cloud Computing Centers Using M/G/m/m+r Queuing Systems

Queueing models can usefully represent production systems experiencing congestion due to irregular flows, but exact analyses of these queueing models can be difficult. Thus it is natural to seek relatively simple approximations that are suitably accurate for engineering purposes. Here approximations for a basic queueing model are developed and evaluated. The model is the GI/G/m queue, which has m identical servers in parallel, unlimited waiting room, and the first-come first-served queue discipline, with service and interarrival times coming from independent sequences of independent and identically distributed random variables with general distributions. The approximations depend on the general interarrival-time and service-time distributions only through their first two moments. The main focus is on the expected waiting time and the probability of having to wait before beginning service, but approximations are also developed for other congestion measures, including the entire distributions of waiting time, queue-length and number in system. These relatively simple approximations are useful supplements to algorithms for computing the exact values that have been developed in recent years. The simple approximations can serve as starting points for developing approximations for more complicated systems for which exact solutions are not yet available. These approximations are especially useful for incorporating GI/G/m models in larger models, such as queueing networks, wherein the approximations can be components of rapid modeling tools.

APPROXIMATIONS FOR THE GI/G/m QUEUE

1. Introduction to the Modeling of Manufacturing Systems.- 1.1. Typical Decision Problems in Manufacturing Systems.- 1.2. Performance Evaluation.- 1.3. Models of Manufacturing Systems.- 1.4. Design of Manufacturing Systems.- References.- 2. Tools of Probability.- 2.1. Random Variables and Their Distributions.- 2.2. Joint Distributions.- 2.3. Conditional Distributions.- 2.4. Functions of Random Variables.- 2.5. Moments of Random Variables.- 2.6. Special Distributions.- 2.7. Stochastic Processes.- 2.8. Phase-Type Distributions.- Related Bibliography on Phase-Type Distributions.- References.- Problems.- 3. Analysis of Single Work Stations.- 3.1. Introduction.- 3.2. A Simple Work Station.- 3.3. Distribution of the Number in the System.- 3.4. Work Stations Subject to Failures.- 3.5. Distribution of the Number of Jobs in a Work Station Subject to Failures.- 3.6. The Process-Completion Time.- 3.7. Failure-Repair Policies.- 3.8. The Machine Interference Problem.- Related Literature.- References.- Problems.- 4. Analysis of Flow Lines.- 4.1. Introduction.- 4.2. Characteristics of Flow Lines.- 4.3. Analysis of Two-Station Flow Lines.- 4.4. Analysis of Flow Lines With More Than Two Stations.- 4.5. Deterministic Processing Times.- 4.6. Bounds on the Output Rate.- 4.7. Design Problems in Flow Lines.- Related Literature.- References.- Problems.- 5. Analysis of Transfer Lines.- 5.1. Introduction.- 5.2. A Two-Station Line with No Intermediate Buffer: A Regenerative Approach.- 5.3. Transfer Lines with Work-in-Process Buffers.- 5.4. Buffer Clearing Policies in Transfer Lines.- 5.5. Analysis of a Two-Machine Synchronous Line Under BCP1.- 5.6. Operation Under BCP2.- 5.7. Line Behavior and Design Issues.- 5.8. Analysis of Larger Systems.- 5.9. Transfer Lines with No Scrapping.- Related Bibliography.- References.- Problems.- 6. Assembly Systems.- 6.1. Introduction.- 6.2. An Asynchronous Assembly System.- 6.3. Approximate Analysis of the Two-Station Assembly System.- 6.4. Analysis of Larger Systems.- 6.5. Synchronous Assembly Systems.- 6.6. Relaxing the Assembly Assumption.- Related Bibliography.- References.- Problems.- 7. Pull-Type Manufacturing Systems.- 7.1. Introduction.- 7.2. Production Control Policies.- 7.3. Analysis of Single-Stage, Single-Product Systems.- 7.4. A Single-Stage System with Two Products.- 7.5. Multistage Pull-Type Systems.- 7.6. The Generalized Kanban Scheme.- 7.7. Manufacturing Systems with Inventory Procurement.- 7.8. A Two-Stage, MultiProduct System.- 7.9. The Look-Back Policy.- Related Bibliography.- References.- Problems.- Appendix: A Single-Stage P/I System with PH Delay, Set-up, and Processing Times.

Performance analysis of manufacturing systems

This paper provides a unifying method of generating and/or evaluating approximations for the principal congestion measures in aGI/G/s queueing system. The main focus is on the mean waiting time, but approximations are also developed for the queue-length distribution, the waiting-time distribution and the delay probability for the Poisson arrival case. The approximations have closed forms that combine analytical solutions of simpler systems, and hence they are referred to as system-interpolation approximations or, simply, system interpolations. The method in this paper is consistent with and generalizes system interpolations previously presented for the mean waiting time in theGI/G/s queue.

Approximations for multi-server queues: System interpolations

This paper deals with the M/G/m queueing system in the steady-state. Applying the method of diffusion approximation, we provide approximate formulas for the distributions of the number of customers, the waiting time and the busy period. We also discuss the relation between the heavy traffic approximation and these approximations. In order to examine the accuracy of the approximate formulas, they are numerically tested on some examples.

Diffusion Approximation for an M/G/m Queue

This paper develops a transform-free approximation for the steady-state queue-length distribution in an M/G/s queue with finite waiting spaces. The approximation is obtained by using a conservation law and some heuristics. It is shown that the approximation is exact for the cases with either no extra waiting space, exponential service-time distribution, or a certain two-parameter family of service-time distributions. It is also shown that the approximation has the same light-traffic properties as the known light-traffic limit theorem for the infinite capacity case when the number of waiting spaces is not less than one.

A Transform-Free Approximation for the Finite Capacity M/G/s Queue

Optimal control of an M/G/1 queuing system with removable server via diffusion approximation

We provide a simple two-moment approximation formula for the mean waiting time in a GI/G/s queue. This formula has the form of a certain combination of the exact mean waiting times for the D/M/s, M/D/s and M/M/s queues, and hence it can be easily calculated by using some queueing tables. The quality of the approximation is tested by comparing it with known solutions in particular cases.

Toshikazu Kimura

Papers

Approximations for multi-server queues: System interpolations

Diffusion Approximation for an M/G/m Queue

A Transform-Free Approximation for the Finite Capacity M/G/s Queue

Optimal control of an M/G/1 queuing system with removable server via diffusion approximation

A two-moment approximation for the mean waiting time in the GI/G/s queue