Sample-Path Analysis of Queueing Systems
Contents
1. Introduction and Overview
- Introduction
- Elementary Properties of Point Processes:
Y =
X
- Little's Formula:
L = W
- Stability and Imbedded Properties of Input-Output Systems
- Busy-Period Analysis
- Conditional Properties of Queues
- Comments and References
2. Background and Fundamental Results
- Introduction
- Background on Point Processes:
Y = X
- Cumulative Processes
- Rate-Conservation Law
- Fundamental Lemma of Maxima
- Time-Averages and Asymptotic Frequency Distributions
- Comments and References
3. Processes with General State Space
- Introduction
- Relations between Frequencies for a Process with an Imbedded Point Process
- Applications to the G/G/1 Queue
- Relations between Frequencies for a Process with an Imbedded Cumulative Process (Fluid Model)
- Martingale ASTA
- Comments and References
4. Processes with Countable State Space
- Introduction
- Basic Relations
- Networks of Queues: The Arrival Theorem
- One-Dimensional Input-Output Systems
- Applications to Stochastic Models
- Relation to Operational Analysis
- Comments and References
5. Sample-Path Stability
- Introduction
- Characterization of Stability
- Rate Stability for Multiserver Models
- Rate Stability for Single-Server Models
-
-Rate Stability
- Comments and References
6. Little's Formula and Extensions
- Introduction
- Little's Formula:
L = W
- Little's Formula for Stable Queues
- Generalization of Little's Formula:
H = G
- Fluid Version of Little's Formula
- Fluid Version of
H = G
- Generalization of
H = G
- Applications to Stochastic Models
- Comments and References
7. Insensitivity of Queueing Networks
- Introduction
- Preliminary Result
- Definitions and Assumptions
- Infinite Server Model
- Erlang Loss Model
- Round Robin Model
- Comments and References
8. Sample-Path Approach to Palm Calculus
- Introduction
- Two Basic Results
- Extended Results
- Relation to Stochastic Models
- Comments and References
Appendices
A. Ergodic Theory and Random Marked Point Processes
- Introduction
- Strong Law of Large Numbers
- The Ergodic Theorem in Discrete Time
- The Ergodic Theorem in Continuous Time
- Stationary Marked Point Processes
- Comments and References
B. Limit Theorems for Markov and Regenerative Processes
- Markov Processes
- Regenerative Processes
C. Stability in Stochastic Models
- Introduction
- Markov Processes
- Regenerative Processes
- Stationary Processes
- Other Models and Definitions of Stability