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E-raamat: Applied Discrete-Time Queues

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  • Ilmumisaeg: 26-Dec-2015
  • Kirjastus: Springer-Verlag New York Inc.
  • Keel: eng
  • ISBN-13: 9781493934201
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Applied Discrete-Time QueuesSuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 26-Dec-2015
  • Kirjastus: Springer-Verlag New York Inc.
  • Keel: eng
  • ISBN-13: 9781493934201
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This book introduces the theoretical fundamentals for modeling queues in discrete-time, and the basic procedures for developing queuing models in discrete-time. There is a focus on applications in modern telecommunication systems.

It presents how most queueing models in discrete-time can be set up as discrete-time Markov chains. Techniques such as matrix-analytic methods (MAM) that can used to analyze the resulting Markov chains are included. This book covers single node systems, tandem system and queueing networks. It shows how queues with time-varying parameters can be analyzed, and illustrates numerical issues associated with computations for the discrete-time queueing systems. Optimal control of queues is also covered.

Applied Discrete-Time Queues targets researchers, advanced-level students and analysts in the field of telecommunication networks. It is suitable as a reference book and can also be used as a secondary text book in computer engineering and computer science.  Examples and exer

cises are included.

1 Introduction
1(22)
1.1 Introduction
1(6)
1.1.1 Examples of Transparent Queues
2(3)
1.1.2 Examples of Non-Transparent Queues
5(1)
1.1.3 Examples of Queues Not Involving Humans Directly
6(1)
1.2 A single node queue
7(2)
1.3 A tandem queueing systems
9(1)
1.4 A network system of queues
10(1)
1.5 Some Well-known Application Areas for Queueing Models
11(5)
1.5.1 Queues in Transportation and Traffic Systems
11(1)
1.5.2 Queues in Manufacturing Systems
12(1)
1.5.3 Queues in the Health Care Systems
13(1)
1.5.4 Queues in Communication networks
14(2)
1.6 Why are queueing systems of interest?
16(1)
1.7 Performance Measures
17(2)
1.7.1 Performance Measures
18(1)
1.8 Characterization of Queues
19(1)
1.9 Discrete time vs Continuous time analyses
20(3)
1.9.1 Time
20(2)
References
22(1)
2 Arrival and Service Processes
23(34)
2.1 Introduction
23(1)
2.2 Review of Probability for Discrete Random Variables and Matrices
23(8)
2.2.1 The z transform
24(1)
2.2.2 Bivariate Cases
24(1)
2.2.3 Some very Common Discrete Distributions
25(3)
2.2.4 Brief Summary of Required Material from Matrices
28(1)
2.2.5 Examples of Simple Representations of Discrete Distributions Using Matrices
29(2)
2.2.6 Matrix Representation
31(1)
2.3 Arrival and Service Processes
31(1)
2.4 Renewal Process
32(2)
2.4.1 Example
33(1)
2.4.2 Number of renewals
34(1)
2.5 Special Arrival and Service Processes in Discrete Time
34(19)
2.5.1 Bernoulli Process
34(1)
2.5.2 Geometric Distribution
35(2)
2.5.3 Phase Type Distribution
37(7)
2.5.4 The infinite phase distribution (IPH)
44(1)
2.5.5 General Inter-event Times
45(1)
2.5.6 Markovian Arrival Process
46(5)
2.5.7 Marked Markovian Arrival Process
51(1)
2.5.8 Semi Markov Processes
51(1)
2.5.9 Data Fitting for PH and MAP
52(1)
2.6 Service Times: What does this really mean?
53(1)
2.7 Problems
53(4)
References
54(3)
3 Discrete-Time Markov Chains
57(34)
3.1 Introduction
57(1)
3.2 Stochastic Processes
57(2)
3.3 Markov Processes
59(1)
3.4 Discrete Time Markov Chains
60(12)
3.4.1 Examples
61(3)
3.4.2 State of DTMC at arbitrary times
64(4)
3.4.3 Classification of States
68(3)
3.4.4 Classification of Markov Chains
71(1)
3.5 First Passage Time
72(6)
3.5.1 Examples
75(1)
3.5.2 Some Key Information Provided by First Passage
76(1)
3.5.3 Example
77(1)
3.5.4 Mean first recurrence time
78(1)
3.6 Absorbing Markov Chains
78(4)
3.6.1 Characteristics of an absorbing Markov chain
79(3)
3.7 Censored Markov Chains
82(1)
3.8 Transient Analysis
83(2)
3.8.1 Direct Algebraic Operations
83(1)
3.8.2 Transient Analysis Based on the z-Transform
84(1)
3.9 Limiting Behaviour of Markov Chains
85(3)
3.9.1 Ergodic Chains
85(1)
3.9.2 Non-Ergodic Chains
86(1)
3.9.3 Mean First Recurrence Time and Steady State Distributions
87(1)
3.10 Bivariate Discrete Time Markov Chains
88(1)
3.11 Problems
88(3)
References
90(1)
4 Numerical Computations with Discrete-Time Markov Chains
91(48)
4.1 Introduction
91(1)
4.2 Numerical Computations of the Invariant Vectors
91(1)
4.3 Finite State Markov Chains
92(6)
4.3.1 Direct Methods
93(2)
4.3.2 Iterative Methods
95(3)
4.4 Bivariate Discrete Time Markov Chains
98(4)
4.4.1 Computing Stationary Distribution for the Finite Bivariate DTMC
99(3)
4.5 Special Finite State DTMC
102(2)
4.6 Infinite State Markov Chains
104(1)
4.7 Some Results for Infinite State Markov Chains with Repeating Structure
105(2)
4.8 Matrix-Analytical Method for Infinite State Markov Chains
107(18)
4.8.1 The GI/M/1 Type
107(3)
4.8.2 Key Measures
110(3)
4.8.3 Some Special Structures of the Matrix R often Encountered
113(1)
4.8.4 The M/G/1 Type
114(2)
4.8.5 Computing Matrix G
116(3)
4.8.6 Some Special Structures of the Matrix G often Encountered
119(1)
4.8.7 QBD
120(3)
4.8.8 Computing the matrices R and G
123(2)
4.8.9 Some Special Structures of the Matrix R and the matrix G
125(1)
4.9 Other special QBDs of interest
125(10)
4.9.1 Level-dependent QBDs
125(1)
4.9.2 Tree structured QBDS
126(3)
4.9.3 Re-blocking of transition matrices
129(3)
4.9.4 Time-inhomogeneous Discrete Time Markov Chains
132(2)
4.9.5 Time-inhomogeneous and spatially-homogeneous QBD
134(1)
4.10 Software Tools for Matrix-Analytic Methods
135(1)
4.11 Problems
136(3)
References
137(2)
5 Basic Single Node Queueing Models with Infinite Buffers
139(70)
5.1 Introduction
139(1)
5.2 Birth-and-Death Process
140(1)
5.3 Discrete time B-D Process
141(2)
5.4 Geo/Geo/1 Queues
143(12)
5.4.1 Algebraic Approach
144(1)
5.4.2 Transform Approach
144(1)
5.4.3 Matrix-geometric Approach
145(1)
5.4.4 Performance Measures
146(8)
5.4.5 Discrete time equivalent of Little's Law
154(1)
5.5 Geo/Geo/1 System with Start-up Times
155(1)
5.6 Geo/G/1 Queues
155(11)
5.6.1 Supplementary Variable Technique
156(3)
5.6.2 Imbedded Markov Chain Approach
159(5)
5.6.3 Mean-Value approach
164(2)
5.7 GI/Geo/1 Queues
166(5)
5.7.1 Supplementary Variable Technique
166(2)
5.7.2 Imbedded Markov Chain Approach
168(3)
5.8 Geo/PH/1 Queues
171(3)
5.8.1 Waiting Times
173(1)
5.9 PH/Geo/1 Queues
174(2)
5.9.1 Waiting Times
175(1)
5.10 PH/PH/1 Queues
176(14)
5.10.1 Examples
178(3)
5.10.2 Waiting Time Distribution
181(3)
5.10.3 PH/PH/1 Queues at points of events
184(6)
5.11 PH/PH/1 System with Start-up Times
190(1)
5.12 GI/G/1 Queues
191(7)
5.12.1 The (RT,RT) representation for the GI/G/1 queue
191(2)
5.12.2 New algorithm for the GI/G/1 system
193(3)
5.12.3 The (ET,ET) representation for the GI/G/1 queue
196(2)
5.13 MAP/PH/1 Queues
198(1)
5.14 Batch Queues
198(7)
5.14.1 GeoX/Geo/1 queue
199(3)
5.14.2 The BMAP/PH/1 System
202(2)
5.14.3 Geo/GeoY/1 queue
204(1)
5.15 Problems
205(4)
References
206(3)
6 Basic Single Node Queueing Models with Finite Buffers
209(18)
6.1 Introduction
209(1)
6.2 Geo/Geo/1/K Queues
209(3)
6.2.1 Busy Period
210(1)
6.2.2 Waiting Time in the Queue
211(1)
6.2.3 Departure Process
212(1)
6.3 Geo/G/1/K-1 Queues
212(1)
6.4 GI/Geo/1/K-1 Queues
213(1)
6.5 GI/G/1/K-1 Queues
214(7)
6.5.1 GI/G/1/K-1 Queues -- Model I
214(5)
6.5.2 GI/G/1/K-1 Queue -- Model II
219(2)
6.6 Queues with Very Large Buffers
221(3)
6.6.1 When Traffic Intensity is Less Than 1
222(1)
6.6.2 When Traffic Intensity is More Than 1
223(1)
6.7 Problems
224(3)
References
226(1)
7 Multiserver Single Node Queueing Models
227(34)
7.1 Introduction
227(1)
7.2 Geo/Geo/k Systems
227(9)
7.2.1 As GI/M/1 type
228(3)
7.2.2 As a QBD
231(2)
7.2.3 An Example
233(1)
7.2.4 Waiting Times
233(3)
7.3 GI/Geo/k Systems
236(4)
7.3.1 Using MAM - Same as using supplementary variables
236(3)
7.3.2 Numerical Example
239(1)
7.4 PH/PH/k Systems
240(2)
7.5 Geo/D/k Systems
242(3)
7.5.1 Waiting Times
244(1)
7.6 MAP/D/k Systems
245(4)
7.7 BMAP/D/k Systems
249(3)
7.8 Geo/G/∞ Systems
252(7)
7.8.1 Preliminaries
253(1)
7.8.2 The Number in the System at Arbitrary Times
254(1)
7.8.3 Special Case of Geo/D/∞
255(1)
7.8.4 The Limiting Distribution of the Number in the System
256(3)
7.8.5 Extension to the PH/G/∞ Case
259(1)
7.9 Problems
259(2)
References
260(1)
8 Single Node Queueing Models with Server Vacations
261(22)
8.1 Vacation Queues
261(1)
8.2 Geo/G/1 vacation systems
262(6)
8.2.1 Single vacation system
262(3)
8.2.2 Multiple vacations system
265(3)
8.3 GI/Geo/1 vacation systems
268(2)
8.4 MAP/PH/1 vacation systems
270(5)
8.4.1 Exhaustive single vacation
270(3)
8.4.2 Stationary Distribution
273(1)
8.4.3 Example of the Geo/Geo/1 vacation
273(1)
8.4.4 Exhaustive multiple vacations
274(1)
8.5 MAP/PH/1 vacation queues with number limited service
275(2)
8.6 MAP/PH/1 vacation queues with time limited service
277(2)
8.7 Random Time Limited Vacation Queues/Queues with Server Breakdowns and Repairs
279(2)
8.8 Problems
281(2)
References
281(2)
9 Single Node Queueing Models with Priorities
283(24)
9.1 Introduction
283(1)
9.2 Geo/Geo/1 Preemptive
283(9)
9.2.1 Stationary Distribution
287(5)
9.3 Geo/Geo/1 Non-preemptive
292(4)
9.3.1 Stationary Distribution
294(2)
9.4 MAP/PH/1 Preemptive
296(5)
9.4.1 Stationary Distribution
298(3)
9.5 MAP/PH/1 Non-preemptive
301(4)
9.5.1 Stationary Distribution
304(1)
9.6 Problems
305(2)
References
306(1)
10 Special Single Node Queueing Models
307(16)
10.1 Introduction
307(1)
10.2 Queues with Multiclass of Packets
307(8)
10.2.1 Multiclass systems -- with FCFS - the MMAP [ K]/PH[ K]/1
308(2)
10.2.2 Multiclass systems -- with LCFS - the MMAP [ K]/PH[ K]/1
310(5)
10.3 Waiting Time Distribution for the LCFS Geo/Geo/1 System
315(1)
10.4 Waiting Time Distribution for the Geo/Geo/1 System with SIRO
316(2)
10.5 Pair Formation Queues (aka Double-ended Queues)
318(2)
10.5.1 Pair formation bounded at one end
319(1)
10.6 Finite Source Queues
320(1)
10.7 Problems
321(2)
References
322(1)
11 Queues with Time Varying Parameters
323(16)
11.1 Introduction
323(1)
11.2 Discrete Time Time-Varying B-D Process
323(3)
11.2.1 The Method of Rectangular Matrix
324(1)
11.2.2 The Use of z-Transforms for the Geon/Geon/1 system
325(1)
11.3 The Geon/Geon/k system
326(2)
11.4 The Geon/GeoYn/1 system
328(2)
11.5 The PHn/PHn/1 System
330(2)
11.6 TheGeon/G/1 System
332(1)
11.7 The Geo(Xn)n /G/1 system
333(4)
11.8 Problems
337(2)
References
337(2)
12 Tandem Queues and Queueing Networks
339(28)
12.1 Introduction
339(1)
12.2 Two Queues in Tandem -- The Geometric Case
339(11)
12.2.1 Second Queue with Infinite Buffer
340(7)
12.2.2 Second Queue with Finite Buffer
347(3)
12.3 Two Queues in Tandem -- The Phase Type Case
350(7)
12.3.1 Second Queue with Infinite Buffer
350(2)
12.3.2 Second Queue with Finite Buffer
352(4)
12.3.3 Both First and Second Queues with Finite Buffers
356(1)
12.4 Approximating (N <2) Tandem Queues Using Above Results
357(4)
12.4.1 Decomposition to N single queues
357(3)
12.4.2 Decomposition to N -- 1 two tandem queues
360(1)
12.5 Queueing Networks
361(6)
12.5.1 The Challenges of Discrete Time Queueing Networks
362(2)
12.5.2 Approximations for Discrete Time Queueing Networks
364(1)
12.5.3 Simple Tandem
364(1)
12.5.4 Merge
364(1)
12.5.5 Split
365(1)
12.5.6 Merge-Split
365(1)
References
365(2)
13 Optimization and Control of Discrete-Time Queues
367(12)
13.1 Introduction
367(1)
13.2 Controlling the Arrival Process
368(7)
13.2.1 One Queue Length Threshold Problem
368(2)
13.2.2 Two Queue Length Thresholds
370(2)
13.2.3 Maximizing Throughput, While Minimizing Loss Probability in Finite Buffer Systems
372(1)
13.2.4 Finding the Maximum Profit, When Costs and Revenues are Associated
373(2)
13.3 Controlling the Service Process
375(1)
13.3.1 Selection of Appropriate Number of Servers
376(1)
13.4 Problems
376(3)
References
377(2)
A Appendix
379(2)
A.1 The z---Transform
379(1)
A.2 Kronecker Product
380(1)
Index 381
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