Perturbed Semi-Markov Type Processes I: Limit Theorems for Rare-Event Times and Processes

Preface
Contents
List of symbols
1 Introduction
	1.1 Part I: First-Rare-Event Times for Regularly Perturbed Semi-Markov Processes
		1.1.1 Part I: Contents, Examples, Models, Results
		1.1.2 Part I: Contents by Chapters
	1.2 Part II: Hitting Times and Phase Space Reduction for Perturbed Semi-Markov Processes
		1.2.1 Part II: Contents, Examples, Models, Results
		1.2.2 Part II: Contents by Chapters
	1.3 Appendices and Conclusion
		1.3.1 Appendix A: Limit Theorems for Randomly Stopped Stochastic Processes
		1.3.2 Appendix B: Methodological and Bibliographical Notes
		1.3.3 Conclusion
Part I First-Rare-Event Times for Regularly Perturbed Semi-Markov Processes
	2 Asymptotics of First-Rare-Event Times for Regularly Perturbed Semi-Markov Processes
		2.1 First-Rare-Event Times for Perturbed Semi-Markov Processes
			2.1.1 First-Rare-Event Times
			2.1.2 Asymptotically Uniformly Ergodic Markov Chains
			2.1.3 Necessary and Sufficient Conditions for Convergence in Distribution of First-Rare-Event Times
		2.2 Asymptotics of Step-Sum Reward Processes
			2.2.1 Necessary and Sufficient Conditions for Convergence in Distribution for Step-Sum Reward Processes
			2.2.2 Examples of Step-Sum Reward Processes
		2.3 Asymptotics of First-Rare-Event Times for Perturbed Markov Chains
			2.3.1 Convergence in Distribution for First-Rare-Event Times for Perturbed Markov Chains
			2.3.2 J-Convergence for First-Rare-Event Time Processes for Perturbed Markov Chains
		2.4 Asymptotics of First-Rare-Event Times for Perturbed Semi-Markov Processes
			2.4.1 Convergence in Distribution for First-Rare-Event Times for Perturbed Semi-Markov Processes
			2.4.2 J-Convergence for First-Rare-Event Processes for Perturbed Semi-Markov Process
	3 Flows of Rare Events for Regularly Perturbed Semi-Markov Processes
		3.1 Counting Processes Generated by Flows of Rare Events
			3.1.1 Counting Processes for Rare Events
			3.1.2 Necessary and Sufficient Conditions of Convergence for Counting Processes Generated by Flows of Rare Events
		3.2 Markov Renewal Processes Generated by Flows of Rare Events
			3.2.1 Return Times and Rare Events
			3.2.2 Necessary and Sufficient Conditions of Convergence for Markov Renewal Processes Generated by Flows of Rare Events
		3.3 Vector Counting Processes Generated by Flows of Rare Events
			3.3.1 Vector Counting Processes for Rare Events
			3.3.2 Necessary and Sufficient Conditions of Convergence for Vector Counting Process Generated by Flows of Rare Events
	4 Generalisations of Limit Theorems for First-Rare-Event Times
		4.1 Modifications of First-Rare-Event Times and Rare-Event Time Processes
			4.1.1 Vector First-Rare-Event Times and Rewards
			4.1.2 Upper and Lower First-Rare-Event Times
			4.1.3 First-Rare-Event Times for Markov Renewal Processes with Transition Periods
			4.1.4 First-Rare-Event Times for Markov Renewal Processes with Extending Phase Spaces
		4.2 First-Rare-Event Times and Hitting Times
			4.2.1 Necessary and Sufficient Conditions of Convergence in Distribution for Standard Hitting Times
			4.2.2 Necessary and Sufficient Conditions of Convergence in Distribution for Directed Hitting Times
	5 First-Rare-Event Times for Perturbed Risk Processes
		5.1 Stable Asymptotics of First-Rare-Event Times
		5.2 Necessary and Sufficient Conditions for Stable Approximation of Non-ruin Distribution Functions
	6 First-Rare-Event Times for Perturbed Closed Queuing Systems
		6.1 Queuing Systems with Rare Events Caused by Too Long Service Times
			6.1.1 Asymptotic Uniform Ergodicity for Birth–Death Markov Chains
			6.1.2 M/M-Type Queuing System with Rare Events Caused by Too Long Service Times
		6.2 Queuing Systems with Highly Reliable Servers
	7 First-Rare-Event Times for Perturbed M/M-Type Queuing Systems
		7.1 Rare Events for Queuing Systems with Bounded Buffers
			7.1.1 Queuing Systems with a Buffer of Fixed Size
			7.1.2 Queuing Systems with Asymptotically Unbounded Buffers
		7.2 Rare Events for Queuing Systems with Unbounded Buffers
Part II Hitting Times and Phase Space Reduction for Perturbed Semi-Markov Processes
	8 Asymptotically Comparable Functions
		8.1 Complete Families of Asymptotically Comparable Functions
			8.1.1 Definitions for Families of Asymptotically Comparable Functions
			8.1.2 Operating Rules for Asymptotically Comparable Functions
		8.2 Examples of Complete Families of Asymptotically Comparable Functions
			8.2.1 Asymptotically Comparable Power-Type Functions
			8.2.2 Asymptotically Comparable Power-Exponential-Type Functions
			8.2.3 Asymptotically Comparable Power-Logarithmic-Type Functions
	9 Perturbed Semi-Markov Processes and Reduction of Phase Space
		9.1 Perturbed Semi-Markov Processes
			9.1.1 Perturbed Semi-Markov Processes
			9.1.2 Perturbation Conditions
			9.1.3 Additional Asymptotic Comparability Conditions
			9.1.4 Perturbed Markov Chains
		9.2 Removing of Virtual Transitions for Perturbed Semi-Markov Processes
			9.2.1 Semi-Markov Processes with Removed Virtual Transitions
			9.2.2 Perturbation Conditions for Semi-Markov Processes with Removed Virtual Transitions
		9.3 One-State Reduction of Phase Space for Perturbed Semi-Markov Processes
			9.3.1 A Procedure of One-Step Phase Space Reduction for Perturbed Semi-Markov Processes
			9.3.2 Perturbation Conditions for Reduced Semi-Markov Processes
		9.4 Recurrent Reduction of Phase Space for Perturbed Semi-Markov Processes
			9.4.1 The Recurrent Phase Space Reduction Algorithm
			9.4.2 Summary of Recurrent Phase Space Reduction Algorithm
	10 Asymptotics of Hitting Times for Perturbed Semi-Markov Processes
		10.1 Hitting Times for Perturbed Semi-Markov Processes
			10.1.1 Hitting Times and Related Asymptotic Problems
			10.1.2 Hitting Times and Reduced Perturbation Conditions for Semi-Markov Processes
			10.1.3 Hitting Times for Semi-Markov Processes with Reduced Phase Spaces
			10.1.4 Recurrent Relations for Distributions and Laplace Transforms of Hitting Times
		10.2 Weak Asymptotics for Distributions of Hitting Times
			10.2.1 Asymptotics for Hitting Probabilities
			10.2.2 Weak Asymptotic for Distributions of Hitting Times for the Case Where the Initial State Belongs to Domain
			10.2.3 Admissible Normalisation Functions and Phase Types of Limiting Distributions
		10.3 Weak Asymptotics for Distributions of Return Times
			10.3.1 Hitting and Return Times
			10.3.2 Weak Asymptotics for Distributions of Hitting and Return Times
	11 Asymptotics for Expectations of Hitting Times for Perturbed Semi-Markov Processes
		11.1 Expectations of Hitting Times
			11.1.1 Recurrent Relations for Expectations of HittingTimes
			11.1.2 Recurrent Relations for Limits of Expectations of Hitting Times
		11.2 Asymptotics of Expectations of Hitting Times
			11.2.1 Asymptotics for Expectations of Hitting Times for the Case Where an Initial State Belongs to Domain
			11.2.2 Conditions of Simultaneous Convergence for Distributions and Expectations of Hitting Times
		11.3 Asymptotics for Expectations of Return Times
			11.3.1 Asymptotic for Expectations of Hitting Times in the Case Where an Initial State Belongs to Domain D
			11.3.2 Asymptotics of Expectations for Return Times to Domain D
	12 Generalisations and Examples of Limit Theorems for Hitting Times
		12.1 Generalisations of Limit Theorems for Hitting Times
			12.1.1 More General Perturbation Conditions
			12.1.2 More General Hitting Reward Functionals
		12.2 Hitting Times for Perturbed Birth–Death-Type Semi-Markov Processes
			12.2.1 Perturbed Birth–Death-Type Semi-Markov Processes
			12.2.2 Phase Space Reduction for Perturbed Birth–Death-Type Semi-Markov Processes
		12.3 Numerical Examples
			12.3.1 An Example of Singularly Perturbed Semi-Markov Processes with Two-State Domain
			12.3.2 A One-Step Asymptotic Reduction of Phase Space
			12.3.3 Asymptotics for Distributions and Expectations of Hitting Times
Limit Theorems for Randomly Stopped Stochastic Processes
	A.1 Functional Limit Theorems for Càdlàg Stochastic Processes
		A.1.1 Convergence of Càdlàg Stochastic Processes in Topology U
		A.1.2 Convergence of Càdlàg Stochastic Processes in Topology J
		A.1.3 Convergence of Step-Sum Processes with Independent Increments in Topology J
	A.2 Limit Theorems for Randomly Stopped Stochastic Processes and Superpositions of Stochastic Processes
		A.2.1 Convergence in Distribution for Randomly Stopped Stochastic Processes
		A.2.2 Convergence in Distribution for Superpositions of Càdlàg Processes
		A.2.3 Convergence of Superpositions of Càdlàg Processes in Topology J
	A.3 Supplementary Asymptotic Results
		A.3.1 Slutsky Theorem and Related Results
		A.3.2 Functional Analogues of Slutsky Theorem
Methodological and Bibliographical Notes
	B.1 Methodological Notes
	B.2 General Bibliographical Remarks
References
Index