Dr. James J Solberg is the Ransburg Professor of Manufacturing and Industrial Engineering and Director of the Purdue University Center for Collaborative Manufacturing (National Science Foundation Engineering Research Center). Dr. Solberg is the U.S. Regional Coordinator for the International Joint Research Committee on Metamorphic Material Handling Systems is a member of the Executive Committee of the Material Handling Engineering Division of ASME, the chair of the Technical Committee on Material Transport Systems. He also currently serves on the Editorial Board, International Journal of Production Economics. His research interests include manufacturing systems, stochastic models, and queuing networks.

• Preface ix1 Probability Review 11.1 Interpreting and Using Probabilities 21.2 Sample Spaces and Events 31.3 Probability 41.4 Random Variables 61.5 Probability Distributions 61.6 Joint, Marginal, and Conditional Distributions 111.7 Expectation 141.8 Variance and Other Moments 161.9 The Law of Total Probability 181.10 Discrete Probability Distributions 201.11 Continuous Probability Distributions 231.12 Where Do Distributions Come From? 261.13 The Binomial Process 281.14 Recommended Reading 292 Formulating Markov Chain Models 322.1 An Example 332.2 Modeling the Progress of Time 342.3 Modeling Possibilities as States 362.4 Simplifying Assumptions 382.5 Modeling Changes of State as Transitions 402.6 Obtaining the Data 452.7 Another Example 462.8 A Case Study 472.9 Higher Order Markov Chains 502.10 Reducing the Number of States 522.11 Nonstationary Markov Chains 532.12 Other Example 542.13 Summary 672.14 Recommended Reading 673 Markov Chain Calculations 723.1 Walk Probabilities 733.2 Transition Probabilities 743.3 State Probabilities 783.4 A Numerical Example 793.5 Expected Number of Visits 803.6 Sojourn Times 823.7 First Passage and Return Probabilities 833.8 Computational Formulas for All Markov Chains 863.9 Special Classes of Markov Chains 863.10 Steady-State Probabilities 873.11 The Uses of Steady-State Results 923.12 Mean First Passage Times 933.13 Computational Formulas for Ergodic Markov Chains 963.14 Terminating Markov Chains 963.15 Expected Number of Visits 983.16 Expected Duration of a Terminating Process 993.17 Absorption Probabilities 1003.18 Hit Probabilities 1023.19 Conditional Mean First Passage Times to Absorbing States 1033.20 Computational Formulas for Terminating Processes 1053.21 Call Center Calculations 1053.22 Classification Terminology 1063.23 Additional Complications in Infinite Chains 1113.24 Dealing with a Reducible Process 1123.25 Periodic Chains 1133.26 Ergodic Chains 1143.27 Recommended Reading 1154 Rewards on Markov Chains 1194.1 Formulation 1204.2 A Numerical Example 1204.3 Expected Total Reward 1214.4 Random Variable Rewards 1244.5 Semi-Markov Processes 1264.6 Limiting Results for Ergodic Processes 1264.7 Total Reward for Terminating Processes 1304.8 Case Study 1324.9 Discounting 1334.10 Case Study 1354.11 Recommended Reading 1375 Continuous Time Markov Processes 1405.1 An Example 1415.2 Interpreting Transition Rates 1465.3 The Assumptions Reconsidered 1495.4 Aging Does Not Affect the Transition Time 1505.5 Competing Transitions 1525.6 Sojourn Times 1535.7 Embedded Markov Chains 1545.8 Deriving the Differential Equations 1555.9 Solving the Differential Equations 1575.10 State Probabilities 1595.11 First Passage Probability Functions 1595.12 State Classification 1605.13 Steady-State Probabilities 1615.14 Other Computable Quantities 1635.15 Case Study 1655.16 Birth-Death Processes 1675.17 The Poisson Process 1695.18 Properties of Poisson Processes 1715.19 Khintchine’s Theorem 1725.20 Phase-Type Distributions 1735.21 Conclusions 1755.22 Recommended Reading 1756 Queueing Models 1796.1 An Example 1806.2 General Characteristics 1826.3 Performance Measures 1866.4 Relations Among Performance Measures 1886.5 Little’s Formula 1906.6 Markovian Queueing Models 1916.7 The M/M/1 Model 1936.8 The Significance of Traffic Intensity 1986.9 Unnormalized Solutions 2006.10 Limited Queue Capacity 2026.11 Multiple Servers 2046.12 Is It Better to Merge or Separate Servers? 2076.13 Which is Better: More Servers or Faster Servers 2086.14 Case Study: A Grain Elevator 2096.15 The M/M/c/c and M/M/1 Models 2106.16 Finite Sources 2126.17 The Machine Repairmen Model 2146.18 Numerical Calculations Using a Spreadsheet 2146.19 Queue Discipline Variations 2176.20 Non-Markovian Queues 2186.21 The M/G/1 Model 2196.22 Approximate Solutions for Other Models 2206.23 Conclusion 2216.24 Recommended Reading 2217 Networks of Queues 2257.1 Open Networks of Markovian Queues 2267.2 An Example Open Network 2277.3 Extensions 2287.4 Closed Networks 2297.5 A Preliminary Example 2297.6 Relative Arrival Rates 2307.7 Unnormalized Solutions for Individual Stations 2327.8 Assembling the Pieces of the Solution 2347.9 Calculating the Normalization Constant 2357.10 Performance Measures for Closed Networks 2377.11 Creating a Closed Model 2397.12 Case Study 2427.13 Extensions 2477.14 Approximate Methods 2477.15 Recommended Reading 2488 Using the Transition Diagram to Compute 2518.1 An Example 2528.2 Definitions 2548.3 Steady-State Probabilities 2588.4 How to Generate All In-trees 2598.5 Check Your Understanding 2628.6 Generalization to Other Quantities 2638.7 Mean First Passage Times 2648.8 Results for Terminating Processes 2658.9 How to Simplify the Arithmetic 2668.10 How to Systematically Generate r-Forests 2678.11 Summary of Results 2678.12 How to Remember the Formulas 2688.13 Advanced Topics 2688.14 Recommended Reading 269Appendix 1 271Appendix 2 278Index 300

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Dr. James J Solberg is the Ransburg Professor of Manufacturing and Industrial Engineering and Director of the Purdue University Center for Collaborative Manufacturing (National Science Foundation Engineering Research Center). Dr. Solberg is the U.S. Regional Coordinator for the International Joint Research Committee on Metamorphic Material Handling Systems is a member of the Executive Committee of the Material Handling Engineering Division of ASME, the chair of the Technical Committee on Material Transport Systems. He also currently serves on the Editorial Board, International Journal of Production Economics. His research interests include manufacturing systems, stochastic models, and queuing networks.

• Preface ix1 Probability Review 11.1 Interpreting and Using Probabilities 21.2 Sample Spaces and Events 31.3 Probability 41.4 Random Variables 61.5 Probability Distributions 61.6 Joint, Marginal, and Conditional Distributions 111.7 Expectation 141.8 Variance and Other Moments 161.9 The Law of Total Probability 181.10 Discrete Probability Distributions 201.11 Continuous Probability Distributions 231.12 Where Do Distributions Come From? 261.13 The Binomial Process 281.14 Recommended Reading 292 Formulating Markov Chain Models 322.1 An Example 332.2 Modeling the Progress of Time 342.3 Modeling Possibilities as States 362.4 Simplifying Assumptions 382.5 Modeling Changes of State as Transitions 402.6 Obtaining the Data 452.7 Another Example 462.8 A Case Study 472.9 Higher Order Markov Chains 502.10 Reducing the Number of States 522.11 Nonstationary Markov Chains 532.12 Other Example 542.13 Summary 672.14 Recommended Reading 673 Markov Chain Calculations 723.1 Walk Probabilities 733.2 Transition Probabilities 743.3 State Probabilities 783.4 A Numerical Example 793.5 Expected Number of Visits 803.6 Sojourn Times 823.7 First Passage and Return Probabilities 833.8 Computational Formulas for All Markov Chains 863.9 Special Classes of Markov Chains 863.10 Steady-State Probabilities 873.11 The Uses of Steady-State Results 923.12 Mean First Passage Times 933.13 Computational Formulas for Ergodic Markov Chains 963.14 Terminating Markov Chains 963.15 Expected Number of Visits 983.16 Expected Duration of a Terminating Process 993.17 Absorption Probabilities 1003.18 Hit Probabilities 1023.19 Conditional Mean First Passage Times to Absorbing States 1033.20 Computational Formulas for Terminating Processes 1053.21 Call Center Calculations 1053.22 Classification Terminology 1063.23 Additional Complications in Infinite Chains 1113.24 Dealing with a Reducible Process 1123.25 Periodic Chains 1133.26 Ergodic Chains 1143.27 Recommended Reading 1154 Rewards on Markov Chains 1194.1 Formulation 1204.2 A Numerical Example 1204.3 Expected Total Reward 1214.4 Random Variable Rewards 1244.5 Semi-Markov Processes 1264.6 Limiting Results for Ergodic Processes 1264.7 Total Reward for Terminating Processes 1304.8 Case Study 1324.9 Discounting 1334.10 Case Study 1354.11 Recommended Reading 1375 Continuous Time Markov Processes 1405.1 An Example 1415.2 Interpreting Transition Rates 1465.3 The Assumptions Reconsidered 1495.4 Aging Does Not Affect the Transition Time 1505.5 Competing Transitions 1525.6 Sojourn Times 1535.7 Embedded Markov Chains 1545.8 Deriving the Differential Equations 1555.9 Solving the Differential Equations 1575.10 State Probabilities 1595.11 First Passage Probability Functions 1595.12 State Classification 1605.13 Steady-State Probabilities 1615.14 Other Computable Quantities 1635.15 Case Study 1655.16 Birth-Death Processes 1675.17 The Poisson Process 1695.18 Properties of Poisson Processes 1715.19 Khintchine’s Theorem 1725.20 Phase-Type Distributions 1735.21 Conclusions 1755.22 Recommended Reading 1756 Queueing Models 1796.1 An Example 1806.2 General Characteristics 1826.3 Performance Measures 1866.4 Relations Among Performance Measures 1886.5 Little’s Formula 1906.6 Markovian Queueing Models 1916.7 The M/M/1 Model 1936.8 The Significance of Traffic Intensity 1986.9 Unnormalized Solutions 2006.10 Limited Queue Capacity 2026.11 Multiple Servers 2046.12 Is It Better to Merge or Separate Servers? 2076.13 Which is Better: More Servers or Faster Servers 2086.14 Case Study: A Grain Elevator 2096.15 The M/M/c/c and M/M/1 Models 2106.16 Finite Sources 2126.17 The Machine Repairmen Model 2146.18 Numerical Calculations Using a Spreadsheet 2146.19 Queue Discipline Variations 2176.20 Non-Markovian Queues 2186.21 The M/G/1 Model 2196.22 Approximate Solutions for Other Models 2206.23 Conclusion 2216.24 Recommended Reading 2217 Networks of Queues 2257.1 Open Networks of Markovian Queues 2267.2 An Example Open Network 2277.3 Extensions 2287.4 Closed Networks 2297.5 A Preliminary Example 2297.6 Relative Arrival Rates 2307.7 Unnormalized Solutions for Individual Stations 2327.8 Assembling the Pieces of the Solution 2347.9 Calculating the Normalization Constant 2357.10 Performance Measures for Closed Networks 2377.11 Creating a Closed Model 2397.12 Case Study 2427.13 Extensions 2477.14 Approximate Methods 2477.15 Recommended Reading 2488 Using the Transition Diagram to Compute 2518.1 An Example 2528.2 Definitions 2548.3 Steady-State Probabilities 2588.4 How to Generate All In-trees 2598.5 Check Your Understanding 2628.6 Generalization to Other Quantities 2638.7 Mean First Passage Times 2648.8 Results for Terminating Processes 2658.9 How to Simplify the Arithmetic 2668.10 How to Systematically Generate r-Forests 2678.11 Summary of Results 2678.12 How to Remember the Formulas 2688.13 Advanced Topics 2688.14 Recommended Reading 269Appendix 1 271Appendix 2 278Index 300

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Dr. James J Solberg is the Ransburg Professor of Manufacturing and Industrial Engineering and Director of the Purdue University Center for Collaborative Manufacturing (National Science Foundation Engineering Research Center). Dr. Solberg is the U.S. Regional Coordinator for the International Joint Research Committee on Metamorphic Material Handling Systems is a member of the Executive Committee of the Material Handling Engineering Division of ASME, the chair of the Technical Committee on Material Transport Systems. He also currently serves on the Editorial Board, International Journal of Production Economics. His research interests include manufacturing systems, stochastic models, and queuing networks.

• Preface ix1 Probability Review 11.1 Interpreting and Using Probabilities 21.2 Sample Spaces and Events 31.3 Probability 41.4 Random Variables 61.5 Probability Distributions 61.6 Joint, Marginal, and Conditional Distributions 111.7 Expectation 141.8 Variance and Other Moments 161.9 The Law of Total Probability 181.10 Discrete Probability Distributions 201.11 Continuous Probability Distributions 231.12 Where Do Distributions Come From? 261.13 The Binomial Process 281.14 Recommended Reading 292 Formulating Markov Chain Models 322.1 An Example 332.2 Modeling the Progress of Time 342.3 Modeling Possibilities as States 362.4 Simplifying Assumptions 382.5 Modeling Changes of State as Transitions 402.6 Obtaining the Data 452.7 Another Example 462.8 A Case Study 472.9 Higher Order Markov Chains 502.10 Reducing the Number of States 522.11 Nonstationary Markov Chains 532.12 Other Example 542.13 Summary 672.14 Recommended Reading 673 Markov Chain Calculations 723.1 Walk Probabilities 733.2 Transition Probabilities 743.3 State Probabilities 783.4 A Numerical Example 793.5 Expected Number of Visits 803.6 Sojourn Times 823.7 First Passage and Return Probabilities 833.8 Computational Formulas for All Markov Chains 863.9 Special Classes of Markov Chains 863.10 Steady-State Probabilities 873.11 The Uses of Steady-State Results 923.12 Mean First Passage Times 933.13 Computational Formulas for Ergodic Markov Chains 963.14 Terminating Markov Chains 963.15 Expected Number of Visits 983.16 Expected Duration of a Terminating Process 993.17 Absorption Probabilities 1003.18 Hit Probabilities 1023.19 Conditional Mean First Passage Times to Absorbing States 1033.20 Computational Formulas for Terminating Processes 1053.21 Call Center Calculations 1053.22 Classification Terminology 1063.23 Additional Complications in Infinite Chains 1113.24 Dealing with a Reducible Process 1123.25 Periodic Chains 1133.26 Ergodic Chains 1143.27 Recommended Reading 1154 Rewards on Markov Chains 1194.1 Formulation 1204.2 A Numerical Example 1204.3 Expected Total Reward 1214.4 Random Variable Rewards 1244.5 Semi-Markov Processes 1264.6 Limiting Results for Ergodic Processes 1264.7 Total Reward for Terminating Processes 1304.8 Case Study 1324.9 Discounting 1334.10 Case Study 1354.11 Recommended Reading 1375 Continuous Time Markov Processes 1405.1 An Example 1415.2 Interpreting Transition Rates 1465.3 The Assumptions Reconsidered 1495.4 Aging Does Not Affect the Transition Time 1505.5 Competing Transitions 1525.6 Sojourn Times 1535.7 Embedded Markov Chains 1545.8 Deriving the Differential Equations 1555.9 Solving the Differential Equations 1575.10 State Probabilities 1595.11 First Passage Probability Functions 1595.12 State Classification 1605.13 Steady-State Probabilities 1615.14 Other Computable Quantities 1635.15 Case Study 1655.16 Birth-Death Processes 1675.17 The Poisson Process 1695.18 Properties of Poisson Processes 1715.19 Khintchine’s Theorem 1725.20 Phase-Type Distributions 1735.21 Conclusions 1755.22 Recommended Reading 1756 Queueing Models 1796.1 An Example 1806.2 General Characteristics 1826.3 Performance Measures 1866.4 Relations Among Performance Measures 1886.5 Little’s Formula 1906.6 Markovian Queueing Models 1916.7 The M/M/1 Model 1936.8 The Significance of Traffic Intensity 1986.9 Unnormalized Solutions 2006.10 Limited Queue Capacity 2026.11 Multiple Servers 2046.12 Is It Better to Merge or Separate Servers? 2076.13 Which is Better: More Servers or Faster Servers 2086.14 Case Study: A Grain Elevator 2096.15 The M/M/c/c and M/M/1 Models 2106.16 Finite Sources 2126.17 The Machine Repairmen Model 2146.18 Numerical Calculations Using a Spreadsheet 2146.19 Queue Discipline Variations 2176.20 Non-Markovian Queues 2186.21 The M/G/1 Model 2196.22 Approximate Solutions for Other Models 2206.23 Conclusion 2216.24 Recommended Reading 2217 Networks of Queues 2257.1 Open Networks of Markovian Queues 2267.2 An Example Open Network 2277.3 Extensions 2287.4 Closed Networks 2297.5 A Preliminary Example 2297.6 Relative Arrival Rates 2307.7 Unnormalized Solutions for Individual Stations 2327.8 Assembling the Pieces of the Solution 2347.9 Calculating the Normalization Constant 2357.10 Performance Measures for Closed Networks 2377.11 Creating a Closed Model 2397.12 Case Study 2427.13 Extensions 2477.14 Approximate Methods 2477.15 Recommended Reading 2488 Using the Transition Diagram to Compute 2518.1 An Example 2528.2 Definitions 2548.3 Steady-State Probabilities 2588.4 How to Generate All In-trees 2598.5 Check Your Understanding 2628.6 Generalization to Other Quantities 2638.7 Mean First Passage Times 2648.8 Results for Terminating Processes 2658.9 How to Simplify the Arithmetic 2668.10 How to Systematically Generate r-Forests 2678.11 Summary of Results 2678.12 How to Remember the Formulas 2688.13 Advanced Topics 2688.14 Recommended Reading 269Appendix 1 271Appendix 2 278Index 300

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