Queueing Theory 1. Nikolaos Limnios

Queueing Theory 1

9.2. Preliminary and notations 9.3. Strong stability of queueing systems 9.4. Conclusion and further directions 9.5. References

14 10 Time-varying Queues: a Two-time-scale Approach 10.1. Introduction 10.2. Time-varying queues 10.3. Main results 10.4. Concluding remarks 10.5. References

15 List of Authors

16 Index

17 End User License Agreement

List of Illustrations

1 Chapter 2Figure 2.1. A classical busy periodFigure 2.2. A period under level x Figure 2.3. A busy period with lossesFigure 2.4. A period of level greater than z

2 Chapter 4Figure 4.1.

3 Chapter 5Figure 5.1. Cor(Λ(t), X₁ (t)) for μ₁ = 1, γ = 2 and D_Λ = 2. For a color version ...Figure 5.2. Estimated distribution of Xi for several phases i, with Λ ~ U (0, 6)...Figure 5.3.Histogram estimating the distribution of X, with Λ ~ U (0, 6) and γ =...Figure 5.4. Estimated distribution of X for different values of γ, with Λ ~ U(0,...Figure 5.5.Cor(X,Λ) for DΛ = 1 andγ = 0.5. For a color version of this figure, s...

4 Chapter 6Figure 6.1. A matrix determining the base vectors of some subspace

5 Chapter 8Figure 8.1. Retrial queuing system of type M/GI/1//N with collisions of the cust...Figure 8.2. Density function of the gamma distribution. For a color version of t...Figure 8.3. Bimodal probability distribution of the number of customers in the s...Figure 8.4. Steady-state distributions. For a color version of this figure, see ...Figure 8.5. vs λ∕N,

Figure 8.6. vs λ∕N,

Figure 8.7. Reliable no conflict, reliable with conflict, and unreliable with co...Figure 8.8. Comparison of steady-state distributions. For a color version of thi...Figure 8.9. Mean waiting time versus intensity of incoming customers. For a colo...Figure 8.10. Steady-state distributions of gamma distributed interarrival times....Figure 8.11. Mean waiting time versus shape parameter. For a color version of th...Figure 8.12. Mean interrupted service time versus shape parameter. For a color v...Figure 8.13. Steady-state distributions of gamma distributed retrial times. For ...Figure 8.14. Comparison of steady-state distributions for mode 2. For a color ve...Figure 8.15. Mean waiting time versus intensity of incoming customers of Case 1....Figure 8.16. Mean waiting time versus intensity of incoming customers of Case 2....Figure 8.17. Mean waiting time versus intensity of incoming customers of Case 3....Figure 8.18. Mean successful service time versus intensity of incoming customers...Figure 8.19. Mean successful service time versus intensity of incoming customers...Figure 8.20. Mean successful service time versus intensity of incoming customers...Figure 8.21. Mean waiting time versus intensity of failure rate. For a color ver...Figure 8.22. Steady-state distributions of scenario B. For a color version of th...Figure 8.23. Mean waiting time versus shape parameter, α = β = 0.5. For a color ...Figure 8.24. Mean waiting time versus shape parameter, α = β = 1. For a color ve...Figure 8.25. Mean waiting time versus shape parameter, α = β = 2. For a color ve...Figure 8.26. Mean successful service time versus shape parameter, α = β = 0.5. F...Figure 8.27. Mean successful service time versus shape parameter, α = β = 1. For...Figure 8.28. Mean successful service time versus shape parameter, α = β = 2. For...

6 Chapter 10Figure 10.1. Function H(y) = H(y,t)

List of Tables

1 Chapter 1Table 1.1. Mean number in system for the three casesTable 1.2. Complementary Cumulative distributions of interarrival times

2 Chapter 6Table 6.1. n = 50, k = 2, q = 2⁵ , ε = 1%Table 6.2. n = 50, ω= 2, q = 2⁵ ,ε = 1%Table 6.3. n = 50, k = 10, q = 2⁵, ε = 1%Table 6.4. Estimates of the number of “good” permutations by the fast simulation...Table 6.5. Estimates for the parameters α(s) and β (s)Table 6.6. Comparison of exact values with statistical estimates

3 Chapter 8Table 8.1. Mean sojourn time E(TS) of the customer under service at various valu...Table 8.2. Mean number of retrials for various values of λ and α = βTable 8.3. Numerical values of model parametersTable 8.4. Kolmogorov distance between prelimit distribution P(i) and the asympt...Table 8.5. Kolmogorov distance between prelimit distribution P(i) and the asympt...Table 8.6. Kolmogorov distance between distribution Ps(i) and approximation of t...Table 8.7. Mean number of retrials in prelimiting and limiting situations for va...Table 8.8. Numerical values of model parametersTable 8.9. Simulation resultsTable 8.10. Numerical values of parameters of gamma distributed interarrival tim...Table 8.11. Simulation resultsTable 8.12. Numerical values of parameters of gamma distributed retrial timesTable 8.13. Simulation resultsTable 8.14. Numerical values of model parametersTable 8.15. Numerical results of scenario ATable 8.16. Numerical values of parameters of scenario BTable 8.17. Numerical results of scenario B

Guide

1 Cover

2 Table of Contents

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