1. 1 TCOM 501: Networking Theory & Fundamentals Lectures 9 & 10 M/G/1 Queue Prof. Yannis A. Korilis

2. 10-2 Topics M/G/1 Queue Pollaczek-Khinchin (P-K) Formula Embedded Markov Chain Observed at Departure Epochs Pollaczek-Khinchin Transform Equation Queues with Vacations Priority Queueing

3. 10-3 M/G/1 Queue Arrival Process: Poisson with rate λ Single server, infinite waiting room Service times: Independent identically distributed following a general distribution Independent of the arrival process Main Results: Determine the average time a customer spends in the queue waiting service (Pollaczek-Khinchin Formula) Calculation of stationary distribution for special cases only

4. 10-4 M/G/1 Queue – Notation

5. 10-5 M/G/1 Queue

6. 10-6 A Result from Probability Theory

7. 10-7 Pollaczek-Khinchin Formula

8. 10-8 Average Residual Time

9. 10-9 Average Residual Time (cont.)

10. 10-10 P-K Formula

11. 10-11 P-K Formula: Examples

12. 10-12 Distribution Upon Arrival or Departure

13. 10-13 Distribution Upon Arrival or Departure (cont.)

14. 10-14 Embedded Markov Chain

15. 10-15 Number of Arrivals During a Service Time

16. 10-16 Embedded Markov Chain

17. 10-17 Calculating the Stationary Distribution

18. 10-18 Moment Generating Functions

19. 10-19 Z-Transforms of Discrete Random Variables

20. 10-20 Continuous Random Variables

21. 10-21 Discrete Random Variables

22. 10-22 P-K Transform Equation

23. 10-23 P-K Transform Equation

24. 10-24 P-K Transform Equation

25. 10-25 Expansion in Partial Fractions

26. 10-26 Expansion in Partial Fractions (cont.)

27. 10-27 M/G/1 Queue with Priority Classes

28. 10-28 Non-Preemptive Priority

29. 10-29 Non-Preemptive Priority

30. 10-30 Non-Preemptive Priority

31. 10-31 Preemptive Priority

32. 10-32 Preemptive Priority