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

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

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

10-4 M/G/1 Queue – Notation

10-5 M/G/1 Queue

10-6 A Result from Probability Theory

10-7 Pollaczek-Khinchin Formula

10-8 Average Residual Time

10-9 Average Residual Time (cont.)

10-10 P-K Formula

10-11 P-K Formula: Examples

10-12 Distribution Upon Arrival or Departure

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

10-14 Embedded Markov Chain

10-15 Number of Arrivals During a Service Time

10-16 Embedded Markov Chain

10-17 Calculating the Stationary Distribution

10-18 Moment Generating Functions

10-19 Z-Transforms of Discrete Random Variables

10-20 Continuous Random Variables

10-21 Discrete Random Variables

10-22 P-K Transform Equation

10-23 P-K Transform Equation

10-24 P-K Transform Equation

10-25 Expansion in Partial Fractions

10-26 Expansion in Partial Fractions (cont.)

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

10-28 Non-Preemptive Priority

10-29 Non-Preemptive Priority

10-30 Non-Preemptive Priority

10-31 Preemptive Priority

10-32 Preemptive Priority