Shanghai Jiao Tong University MG1 QUEUE Communication Networks

Shanghai Jiao Tong University M/G/1 QUEUE Communication Networks

M/G/1 Queue The M/G/1 queue Communication Networks 2

Pollaczek-Khinchin (P-K) Formula • Communication Networks 3

Evolution of M/G/1 Queues • General service time distribution → memory • Analysis technique of M/M/x cannot be applied to M/G/1 Departures Not memoryless Arrivals Communication Networks 4

Two Possible Methods to Solve M/G/1 • Embedded Markov Chain • Graphic method Communication Networks 5

State of M/G/1 Queues • Departures Residual service time Arrivals Communication Networks 6

Embedded Points • Departures Arrivals Communication Networks 7

Embedded Markov Chain • Communication Networks 8

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Distribution at Embedded Points • Communication Networks 11

Distribution at Arbitrary Time Point • L. Kleinrock and R. Gail, “Queuing Systems: Problems and Solutions”, Problem 5. 6, John Wiley & Sons, 1996 Communication Networks 12

Two Possible Method to Solve M/G/1 • Embedded Markov Chain • Graphic method Communication Networks 13

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Stable Condition of M/G/1 Queues • Communication Networks 19

Examples • Communication Networks 20

M/M/1 vs. M/D/1 Communication Networks 21

M/G/1 with Vacations • Once the system becomes empty, the server takes a vacation – If system is still empty after the vacation, the server takes another vacation – Useful in analyzing some polling and reservation systems • Vacation times are i. i. d. and independent of service times and arrival times • The only impact on analysis is that a customer may enter to find the server on vacation, and must wait until end of that vacation Communication Networks 22

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Example: Slotted M/D/1 system • Communication Networks 25

Homework Communication Networks 26

Homework Communication Networks 27

Homework Communication Networks 28