1. 1 Chapter 4 Fundamental Queueing System

2. 2

3. 3

4. 4

5. 5

6. 6

7. 7

8. 8

9. 9

10. 10

11. 11

12. 12

13. 13

14. 14 net arrival rate= = r The through pat r/μ=

15. 15

16. 16

17. 17

18. 18 Example 1 Statistical Multiplexing Compared with TDM and FDM Assume m statistically iid Poisson packet streams each with an arrival rate of packets/sec. The packet lengths for all streams are independent and exponentially distributed. The average transmission time is. If the streams are merged into a single Poisson stream, with rate, the average delay per packet is If, the transmission capacity is divided into m equal portions, as in TDM and FDM, each portion behaves like an M/M/1 queue with arrival rate and average service rate. Therefore, the average delay per packet is.

19. 19 Example 2 Using One vs. Using Multiple Channels Statistical MUX(1) A communication link serving m independent Poisson traffic streams with overall rate. Packet transmission times on each channel are exponentially distributed with mean. The system can be modeled by the same Markov chain as the M/M/m queue. The average delay per packet is given by An M/M/1 system with the same arrival rate and service rate (statistical multiplexing with one channel having m times larger capacity), the average delay per packet is and denote the queueing probability

20. 20 When << 1 (light load),, and At light load, statistical MUX with m channels produces a delay almost m times larger than the delay of statistical MUX with the m channels combined in one. When,,, <<, and At heavy load, the ratio of the two delays is close to 1.

21. 21 4.6

22. 22

23. 23

24. 24

25. 25

26. 26 A.

27. 27 second moment of service time and load IF

28. 28

29. 29

30. 30

31. 31

32. 32 Roll-call Polling Stations are interrogated sequentially, one by one, by the central system, which asks if they have any messages to transmit. : walk time : frame transmission time

33. 33 The scan or cycle time is given by The average scan time, are the ave. walk time and the ave. time to transmit pkt at station. L is the total walk time of the complete poling system.

34. 34 For station, let : the ave. pkt arrival rate : the ave. packet length : the number of overhead bits C : the channel capacity in bps : the ave. frame length in time The average number of packets waiting to be transmitted when station is polled is, the time required to transmit is With the traffic intensity, the average scan time is given With representing the total traffic intensity on the common channel.

35. 35 For small the average access delay should be. Assume that each station has the same, same frame-length statistics, and the same. The average access delay is is the second moment of the frame length,. The access delay is the average time a packet must wait at a station from the time it first arrives until the time transmission begins. Access delay is thus the average wait time in an M/G/1 queue. Ref: Mischa Schwarty: “Telecommunication Networks, Protocols Modeling and Analysis”, Addison-Wesley Publishing Company, 1988, PP. 408-422

36. 36 Hub Polling Control is passed sequentially from one station to another. Let the polling message be a fixed value, t p sec in length. The time required per station to synchronize to a polling message is t s sec. The total propagation delay for the entire N-station system is sec.

37. 37 Hub Polling (Cont’) The total walk time for roll-call polling is. Let the stations all be equally spaced, and the round-trip propagation delay between the controller and station N be τ sec. The overall propagation delay is just The analysis of the hub-polling strategy is identical to that of roll-call polling. The only difference is that the walk time L is reduced through the use of hub polling. For hub polling,.