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Chapter 4 Fundamental Queueing System 1 2 3 4 p T T m 5 6 7 8 9 10 11 12 13 =r net arrival rate= The through pat r/μ= (1 P0 ) 1 1 P P(1 P N ) (1 P N 1 ) 1 P N 1 14 15 i i p m 16 WQ T 1 W 1 NQ PQ m 17 Example 1 Statistical Multiplexing Compared with TDM and FDM Assume m statistically iid Poisson packet streams each with an arrival rate of m packets/sec. The packet lengths for all streams are independent and exponentially distributed. The average transmission time is 1 . If the streams are merged into a single Poisson stream, with rate , the average delay per packet is T 1 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 m and average service rate m . Therefore, the average delay per m packet is T 18 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 1 . The system can be modeled by the same Markov chain as the M/M/m queue. The average delay per packet is given by PQ 1 T m An M/M/1 system with the same arrival rate and service rate m (statistical multiplexing with one channel having m times larger capacity), the average delay per packet is PˆQ 1 Tˆ m m PQ and pˆ Q denote the queueing probability 19 When << 1 (light load) PQ 0 , pˆ Q 0 , and T m ˆ T 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 1 , PQ 1 , pˆ Q 1, 1 << 1 (m ) , and T 1 ˆ T At heavy load, the ratio of the two delays is close to 1. 20 4.6 21 22 23 24 W R W W R W 25 A. 26 second moment of service time and load IF X2 W 27 2 28 29 30 31 Roll-call Polling Stations are interrogated sequentially, one by one, by the central system, which asks if they have any messages to transmit. time Wi : walk time ti : frame transmission time 32 The scan or cycle time t c is given by The average scan time tc wi , t are the ave. walk time and the ave. time to transmit pkt at station i . i L is the total walk time of the complete poling system. 33 For station i , let i : the ave. pkt arrival rate : the ave. packet length : the number of overhead bits C : the channel capacity in bps mi : the ave. frame length in time mi ( ) / C The average number of packets waiting to be transmitted when station polled is t , the time required to transmit is i With c is ti i tc mi i t c i i mi N i the traffic intensity, the average scan time t c is given N With i i mi representing the total traffic intensity on the common i 1 channel. i 1 34 For small the average access delay should be tc / 2 . Assume that each station has the same and the same w . , same frame-length statistics, The average access delay is Nm m 2 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 35 Hub Polling • Control is passed sequentially from one station to another. • Let the polling message be a fixed value, tp sec in length. • The time required per station to synchronize to a polling message is ts sec. • The total propagation delay for the entire N-station system is sec. 36 Hub Polling (Cont’) • The total walk time for roll-call polling is L Nt p Nts . • 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 (1 N ) 2 • 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, Lhub Nt s . 37