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5.3 Queueing Models for ATM Multiplexers ( 5.3.1 – 5.3.1.3 )

데이터 통신 특론. 5.3 Queueing Models for ATM Multiplexers ( 5.3.1 – 5.3.1.3 ). 3 조 A0111517 장재승 A1111504 김대연 A1314502 이재곤. Contents. Introduction to Queueing Models Little’s Fomula The M/M/1 Queueing System Poisson Arrivals Set Time Averages. Introduction to Queueing Models.

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5.3 Queueing Models for ATM Multiplexers ( 5.3.1 – 5.3.1.3 )

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  1. 데이터 통신 특론 5.3 Queueing Models for ATM Multiplexers( 5.3.1 – 5.3.1.3 ) 3조 A0111517 장재승 A1111504 김대연 A1314502 이재곤

  2. Contents • Introduction to Queueing Models • Little’s Fomula • The M/M/1 Queueing System • Poisson Arrivals Set Time Averages HONGIK UNIV. Automated Design LAB.

  3. Introduction to Queueing Models • Little’s and Pollaczek-Khintchine’s formula • Poisson arrivals and time averages • The continuous time M/M/1, M/G/1, M/D/1 queueing system • The discrete-time Geo/D/1 model The main focus is quasi-only devoted to the queue length distribution which leads to the calculation of the cell probability and the random transfer delay. HONGIK UNIV. Automated Design LAB.

  4. Little’s Formula • System begin at t=0 • Si : customer arrival times • A(t): # of customer arrivals up to time t • D(t): # of customer departures up to time t • N(t) = A(t) – D(t): # of customers in system i th customers spends time Ti in the system, and then departs at time Di = Si + Ti HONGIK UNIV. Automated Design LAB.

  5. Little’s Formula • Time average of N(t) • The above of integral is exactly given by the sum of the Ti of the first A(t) so. HONGIK UNIV. Automated Design LAB.

  6. Little’s Formula • The average arrival rate up to time t is given by The average of the time spent in system by the first A(t) HONGIK UNIV. Automated Design LAB.

  7. Little’s Formula • The cumulative cross-hatched area in between arrival and departure curves HONGIK UNIV. Automated Design LAB.

  8. The M/M/1 Queueing System • N(t): The number of customer • A(t): The number of arrivals 1. The probability of one arrival in an interval of length  2. Similarly, the probability of more than one arrival is HONGIK UNIV. Automated Design LAB.

  9. The M/M/1 Queueing System 3. :exponential random variable 4. The probability of one arrival and one departure in an interval of length HONGIK UNIV. Automated Design LAB.

  10. The M/M/1 Queueing System • The global balance equation for the steady state probability (j = 1,2….) when The mean number of customers in the system HONGIK UNIV. Automated Design LAB.

  11. The M/M/1 Queueing System • The mean total customer delay in the system The mean waiting time in queue HONGIK UNIV. Automated Design LAB.

  12. The M/M/1 Queueing System • The mean number in queue The server utilization HONGIK UNIV. Automated Design LAB.

  13. The M/M/1 Queueing System • The average queue length as the load HONGIK UNIV. Automated Design LAB.

  14. Poisson Arrivals See Time Averages • To perceive the whole general sense of Poisson arrivals see time average • The sojourn time • The fraction of Poisson arrivals N(t): spends in some state to the Poisson process of rate A(t): which sees the system in that state U(t): An arbitrary state B in the state-space of N(t) = 1 if N(t) B U(t) = 0 O.W. HONGIK UNIV. Automated Design LAB.

  15. Poisson Arrivals See Time Averages • The fraction of time V(t) • The number of arrivals Y(t) • The fraction Z(t) of arrivals HONGIK UNIV. Automated Design LAB.

  16. Poisson Arrivals See Time Averages • Calculating the expectation on Y(t) • R(t): The intermediate process the later property of R(t) HONGIK UNIV. Automated Design LAB.

  17. Poisson Arrivals See Time Averages • To be further convinced 1. R(t) is a continuous-time martingle • The ratio tends to 0 when t tends to The theorem according to Poisson arrivals see time averages holds also for Bernoulli and the non-stationary Poisson arrivals. HONGIK UNIV. Automated Design LAB.

  18. 참 고 자 료 • Othma Kyas, ‘ATM Networks’, International Thomson Publishing Asia, 1997 • Tominaga Hideyoshi, ‘표준 ATM’, 교보문고, 1996 • Alberto Leon-Garcia, ‘Probability and Random Processes for Electrical Engineering’, Addison Wesley, 1994 • http://osl7.kaist.ac.kr/books/stochastic_modeling/sm.htm • http://www.terms.co.kr/queueingtheory.htm • http://kices.knu.ac.kr/courses/qtheory/html/week13.html HONGIK UNIV. Automated Design LAB.

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