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Chap.3 : Birth-Death Queueing Systems in Equilibrium

Chap.3 : Birth-Death Queueing Systems in Equilibrium. S.Y. Yang. General Equilibrium Solution. Conservation?. 0. 1. 2. In equilibrium case it is clear that flow must be conserved in the sense that the input flow must equal the output flow from a given state.

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Chap.3 : Birth-Death Queueing Systems in Equilibrium

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  1. Chap.3 : Birth-Death Queueing Systems in Equilibrium S.Y. Yang

  2. General Equilibrium Solution

  3. Conservation? 0 1 2 • In equilibrium case it is clear that flow must be conserved in the sense that the input flow must equal the output flow from a given state.

  4. conservation of flow across any closed boundary

  5. M/M/1: The Classical Queueing System • All birth coefficients equal to a constant • All death coefficients equal to a constant

  6. continued • Average number of customers in the system • Average time spent in the system

  7. continued • The probability of finding at least k customers in the system

  8. Graph of the Average Number of Packets in the M/M/1 Queue

  9. close; clear; x=0:0.001:0.980; y=(x)./((1-x)); plot(x,y); grid on; xlabel('Utilization'); ylabel('Average number in queueing system');

  10. System State Distribution for the M/M/1 Queue

  11. close; clear; x=0:1:20; for i=0:20 y1(i+1) = PrSystemSizeisX(0.2,i); y2(i+1) = PrSystemSizeisX(0.4,i); y3(i+1) = PrSystemSizeisX(0.472,i); y4(i+1) = PrSystemSizeisX(0.6,i); y5(i+1) = PrSystemSizeisX(0.8,i); end semilogy(x,y1,'d',x,y2,'s', x,y3,'v', x,y4,'^',x,y5,'o'); grid on; xlabel('Queue size'); ylabel('Probability'); legend('rho=0.2','rho=0.4','rho=0.472','rho=0.6','rho=0.8',3); function y=PrSystemSizeisX(rho,x) y=(1-rho)*(rho^x);

  12. Average waiting time for M/M/1 Queue(ATM case)

  13. close; clear; for i=0:950 x(i+1) = i/1000; y1(i+1) = MM1tw(x(i+1),2.831); y2(i+1) = MD1tw(x(i+1),2.831); y3(i+1) = 2.831; end %plot(x,y1,x,y2,x,y3); plot(x,y1,x,y3); grid on; xlabel('Utilization'); ylabel('Average waiting time (in microseconds)'); legend('M/M/1','1 cell time=1/u=2.831us, STM-1 case',2); function y=MM1tw(rho,s) y=(rho*s) / ((1-rho)+eps);

  14. Estimate Loss Probability for the M/M/1 Queue

  15. close; clear; k = 24; for i=200:980 x(i+1) = i/1000; y1(i+1) = PrSystemSizeisX (x(i+1),k); y2(i+1) = PrSystemSizeisGTX(x(i+1),k); end semilogy(x,y1,x,y2); %semilogy(x,y2); grid on; xlabel('Utilization'); ylabel('Probability'); legend('Queue size k=24', 'Queue size k>24',2);

  16. Discouraged Arrivals • Birth and Death Coefficients

  17. M/M/Infinite: Responsive Servers • Birth and Death Coefficients

  18. M/M/m: The m-server case • Birth and Death Coefficients

  19. M/M/1/K: Finite Storage • Birth and Death Coefficients

  20. M/M/m/m: m-server Loss System • Birth and Death Coefficients • The fraction of time that all m servers are busy.

  21. M/M/1//M: Finite Customer Population Single Server • Birth and Death Coefficients

  22. M/M/Infinite//M: Finite Customer Population, Infinite Number of Server • Birth and Death Coefficients

  23. M/M/m/K/M: Finite Population, m-Server, Finite Storage • Birth and Death Coefficients

  24. References • Queueing Systems Volume I; Theory, Leonard Kleinrock, 1975, JW&S http://www.lk.cs.ucla.edu/ • Introduction to IP and ATM Design and Performance, 2nd Edition, Pitts & Schormans, 2000, JW&S http://www.elec.qmw.ac.uk/ipatm/ • Lecture Notes from http://vega.icu.ac.kr/~bnec/ written by Professor J.K. Choi.

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