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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 S.Y. Yang
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.
M/M/1: The Classical Queueing System • All birth coefficients equal to a constant • All death coefficients equal to a constant
continued • Average number of customers in the system • Average time spent in the system
continued • The probability of finding at least k customers in the system
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');
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);
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);
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);
Discouraged Arrivals • Birth and Death Coefficients
M/M/Infinite: Responsive Servers • Birth and Death Coefficients
M/M/m: The m-server case • Birth and Death Coefficients
M/M/1/K: Finite Storage • Birth and Death Coefficients
M/M/m/m: m-server Loss System • Birth and Death Coefficients • The fraction of time that all m servers are busy.
M/M/1//M: Finite Customer Population Single Server • Birth and Death Coefficients
M/M/Infinite//M: Finite Customer Population, Infinite Number of Server • Birth and Death Coefficients
M/M/m/K/M: Finite Population, m-Server, Finite Storage • Birth and Death Coefficients
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.