Chap'3 : BirthDeath Queueing Systems in Equilibrium

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Chap.3 Birth-Death Queueing Systems in
Equilibrium

• S.Y. Yang

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General Equilibrium Solution
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Conservation?

• 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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conservation of flow across any closed boundary
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M/M/1 The Classical Queueing System

• All birth coefficients equal to a constant
• All death coefficients equal to a constant

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continued

• Average number of customers in the system
• Average time spent in the system

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continued

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

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Graph of the Average Number of Packets in the
M/M/1 Queue
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close clear x00.0010.980 y(x)./((1-x)) pl
ot(x,y) grid on xlabel('Utilization') ylabel('A
verage number in queueing system')
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System State Distribution for the M/M/1 Queue
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close clear x0120 for i020 y1(i1)
PrSystemSizeisX(0.2,i) y2(i1)
PrSystemSizeisX(0.4,i) y3(i1)
PrSystemSizeisX(0.472,i) y4(i1)
PrSystemSizeisX(0.6,i) y5(i1)
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') l
egend('rho0.2','rho0.4','rho0.472','rho0.6','r
ho0.8',3)
function yPrSystemSizeisX(rho,x) y(1-rho)(rho
x)
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Average waiting time for M/M/1 Queue(ATM case)
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close clear for i0950 x(i1) i/1000
y1(i1) MM1tw(x(i1),2.831) y2(i1)
MD1tw(x(i1),2.831) y3(i1) 2.831
end plot(x,y1,x,y2,x,y3) plot(x,y1,x,y3) gri
d on xlabel('Utilization') ylabel('Average
waiting time (in microseconds)') legend('M/M/1','
1 cell time1/u2.831us, STM-1 case',2)
function yMM1tw(rho,s) y(rhos) / ((1-rho)eps)
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Estimate Loss Probability for the M/M/1 Queue
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close clear k 24 for i200980 x(i1)
i/1000 y1(i1) PrSystemSizeisX
(x(i1),k) y2(i1) PrSystemSizeisGTX(x(i1),k)
end semilogy(x,y1,x,y2) semilogy(x,y2)
grid on xlabel('Utilization') ylabel('Probabil
ity') legend('Queue size k24', 'Queue size
kgt24',2)
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Discouraged Arrivals

• Birth and Death Coefficients

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M/M/Infinite Responsive Servers

• Birth and Death Coefficients

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M/M/m The m-server case

• Birth and Death Coefficients

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M/M/1/K Finite Storage

• Birth and Death Coefficients

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M/M/m/m m-server Loss System

• Birth and Death Coefficients
• The fraction of time that all m servers are busy.

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M/M/1//M Finite Customer Population Single Server

• Birth and Death Coefficients

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M/M/Infinite//M Finite Customer Population,
Infinite Number of Server

• Birth and Death Coefficients

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M/M/m/K/M Finite Population, m-Server, Finite
Storage

• Birth and Death Coefficients

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References

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