Flows and Networks Plan for today (lecture 4): • Last time / Questions? • Output simple queue • Tandem network • Jackson network: definition • Jackson network: equilibrium distribution • Partial balance • Kelly/Whittle network • Examples • Summary / Next • Exercises
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Flows and Networks Plan for today (lecture 4): Last time / Questions? Output simple queue Tandem network Jackson network: definition Jackson network: equilibrium.
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Flows and Networks
Plan for today (lecture 4):
• Last time / Questions?
• Output simple queue• Tandem network • Jackson network: definition• Jackson network: equilibrium
• Last time / Questions?• Reversibility, stationarity• Truncation• Kolmogorov’s criteria• Poisson process• PASTA• Output simple queue• Tandem network• Summary / Next• Exercises
Poisson process• Definition : Poisson process :
Let S1,S2,… be a sequence of independent exponential() r.v. Let Tn=S1+…+Sn, T0=0 and N(s)=max{n,Tn≤s}. The counting process {N(s),s≥0} is called Poisson process.
• Theorem : If {N(s),s≥0} is a Poisson process, then(i) N(0)=0,(ii) N(t+s)-N(s)=Poisson( t), and(iii) N(t) has independent increments.Conversely, if (i), (ii), (iii) hold, then {N(s),s≥0} is a Poisson process
Flows and Networks
Plan for today (lecture 3):
• Last time / Questions?• Reversibility, stationarity• Truncation• Kolmogorov’s criteria• Poisson process• PASTA• Output simple queue• Tandem network• Summary / Next• Exercises
PASTA: Poisson Arrivals See Time Averages
• fraction of time system in state n
• probability outside observer sees n customers at time t
• probability that arriving customer sees n
customers at time t (just before arrival at time t there
are n customers in the system)
• in general
)(,' tP nn
)(0,' tP nn
)()( 0,',' tPtP nnnn
PASTA: Poisson Arrivals See Time Averages
• Let C(t,t+h) event customer arrives in (t,t+h)
• For Poisson arrivals q(n,n+1)= so that
• Alternative explanation;
• PASTA holds in general!
PASTA
)()1,(
)()1,(
)()]()1,([
)()]()1,([lim
}')0(|)(Pr{}')0(,)(|),(Pr{
}')0(|)(Pr{}')0(,)(|),(Pr{lim
}')0(),,(|)(Pr{lim)(
,'0
,'
,'0
,'
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tPkkq
tPnnq
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tPhohnnq
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nXntXnXntXhttC
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knk
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knk
nn
h
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hnn
)()( 0,',' tPtP nnnn
Flows and Networks
Plan for today (lecture 3):
• Last time / Questions?• Reversibility, stationarity• Truncation• Kolmogorov’s criteria• Poisson process• PASTA• Output simple queue• Tandem network• Summary / Next• Exercises
Output simple queue
• Simple queue, Poisson() arrivals, exponential() service
• X(t) number of customers in M/M/1 queue:
in equilibrium reversible Markov process.
• Forward process: upward jumps Poisson ()
• Reversed process X(-t): upward jumps Poisson ()
= downward jump of forward process
• Downward jump process of X(t) Poisson () process
Output simple queue (2)
• Let t0 fixed. Arrival process Poisson, thus arrival process
after t0 independent of number in queue at t0.
• For reversed process X(-t): arrival process after –t0
independent of number in queue at –t0
• Reversibility: joint distribution departure process up to t0
and number in queue at t0 for X(t) have same distribution
as arrival process to X(-t) up to –t0 and number in queue
at –t0.
• In equilibrium the departure process from an M/M/1 queue
is a Poisson process, and the number in the queue at time
t0 is independent of the departure process prior to t0
• Holds for each reversible Markov process with Poisson
arrivals as long as an arrival causes the process to
change state
Flows and Networks
Plan for today (lecture 3):
• Last time / Questions?• Reversibility, stationarity• Truncation• Kolmogorov’s criteria• Poisson process• PASTA• Output simple queue• Tandem network• Summary / Next• Exercises
Tandem network of simple queues
• Simple queue, Poisson() arrivals, exponential() service
• Equilibrium distribution
• Tandem of J M/M/1 queues, exp(i) service queue i
• Xi(t) number in queue i at time t
• Queue 1 in isolation: simple queue.
• Departure process queue 1 Poisson,
thus queue 2 in isolation: simple queue
• State X1(t0) independent departure process prior to t0,
but this determines (X2(t0),…, XJ(t0)), hence X1(t0)
independent (X2(t0),…, XJ(t0)). Similar Xj(t0) independent
• Last time / Questions?• Waiting time simple queue• Little• Sojourn time tandem network• Jackson network: mean sojourn time• Summary / Next• Exercises
Waiting time simple queue (1)
• Consider simple queue FCFS discipline– W : waiting time typical customer in
M/M/1(excludes service time)
– N customers present upon arrival
– Sr (residual) service time of customers present
PASTA
Voor j=0,1,2,…
tkj
k
r
j
r
j
ek
t
tSPjNtWP
jNtWPjtWP
!
)(
)()1|(
)1|()1()(
0
1
1
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t
t
t
tt
kj
jk
kk
k
t
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k
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EXEWEF
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)1(
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0
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111
)1(
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1)1(
!
)()1(
!
)()1()(
Waiting time simple queue (2)
• Thus
• is exponential (-)
Flows and Networks
Plan for today (lecture 5):
• Last time / Questions?• Waiting time simple queue• Little• Sojourn time tandem network• Jackson network: mean sojourn time• Summary / Next• Exercises
Little’s law (1)
• Let– A(t) : number of arrivals entering in (0,t]– D(t) : number of departure from system
(0,t]– X(t) : number of jobs in system at time t
)()()0()( tDtRXtX
• Equilibrium for t∞
• In equilibrium: average number of arrivals per time unit = average number of departures per time unit
)(1
lim)(1
lim
0)(1
lim
tDt
tAt
tXt
tt
t
Little’s law (2)
)()0( tRX
• Fj sojourn time j-th departing job
• S(t) obtained sojourn times jobs in system at t
t
)(tD
)(uX
)()()(
10
tSFduuX j
tD
j
t
• Assume following limits exist(ergodic theory, see SMOR)
• Then
• Little’s law
Little’s law (3)
j
n
jn
tt
t
t
Fn
F
tDt
tAt
duuXt
X
1
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1lim
)(1
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)(1
lim
)()()(
10
tSFduuX j
tD
j
t
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t
)(tD
)(uX
FX
tSt
FtDt
tDduuX
t j
tD
j
t
)(1
)(
1)()(
1 )(
10
Little’s law (4)
• Intuition– Suppose each job pays 1 euro per time
unit in system– Count at arrival epoch of jobs: job pays
at arrival for entire duration in system, i.e., pays EF
– Total average amount paid per time unit EF
– Count as cumulative over time: system receives on average per time unit amount equal to average number in system
– Amount received per time unit EX
• Little’s law valid for general systems irrespective of order of service, service time distribution, arrival process, …
Flows and Networks
Plan for today (lecture 5):
• Last time / Questions?• Waiting time simple queue• Little• Sojourn time tandem network• Jackson network: mean sojourn time• Summary / Next• Exercises
• Recall: In equilibrium the departure process from an M/M/1 queue is a Poisson process, and the number in the queue at time t0 is independent of the departure process prior to t0
• Theorem 2.2: If service discipline at each queue in tandem of J simple queues is FCFS, then in equilibrium the waiting times of a customer at each of the J queues are independent
• Proof: Kelly p. 38
• Tandem M/M/s queues: overtaking
• Distribution sojourn time: Ex 2.2.2
Sojourn time tandem simple queues
Flows and Networks
Plan for today (lecture 4):
• Last time / Questions?• Output simple queue• Tandem network • Jackson network: definition• Jackson network: equilibrium