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
Last time: Reversibility and stationarity; various properties
• Definition: Reversible process: A stochastic process is reversible if for all t1,…,tn,
Skjjkqkkjqj ,),,()(),()(
))(),...,(),((~))(),...,(),(( 2121 nn tXtXtXtXtXtX
• Theorem: A stationary Markov chain is reversible if and only if there exists a collection of positive numbers π(j), jS, summing to unity that satisfy the detailed balance equations
When there exists such a collection π(j), jS, it is the equilibrium distribution
• Theorem 1.13: Kelly’s lemmaLet X(t) be a stationary Markov processwith transition rates q(j,k). If we can find a collection of numbers q’(j,k) such that q’(j)=q(j), jS, and a collection of positive numbers (j), jS, summing to unity, such that
then q’(j,k) are the transition rates of the time-reversed process, and (j), jS, is the equilibrium distribution of both processes.
),(')(),()( jkqkkjqj
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)
• PASTA
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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
distribution• Partial balance• Kelly/Whittle 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 4):
• Last time / Questions?• Output simple queue• Tandem network • Jackson network: definition• Jackson network: equilibrium
distribution• Partial balance• Kelly/Whittle 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
(Xj+1(t0),…, XJ(t0)). Thus X1(t0), X2(t0),…, XJ(t0) mutually
independent, and
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1
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inn
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• Summary / Next• Exercises
Jackson network : Definition
• Simple queues, exponential service queue j, j=1,…,J
• state
move
depart
arrive
• Transition rates
• Traffic equations
• Irreducible, unique solution, interpretation, exercise
• Jackson network: open
• Gordon Newell network: closed
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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
distribution• Partial balance• Kelly/Whittle network• Summary / Next• Exercises
Jackson network : Equilibrium distribution
• Simple queues,
• Transition rates
• Traffic equations
• Closed network
• Open network
• Global balance equations:
• Closed network:
• Open network:
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closed network : equilibrium distribution
• Transition rates
• Traffic equations
• Closed network
• Global balance equations:
• Theorem: The equilibrium distribution for the closed Jackson
network containing N jobs is
• Proof
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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
distribution• Partial balance• Kelly/Whittle network• Summary / Next• Exercises
Partial balance
• Global balance verified via partial balance
Theorem: If distribution satisfies partial balance, then it is
the equilibrium distribution.
• Interpretation partial balance
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Jackson network : Equilibrium distribution
• Transition rates
• Traffic equations
• Open network
• Global balance equations:
• Theorem: The equilibrium distribution for the open Jackson
network containing N jobs is, provided αj<1, j=1,…,J,
Proof
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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
distribution• Partial balance• Kelly/Whittle network• Summary / Next• Exercises
Kelly / Whittle network
• Transition rates
for some functions
:S[0,),
:S(0,)
• Traffic equations
• Open network
• Partial balance equations:
• Theorem: Assume that
then
satisfies partial balance,
and is equilibrium distribution Kelly / Whittle network
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n
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Examples
• Independent service, Poisson arrivals
• Alternative
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Examples
• Simple queue
• s-server queue
• Infinite server queue
• Each station may have different service type
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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
distribution• Partial balance• Kelly/Whittle network• Summary / Next• Exercises
Summary / next:
Equilibrium distributions• Reversibility• Output reversible Markov process• Tandem network• Jackson network• Partial balance• Kelly-Whittle network
NEXT: Sojourn times
Exercises[R+SN] 2.1.1, 2.1.2, 2.3.1, 2.3.4, 2.3.5, 2.3.6,
2.4.1, 2.4.2, 2.4.6, 2.4.7