Networks Plan for today (lecture 8):

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NetworksPlan for today (lecture 8)

• Last time / Questions?
• Quasi reversibility
• Network of quasi reversible queues
• Symmetric queues, insensitivity
• Partial balance vs quasi reversibility
• Proof of insensitivity?
• Summary
• Exercises
• Questions

2
Customer types queue discipline

• Customers ordered at queue
• Consider queue j, containing nj jobs
• Queue j contains jobs in positions 1,, nj
• Operation of the queue j(i) Each job requires
  exponential(1) amount of service.(ii) Total
  service effort supplied at rate ?j(nj)?(iii)
  Proportion ? j(k,nj) of this effort directed to
  job in position k, k1,, nj when this job
  leaves, his service is completed, jobs in
  positions k1,, nj move to positions k,, nj
  -1.(iv) When a job arrives at queue j he moves
  into position k with probability ?j(k,nj 1),
  k1,, nj 1 jobs previously in positions k,,
  nj move to positions k1,, nj 1.

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NetworksPlan for today (lecture 8)

• Last time / Questions?
• Quasi reversibility
• Network of quasi reversible queues
• Symmetric queues, insensitivity
• Partial balance vs quasi reversibility
• Proof of insensitivity?
• Summary
• Exercises
• Questions

4
Quasi-reversibility

• Multi class queueing network, class c ? C
• A queue is quasi-reversible if its state x(t) is
  a stationary Markov process with the property
  that the state of the queue at time t0, x(t0), is
  independent of(i) arrival times of class c
  customers subsequent to time t0(ii) departure
  times of class c custmers prior to time t0.
• TheoremIf a queue is QR then(i) arrival times
  of class c customers form independent Poisson
  processes(ii) departure times of class c
  customers form independent Poisson processes.

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Quasi-reversibility

• Multi class queueing network, class c ? C
• S(c,x) set of states queue contains one more
  class c than in state x
• Arrival rate class c customer
• Departure rate class c customer
• Characterise QR, combine

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NetworksPlan for today (lecture 8)

• Last time / Questions?
• Quasi reversibility
• Network of quasi reversible queues
• Symmetric queues, insensitivity
• Partial balance vs quasi reversibility
• Proof of insensitivity?
• Summary
• Exercises
• Questions

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Quasi-reversibility network

• Multi class queueing network, type i1,,I
• J queues
• Customer type identifies route
• Poisson arrival rate per type??(i), i1,,I
• Route r(i,1), r(i,2),,r(i,S(i))
• Type i at stage s in queue r(i,s)
• State X(t)(x1(t),,xJ(t))
• Construct a network by multiplying the rates for
  the individual queues
• Transition rates
• Arrival of type i causes queue kr(i,1) to change
  at
• Departure type i from queue j r(i,S(i))
• Routing

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Quasi-reversibility network

• Transition rates
• Theorem For an open network of QR queues(i)
  the states of individual queues are
  independent(ii) an arriving customer sees the
  equilibrium distri bution(ii) the equibrium
  distribution for a queue is as it would be in
  isolation with arrivals forming Poisson
  process.(iii) time-reversal another open
  network of QR queues(iv) system is QR, so
  departures form Poisson process
• Proof of part (i)

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Quasi-reversibility network

• Proof of part (i)
• Transition rates
• Transition rates reversed process (guess)

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Quasi-reversibility state aggregation,

• Transition rates
• x may be complicated state, consider only total
  number in component flow equivalent server

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NetworksPlan for today (lecture 8)

• Last time / Questions?
• Quasi reversibility
• Network of quasi reversible queues
• Symmetric queues, insensitivity
• Partial balance vs quasi reversibility
• Proof of insensitivity?
• Summary
• Exercises
• Questions

12
Symmetric queues insensitivity

• Operation of the queue j(i) Each job requires
  exponential(1) amount of service.(ii) Total
  service effort supplied at rate ?j(nj)?(iii)
  Proportion ? j(k,nj) of this effort directed to
  job in position k, k1,, nj when this job
  leaves, his service is completed, jobs in
  positions k1,, nj move to positions k,, nj
  -1.(iv) When a job arrives at queue j he moves
  into position k with probability ?j(k,nj 1),
  k1,, nj 1 jobs previously in positions k,,
  nj move to positions k1,, nj 1.
• Examples infinite server queue, lcfs, ps
• Symmetric queue QR for general service
  requirement
• Instanteneous attention
• Symmetric queue is insensitive

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NetworksPlan for today (lecture 8)

• Last time / Questions?
• Quasi reversibility
• Network of quasi reversible queues
• Symmetric queues, insensitivity
• Partial balance vs quasi reversibility
• Proof of insensitivity?
• Summary
• Exercises
• Questions

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Quasi-reversiblity vs Partial balance

• QR fairly general queues, service disciplines,
  Markov routing, product form equilibrium
  distribution factorizes over queues.
• PB fairly general relation between service rate
  at queues, state-dependent routing (blocking),
  product form equilibrium distribution factorizes
  over service and routing parts.
• Identical for single type queueing network with
  Markov routing

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NetworksPlan for today (lecture 8)

• Last time / Questions?
• Quasi reversibility
• Network of quasi reversible queues
• Symmetric queues, insensitivity
• Partial balance vs quasi reversibility
• Proof of insensitivity?
• Summary
• Exercises
• Questions

16
NetworksPlan for today (lecture 8)

• Last time / Questions?
• Quasi reversibility
• Network of quasi reversible queues
• Symmetric queues, insensitivity
• Partial balance vs quasi reversibility
• Proof of insensitivity?
• Summary
• Exercises
• Questions

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Exercises

• Exercise BlockingConsider a tandem network of
  two simple queues. Let the arrival rate to queue
  1 be Poisson ?, and let the service rate at each
  queue be exponential ?i , i1,2. Let queue 1 have
  capacity N1. For N1 ?, give the equilibrium
  distribution. For N1lt ? formulate three distinct
  protocols that preserve product form, indicate
  graphically what the implication of these
  protocols is on the transition diagram, and proof
  (by partial balance) that the equilbrium
  distribution is of product form.Can the product
  form results also be obtained via Quasi
  reversibility? If so, provide the proof of the
  product form result via quasi reversibility.
• RSN 3.2.3, 3.3.2
• Exercise insensitivityShow that the PS queue is
  insensitive.

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Networks (3TU) summary stochastic networks

• Contents
• Introduction Markov chains
• Birth-death processes Poisson process, simple
  queuereversibility detailed balance
• Output of simple queue Tandem network
  equilibrium distribution
• Jackson networksPartial balance
• Sojourn time simple queue and tandem network
• Performance measures for Jackson
  networksthroughput, mean sojourn time, blocking
• Application service rate allocation for
  throughput optimisationApplication optimal
  routing
• Quasi reversibility, customer types,
  insensitivity
• further readingRSN chapter 3 customer
  types chapter 4 examplesN chapter 10
• Next deterministic network flows