Queueing Theory Delay Models

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Queueing Theory (Delay Models)

• Sunghyun Choi
• Adopted from Prof. Saewoong Bahks material

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Time Reversibility - Burkes theorem

• Birth-death process Markov chain with integer
  states in that transitions occur only between
  neighboring states
• Consider a discrete-time Markov chain Xn, Xn1 ,

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• The backward transition probability
• If for all i,j the chain is time
  reversible!!!

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• Properties of the reversed chain (discrete time)
• (P1) Irreducible, aperiodic, the same stationary
  distribution as the forward chain
• Proof

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• (P2) If there are pi s.t.
• form a transition prob. matrix, i.e.,
  ,
• then pi is the stationary distribution and
• are the transition prob. of reversed
  chain.
• Proof

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• (P3) A chain is time reversible iff
• (P1) (P2) hold even if chain is not time
  reversible!
• Birth-death process (e.g., M/M/1, M/M/m, etc.)
  are time reversible

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• For irreducible CTMC
• (P1) The same stationary distribution as the
  forward chain with
• (P3) The forward chain is time reversible iff
• (detailed balance eq.)

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• (P2) If there are pi s.t. and
  
• then pi is the stationary distribution and
• are the transition rates of reversed
  chain.
• Proof

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time reversible chain
Non-time reversible chain
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In steady-state, forward and reverse systems are
statistically indistinguishable!
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• Burkes theorem Consider an M/M/1, M/M/m, M/M/8
  system with arrival rate ?. Suppose that the
  system starts in steady-state. Then the following
  is true
• The departure process is Poisson with rate ?.
• At each time t, the number of customers in the
  system is independent of the sequence of
  departure times prior to t.

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EX) Two M/M/1 queues in tandem

• - Poisson arrival and exp. service time
• - Assume that the service times of a customer at
  1st and 2nd queues are mutually independent and
  independent of arrival process.