Chapter 3

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Chapter 3 Queueing Systems in Equilibrium

• Leonard Kleinrock, Queueing Systems, Vol I
  Theory
• Nelson Fonseca
• State University of Campinas, Brazil

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3. Birth-Death Queuing Systems in Equilibrium
Equations
Transient solution Solution in equilibrium can
not be generalized
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3.1. Solution in Equilibrium
Whereas pk is no longer a function of t, we are
not claiming that the process does not move from
state to state in this limiting case The
long-run probability of finding the system with k
members will be properly described by pk
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Flow rate into Ek ?k-1pk-1 ?k1pk1 Flow
rate out of Ek (?k ?k)pk In equilibrium
(flow in flow out)
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Rather than surrounding each state we could
choose a sequence of boundaries the first of
which surrounds E0, the second of which surrounds
E0 and E1, and so on, we would have the following
relationship
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Is there a solution in equilibrium?
Ergodic S1 lt ? S2 ? Recurrent null S1
? S2 ? Transient S1 ? S2 lt ?
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The MM1 queue
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Using
we have
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• Probability of exceeding ? geometrically
  decreasing

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• 3.3. Discouraged Arrivals

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• 3.4. MM8 (Infinite Server)

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• 3.5. MMm (m Server)

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• Pqueueing probability that no server is
  available in a system of m servers. Erlangs C
  formula ? C(m,l/m)

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• 3.6. MM1k Finite Storage
• K1 - Blocked calls cleared

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• 3.7. MMmm m-Server Loss System

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• 3.8. MM1M

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• 3.9. MM8 M

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• 3.10. MMmKM

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Problem 3.2

• Consider a Markovian queueing system in which
• Find the equilibrium probability pk of having k
  customers in the system. Express your answer in
  terms of p0.
• Give an expression for p0.

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• Solution

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• So
• Note for 0alt1, this system is always stable.

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Problem 3.5

• Consider a birth-death system with the following
  birth and death coefficients
• Solve for pk. Be sure to express your answers
  explicitly in terms of l, k, and m only.
• Find the average number of customers in the
  system.

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• Solution
• Here we demonstrate the differentiation trick
  for summing series (similar to that on page 69).

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• Since
• we have

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• Thus
• and so

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• (b)