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Queueing Properties

  • In a queue, we can model an arrival process with a deterministic process. However, when aggregated, arrival processes are random events. So we model them with a random process.
  • The study of random arrival processes is part of a discipline called Queueing theory. This study shows that random arrival processes have interesting properties:
    • Traffic burstiness increases delay.
    • Determinism reduces delay

queueing-properties

  • Poisson process: models an aggregation of random events, eg the incoming phone calls at a telecom switch.
  • Generally we can not use the Poisson process with queue arrivals. However, we can use the Poisson process with new flows in some types of events such as Web requests or new user connections
  • Packet arrival on the network is not a Poisson process.
  • Little’s Result, is a simplistic queueing model
    • given:
      • λ the arrival rate,
      • L: the number of packets in the queue waiting and not served,
      • d: the average delay spent by packets waiting in the queue and not being served
      • then L = λ * d

queueing-properties-littles-result

note:

L can also be defined as the number of packets in the queue and being served, and d as the average delay spent by packets in the queue and being served.

  • M/M/1 queue model gives a good intuition about the arrival rate, queue occupancy and the service rate. However, it is not an accurate measure of the queue occupancy.
  • The Poisson process is used in the M/M/1 queue model
    • given:
      • λ: the arrival rate,
      • µ: the service rate
      • then d = 1 / (µ – λ)

and having L = λ* d, in the Little’s Result,

then L = λ / (µ – λ) = (λ / µ)  / (1 – λ/ µ)

queueing-properties-mm1-queue

the M/M/1 Queueing model – © Standford.edu

 

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