Little's Formula and Queueing Delay

• Let T = time spent by a customer in a queueing system (waiting and being served).

• = the average delay for a customer.

• Little's Formula says where is the ``arrival rate for customers eventually served" (which we had called earlier).

• Little's Formula holds for very general queueing systems (not just M/M/1). Even whole networks!

• Little's Formula is either deep or obvious.

• Intuition:
  • Pick a ``typical customer."

  • When it arrives to the queueing system, it should find customers waiting.

  • When it leaves the system, it has been in the system for .
    • Thus, customers should have arrived during it's time in the system.
  • In steady state, the number of customers left behind on departure should equal the number found on arrival, i.e.
  
  

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  • Applying Little's Formula to an M/M/1 Queue
  
  

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  Ken Vastola
  Fri Mar 15 14:12:07 EST 1996