Solving the Balance Equations

• Lets solve the balance equations:

• For n=0 we get ????

• If we let , this becomes

  

• Similarly

  

• And in general

  ????

• We have

• We need to solve for , so we need one more equation. Use

  

• We obtain

  

• So we must have

  

• Note that requiring for stability (i.e. ) makes intuitive sense.
• Also
  
  = probability that the queueing system is NOT empty

  = probability that the server is working.

  So is sometimes called the ``server utilization"

  Finally note that is a geometric distribution.