Introduction to Queuing Theory

1
Chapter 30

• Introduction to Queuing Theory

2
Exercise 30.1

• What can you say about a queue denotes by
  Ek/Gx/5/300/5000/LCFS-PR?
• Time between successive arrival is Erlang
  distribution with parameter k.
• Bulk service with general service time
  distribution.
• 5 servers.
• 300 buffers 5 service 295 waiting
• Total of 5000 jobs that can be serviced.
• Service disciple is fist-come-first served with
  Preempt and Resume.

3
Exercise 30.2

• Why is it not good idea to have an
  Hk/G/12/10/5/LCFS?
• There are 12 server but system capacity is 10,
  that mean 2 server will never be used.
• The system capacity is 10, but the population is
  5 only, in other word, only 5 jobs can be
  services. Therefore, there is not enough jobs to
  fill up the system.

4
Exercise 30.3

• Which queuing system would provide better
  performance an M/M/3/300/100 system or an
  M/M/3/100/100 system?
• The M/M/3/100/100 system would provide better
  performance, since the system capacity is equal
  to the population size. However, M/M/3/300/100,
  the system capacity is large than the population,
  therefore, the system cannot be fully utilized,
  some resource is wasted.

5
Exercise 30.4

• During a one-hour observation interval, the name
  server of a distributed system received 10800
  requests. The mean response time of these
  requests was observed to be one-third of a
  second. What is the mean number of quires in the
  server? What assumptions have you made about the
  system? Would the mean number of quires be
  different if the service time was not
  exponentially distributed?

6
Exercise 30.4 (II)

• By Littles Law
• mean number of quires
• arrival rate x mean time spent in the system
• 10800/3600 x 1/3
• 1
• Assume no job is loss.
• As we take the mean time spent in the system, the
  distribution of the service time will not affect
  the mean number of quires.

7
Exercise 30.5

• When can the arrivals to an Ek/M/1 queue be
  called a Poisson process?
• The Poisson process is the process where the
  interarrival time are (Independent and
  Identically Distributed) IID and exponentially
  distributed.
• When h1, it will be exponentially distributed.