Computer Networking Queueing A Summary from Appendix A

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Computer NetworkingQueueing (A Summary from
Appendix A)

• Dr Sandra I. Woolley

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Queueing

• Source A summary of Appendix A
• Delay analysis and Littles formula
• A basic queueing model

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Delay Analysis

• A basic model for a delay/loss system
• Time spent in system T
• No. customers in system N(t)
• Fraction of arriving customers that are lost or
  blocked Pb
• Long term arrival rate ?
• Average no of messages/second that pass through
  throughput

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Key System Variables

• From time 0 to time t
• Number of arrivals at the system A(t)
• Number of blocked customers B(t)
• Number of departed customers D(t)
• Number of customers in system at time t (assuming
  system was empty at t 0)
• Long term arrival rate is
• Throughput is
• Average system occupancy is
• Fraction of blocked customers is

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Arrival Rates and Traffic Loads
Assuming inter-arrival times are iid
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Littles Formula

• We need to link the average time spent in the
  system to the average number of customers in the
  system
• Assume non-blocking system
• Examine the arrivals and departures in a FIFO
  system

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Arrivals and Departures in a FIFO System

• Time T1 spent by 1st customer in system is the
  time between A(t) going from 0 to 1 to the time
  D(t) going from 0 to 1
• T1 is the area of the first rectangle at the
  lower left hand side in the figure (its height is
  1)

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Arrivals and Departures in a FIFO System

• The time average of the no of customers in the
  system up to time t0 is,
• This equation states that, up to time t0, the
  average number of customers in the system is the
  product of average arrival rate (1st fraction)
  and the arithmetic average of the times spent in
  the system by the first A(t0 ) customers

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Littles Formula

• EN ?ET (without blocking)
• Valid irrespective of whether customers are
  processed in order of arrival!
• EN ?(1 Pb)ET (with blocking)

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Application of Littles Formula

• Consider an entire network of queuing systems

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Application of Littles formula

• Littles formula states that ENnet
  ?netETnet
• Thus, ETnet ENnet/?net
• For the mth switch/multiplexer ENm ?mETm
• The total no of packets in the network is the sum
  of the total no of packets in the switches
  ENnet Sm ENm Sm ?mETm
• Combining the above equations yields,
• ETnet 1/?net Sm ?mETm
• Network delay depends on overall arrival rate in
  network (offered traffic), arrival rate to
  individual routers (determined by routing
  algorithm) and delay in each router (determined
  by arrival rate, switching capacity and
  transmission line capacity)

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Basic Queueing Models
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Arrival Processes

• Arrival rate to system ? 1/Et customers/s
• Deterministic ? constant (equal) inter-arrival
  times
• Exponential ? exponential random variables with
  mean Et 1/? ? Pt gt t exp(?t) for t gt 0
• It can easily be shown that (do this) A(t) is a
  Poisson random arrival process,
• PA(t) k (?t)kexp(?t)/k!

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Service Times

• Resources are denoted by servers because their
  function is to serve customer requests
• The time required to service a customer is called
  the service time, X
• Processing capacity, m, is given by
• m 1/EX customers/s

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Queueing Model Classification
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Queueing system variables

• Useful for calculations
• EN ?(1 Pb)ET
• ENq ?(1 Pb)EW
• ENs ?(1 Pb)EX

• Offered (traffic) load ?/ m Erlangs
• (rate at which work arrives at system)
• Carried load ?/ m (1 Pb)
• (average rate at which system does work)
• Utilisation ?/ m c(1 Pb)
• (average rate at which system does work)

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Thank You