Introduction to Queuing Theory

1
Introduction to Queuing Theory
2
Queuing theory

• View network as collections of queues
• FIFO data-structures
• Queuing theory provides probabilistic analysis of
  these queues
• Examples
• Average length
• Probability queue is at a certain length
• Probability a packet will be lost

3
Littles Law
System
Arrivals
Departures

• Littles Law Mean number tasks in system
  arrival rate x mean reponse time
• Observed before, Little was first to prove
• Applies to any system in equilibrium, as long as
  nothing in black box is creating or destroying
  tasks

4
Proving Littles Law
Arrivals
Packet
Departures
1 2 3 4 5 6 7 8
Time
J Shaded area 9 Same in all cases!
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Definitions

• J Area from previous slide
• N Number of jobs (packets)
• T Total time
• l Average arrival rate
• N/T
• W Average time job is in the system
• J/N
• L Average number of jobs in the system
• J/T

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Proof Method 1 Definition

Time (T)
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Proof Method 2 Substitution
Tautology
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Model Queuing System
Queuing System
l
l
m
Server
Queue
Queuing System
Server System

• Use Littles law on complete system and parts to
  reason about average time in the queue

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Example using Littles law

• Dining hall
• Observe 250 people waiting to be checked into the
  serving line
• count departing rate of being served in food line
  as 23 people per minute
• What is the average time spent in and before the
  food line?

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Kendal Notation

• Six parameters in shorthand
• First three typically used, unless specified
• Arrival Distribution
• Service Distribution
• Number of servers
• Total Capacity (infinite if not specified)
• Population Size (infinite)
• Service Discipline (FCFS/FIFO)

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Distributions

• M Exponential
• D Deterministic (e.g. fixed constant)
• Ek Erlang with parameter k
• Hk Hyperexponential with param. k
• G General (anything)
• M/M/1 is the simplest realistic queue

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Kendal Notation Examples

• M/M/1
• Exponential arrivals and service, 1 server,
  infinite capacity and population, FCFS (FIFO)
• M/M/m
• Same, but M servers
• G/G/3/20/1500/SPF
• General arrival and service distributions, 3
  servers, 17 queue slots (20-3), 1500 total jobs,
  Shortest Packet First

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Analysis of M/M/1 queue

• Goal closed form expression of the probability
  of the number of jobs in the queue (Pi) given
  only l and m

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M/M/1 queue model
L

Lq


Wq
W
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Solving queuing systems

• Given
• l Arrival rate of jobs (packets)
• m Service rate of the server (output link)
• Solve
• L average number in queuing system
• Lq ave. number in the queue
• W ave. waiting time in whole system
• Wq ave. waiting time in the queue
• 4 unknowns need 4 equations

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Solving queuing systems

• 4 unknowns L, Lq W, Wq
• Relationships
• LlW
• LqlW (steady-state argument)
• W Wq (1/m)
• If we know any 1, can find the others
• Finding L is hard or easy depending on the type
  of system. In general

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Solving for P0 and Pn
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Solving for P0 and Pn

• Step 1
• Step 2

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Solving for P0 and Pn

• Step 3
• Step 4

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Example

• On a network gateway, measurements show that the
  packets arrive at a mean rate of 125 packets per
  second (pps) and the gateway takes about 2
  millisecs to forward them. Assuming an M/M/1
  model, what is the probability of buffer overflow
  if the gateway had only 13 buffers. How many
  buffers are needed to keep packet loss below one
  packet per million?

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Example

• Measurement of a network gateway
• mean arrival rate (l) 125 Packets/s
• mean response time (m) 2 ms
• Assuming exponential arrivals
• What is the gateways utilization?
• What is the probability of n packets in the
  gateway?
• mean number of packets in the gateway?
• The number of buffers so P(overflow) is lt10-6?

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Example

• Arrival rate ?
• Service rate µ
• Gateway utilization ? ?/µ
• Prob. of n packets in gateway
• Mean number of packets in gateway

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Example

• Arrival rate ? 125 pps
• Service rate µ 1/0.002 500 pps
• Gateway utilization ? ?/µ 0.25
• Prob. of n packets in gateway
• Mean number of packets in gateway

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Example

• Probability of buffer overflow
• To limit the probability of loss to less than
  10-6

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Example

• Probability of buffer overflow P(more than
  13 packets in gateway)
• To limit the probability of loss to less than
  10-6

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Example

• Probability of buffer overflow P(more than
  13 packets in gateway) ?13 0.2513
  1.49x10-8 15 packets per billion packets
• To limit the probability of loss to less than
  10-6

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Example

• Probability of buffer overflow P(more than
  13 packets in gateway) ?13 0.2513
  1.49x10-8 15 packets per billion packets
• To limit the probability of loss to less than
  10-6

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Example

• To limit the probability of loss to less than
  10-6
• or

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Example

• To limit the probability of loss to less than
  10-6
• or 9.96

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Properties of poisson processes

• Discuss joining, merging of poisson processes
• poission process exponential arrival time
  between packets

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Queuing networks

• How to solve for networks of queues (M/M/m
  model)
• Definitions m number of servers ?
  arrival rate µ service rate
• Which depend on m?

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Queuing networks

• The state of the system is represented by the
  number of jobs n in the system

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Queuing networks

• The expression for the probability of n jobs in
  the system is

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Queuing networks

• In terms of traffic intensity ??/mµ