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Introduction to Queuing Theory

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(Mathworld) 'The study of the waiting times, lengths, and other properties of queues. ... Mean number tasks in system = mean arrival rate x mean response time ... – PowerPoint PPT presentation

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Title: Introduction to Queuing Theory


1
Introduction to Queuing Theory
2
Queuing theory definitions
  • (Bose) the basic phenomenon of queueing arises
    whenever a shared facility needs to be accessed
    for service by a large number of jobs or
    customers.
  • (Wolff) The primary tool for studying these
    problems of congestions is known as queueing
    theory.
  • (Kleinrock) We study the phenomena of standing,
    waiting, and serving, and we call this study
    Queueing Theory." "Any system in which arrivals
    place demands upon a finite capacity resource may
    be termed a queueing system.
  • (Mathworld) The study of the waiting times,
    lengths, and other properties of queues.

http//www2.uwindsor.ca/hlynka/queue.html
3
Applications of Queuing Theory
  • Telecommunications
  • Traffic control
  • Determining the sequence of computer operations
  • Predicting computer performance
  • Health services (eg. control of hospital bed
    assignments)
  • Airport traffic, airline ticket sales
  • Layout of manufacturing systems.

http//www2.uwindsor.ca/hlynka/queue.html
4
Example application of queuing theory
  • In many retail stores and banks
  • multiple line/multiple checkout system ? a
    queuing system where customers wait for the next
    available cashier
  • We can prove using queuing theory that
    throughput improves increases when queues are
    used instead of separate lines

http//www.andrews.edu/calkins/math/webtexts/prod
10.htmQT
5
Example application of queuing theory
http//www.bsbpa.umkc.edu/classes/ashley/Chaptr14/
sld006.htm
6
Queuing theory for studying networks
  • View network as collections of queues
  • FIFO data-structures
  • Queuing theory provides probabilistic analysis of
    these queues
  • Examples
  • Average length
  • Average waiting time
  • Probability queue is at a certain length
  • Probability a packet will be lost

7
Model Queuing System
  • Use Queuing models to
  • Describe the behavior of queuing systems
  • Evaluate system performance

8
Characteristics of queuing systems
  • Arrival Process
  • The distribution that determines how the tasks
    arrives in the system.
  • Service Process
  • The distribution that determines the task
    processing time
  • Number of Servers
  • Total number of servers available to process the
    tasks

9
Kendall Notation 1/2/3(/4/5/6)
  • 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)

10
Distributions
  • M stands for "Markovian", implying exponential
    distribution for service times or inter-arrival
    times.
  • D Deterministic (e.g. fixed constant)
  • Ek Erlang with parameter k
  • Hk Hyperexponential with param. k
  • G General (anything)

11
Kendall Notation Examples
  • M/M/1
  • Poisson arrivals and exponential service, 1
    server, infinite capacity and population, FCFS
    (FIFO)
  • the simplest realistic queue
  • 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

12
Analysis of M/M/1 queue
  • Given
  • l Arrival rate of jobs (packets on input link)
  • m Service rate of the server (output link)
  • Solve
  • L average number in queuing system
  • Lq average number in the queue
  • W average waiting time in whole system
  • Wq average waiting time in the queue

13
M/M/1 queue model
L

Lq
l
m


Wq
W
14
Littles Law
System
Arrivals
Departures
  • Littles Law Mean number tasks in system mean
    arrival rate x mean response 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

15
Proving Littles Law
Arrivals
Packet
Departures
1 2 3 4 5 6 7 8
Time
J Shaded area 9 Same in all cases!
16
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

17
Proof Method 1 Definition
in System (L)

1 2 3 4 5 6 7 8
Time (T)
18
Proof Method 2 Substitution
Tautology
19
M/M/1 queue model
L

Lq
l
m


Wq
W
L?W Lq?Wq
20
Poisson Process
  • For a poisson process with average arrival rate
    , the probability of seeing n arrivals in time
    interval delta t

21
Poisson process exponential distribution
  • Inter-arrival time t (time between arrivals) in a
    Poisson process follows exponential distribution
    with parameter

22
M/M/1 queue model
L?W Lq?Wq W Wq (1/µ)
23
Solving queuing systems
  • 4 unknowns L, Lq W, Wq
  • Relationships
  • LlW
  • LqlWq
  • W Wq (1/m)
  • If we know any 1, can find the others

24
Analysis of M/M/1 queue
  • Goal A closed form expression of the probability
    of the number of jobs in the queue (Pi) given
    only l and m

25
Equilibrium conditions
Define to be the probability of having
n tasks in the system at time t
26
Equilibrium conditions
l
l
l
l
n1
n
n-1
m
m
m
m
27
Solving for P0 and Pn
  • Step 1
  • Step 2

28
Solving for P0 and Pn
  • Step 3
  • Step 4

29
Solving for L
30
Solving W, Wq and Lq
31
Online M/M/1 animation
  • http//www.dcs.ed.ac.uk/home/jeh/Simjava/queueing/
    mm1_q/mm1_q.html

32
Response Time vs. Arrivals
33
Stable Region
linear region
34
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?

35
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?

36
Example
  • Arrival rate ?
  • Service rate µ
  • Gateway utilization ? ?/µ
  • Prob. of n packets in gateway
  • Mean number of packets in gateway

37
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

38
Example
  • Probability of buffer overflow
  • To limit the probability of loss to less than
    10-6

39
Example
  • Probability of buffer overflow P(more than
    13 packets in gateway)
  • To limit the probability of loss to less than
    10-6

40
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

41
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

42
Example
  • To limit the probability of loss to less than
    10-6
  • or

43
Example
  • To limit the probability of loss to less than
    10-6
  • or 9.96

44
Empirical Example
M/M/m system
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