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Lecture 5 This lecture is about: Introduction to Queuing Theory Queuing Theory Notation Bertsekas/Gallager: Section 3.3 Kleinrock (Book I) Basics of Markov Chains – PowerPoint PPT presentation

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Title: This lecture is about:


1
Lecture 5
  • This lecture is about
  • Introduction to Queuing Theory
  • Queuing Theory Notation
  • Bertsekas/Gallager Section 3.3
  • Kleinrock (Book I)
  • Basics of Markov Chains
  • Bertsekas/Gallager Appendix A
  • Kleinrock (Book I)
  • Markov Chains by J. R. Norris

2
Queuing Theory
  • Queuing Theory deals with systems of the
    following type
  • Typically we are interested in how much queuing
    occurs or in the delays at the servers.

Server Process(es)
Input Process
Output
3
Queuing Theory Notation
  • A standard notation is used in queuing theory to
    denote the type of system we are dealing with.
  • Typical examples are
  • M/M/1 Poisson Input/Poisson Server/1 Server
  • M/G/1 Poisson Input/General Server/1 Server
  • D/G/n Deterministic Input/General Server/n
    Servers
  • E/G/? Erlangian Input/General Server/Inf.
    Servers
  • The first letter indicates the input process, the
    second letter is the server process and the
    number is the number of servers.
  • (M Memoryless Poisson)

4
The M/M/1 Queue
  • The simplest queue is the M/M/1 queue.
  • Recall that a Poisson process has the following
    characteristics
  • Where A(t) is the number of events (arrivals) up
    to time t.
  • Let us assume that the arrival process is a
    Poisson with mean ? and the service process is a
    Poisson with a mean ?

5
Poisson Processes (a refresher)
  • Interarrival times are i.i.d. and exponentially
    distributed with parameter ?.
  • tn is the time of packet n and ?n tn1 - tn
    then
  • For every t ? 0 and ? ? 0

6
Poisson Processes (a refresher)
  • If two or more Poisson processes (A1,A2...Ak)
    with different means(?1, ?2... ?k) are merged
    then the resultant process has a mean ? given by
  • If a Poisson process is split into two (or more)
    by independently assigning arrivals to streams
    then the resultant processes are both Poisson.
  • Because of the memoryless property of the Poisson
    process, an ideal tool for investigating this
    type of system is the Markov chain.

7
On the Buses (a paradoxical property of Poisson
Processes)
  • You are waiting for a bus. The timetable says
    that buses are every 30 minutes. (But who
    believes bus timetables?)
  • As a mathematician, you have observed that, in
    fact, the buses are a Poisson process with a mean
    arrival rate such that the expectation time
    between buses is 30 minutes.
  • You arrived at a random time at the bus stop.
    What is your expected wait for a bus? What is
    the expected time since the last bus?
  • 15 minutes. After all, they are, on average, 30
    minutes apart.
  • 30 minutes. As we have said, a Poisson Process
    is memoryless so logically, the expected waiting
    time must be the same whether we arrive just
    after a previous bus or a full hour since the
    previous bus.

8
Introduction to Markov Chains
  • Some process (or time series) Xn n 0,1,2,...
    takes values in nonnegative integers.
  • The process is a Markov chain if, whenever it is
    in state i, the probability of being in state j
    next is pij
  • This is, of course, another way of saying that a
    Markov Chain is memoryless.
  • pij are the transition probabilities.

9
Visualising Markov Chains (the confused hippy
hitcher example)
A hitchhiking hippy begins at A town. For some
reason he has poor short-term memory and
travels at random according to the probabilities
shown. What is the chance he is back at A after
2 days? What about after 3 days? Where is he
likely to end up?
10
The Hippy Hitcher (continued)
  • After 1 day he will be in B town with probability
    3/4 or C town with probability 1/4
  • The probability of returning to A via B after 1
    day is 3/12 and via C is 2/12 total 5/12
  • We can perform similar
  • calculations for 3 or 4 days
  • but it will quickly
  • become tricky and
  • finding which city he
  • is most likely to end up
  • in is impossible.

11
Transition Matrix
  • Instead we can represent the transitions as a
    matrix

Prob of going to B from A
Prob of going to A from C
12
Markov Chain Transition Basics
  • pij are the transition probabilities of a chain.
    They have the following properties
  • The corresponding probability matrix is

13
Transition Matrix
  • Define ?n as a distribution vector representing
    the probabilities of each state at time step n.
  • We can now define 1 step in our chain as
  • And clearly, by iterating this, after m steps we
    have

14
The Return of the Hippy Hitcher
  • What does this imply for our hippy?
  • We know the initial state vector
  • So we can calculate ?n with a little drudge work.
  • (If you get bored raising P to the power n then
    you can use a computer)
  • But which city is the hippy likely to end up in?
  • We want to know

15
Invariant (or equilibrium) probabilities)
  • Assuming the limit exists, the distribution
    vector ? is known as the invariant or equilibrium
    probabilities.
  • We might think of them as being the proportion of
    the time that the system spends in each state or
    alternatively, as the probability of finding the
    system in a given state at a particular time.
  • They can be found by finding a distribution which
    solves the equation
  • We will formalise these ideas in a subsequent
    lecture.

16
Some Notation for Markov Chains
  • Formally, a process Xn is Markov chain with
    initial distribution ? and transition matrix P
    if
  • PX0i ?i (where ?i is the ith element of
    ?)
  • PXn1j Xni, Xn-1xn-1,...X0x0 PXn1j
    Xni pij
  • For short we say Xn is Markov (?,P)
  • We now introduce the notation for an n step
    transition
  • And note in passing that

This is the Chapman-Kolmogorov equation
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