Queuing

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Queuing
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Problem Solving for performance evaluation

• Analysis use mathematical model to solve
  usually need assumption to make it easy to
  solve
• Can have full domain of solution, I.e. optimal
  solution
• Simulation more realistic assumption
  heuristic solution
• Analysis used to be the main research area, now
  simulation is a popular method, but

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Queuing examples

• Customer service
• Bank, post office , number of clerks vs. waiting
  time , service time
• Computer servers and jobs (requests)
• Traffic flow on freeway or computer networks.
• Why probability ?
• Because everything is unpredictable. We use
  mean , variance to be the metrics.

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Sampling of Problems

• Resource Scheduling
• Demands
• Unpredictable arrivals
• Unpredictable service demands
• Everybody wants immediate service
• Conflict resolution
• Waiting
• Loss
• sharing

Demand
Resource
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Single Resource with waiting (queuing)
Arrivals
departures
queue
server

• Assume
• Arrival
• Each service take
• Service

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Single Resource with waiting (queuing)

• Def
• IF constant inter-arrival times AND constant
  service times.
• IF random inter-arrival times AND random service
  times.

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Sampling of Problems

• Every disk access is equally likely to be to any
  cylinder on the disk
• What is the average distance (in cylinders) of a
  seek ?
• Expected execution
• time variance?

start
t1
yes
branch
no
0.25
exit
t2
0.75
t3
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Sampling of Problems

• fi fraction time link i is down at a random
  time , what is the prob. That there is
• Cherry GOES DOWN TWICE A WEEK. Whats the prob.
  You lost your term report ?

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Probability

• Random experiment.
• Real world experiment.
• outcome not predictable.
• Associated with a random experiment.
• Set of all possible outcomes.
• Mutually exclusive and exhaustive.
• Events
• Event occurs or does not occur for each outcome.
• Defines a subset of outcomes.

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Probability

• Independent trials
• If outcome of any trial does not affect the
  outcome of any other trial
• If trials are performed under identical
  conditions.

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Probability

• Statistical regularity
• n independent trials.
• Def likelihood differing

(????????)
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Probability

• Sample space set of all possible outcomes of a
  random experiment Not required, but often deal
  with equally probable outcomes.
• Suppose event A occurs iff
• Oi1 or Oi2 or Oij occurs

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Probability

• i.e.

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Probability

• Counting Things Combinations
• Product rule
• Let A1 be a set containing n1 objects,
• A2 be a set containing n2 objects.
• The number of ordered pairs
• (a1, a2) s.t
• is n1n2
• Generalizes to
• r sets A1,A2,Ar of cordinalistics n1,n2,...,nr
• Number of distinct r-tuples with i-th componemt
  an object from Ai is n1n2n3nr

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Probability

• Some fundamental cases
• Ways of selecting r out of n objects
• Two dimensions
• Ordered or unordered samples. (sequence v.s. a
  set)
• With or without replacement. (whether can repeat
  an object)
• Ordered sampling with replacement.
• The number of ways of selecting r objects in
  succession from a set of n objects, where each
  object is returned prior to the next selection is
  nr

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Probability

• Ordered sampling without replacement
• The number of ways of selecting r objects in
  succession from a set of n objects (r n) where
  each object selected is removed from the set
  prior to the next selection is, n(n-1)(n-r1)
  n!/(n-r)!
• Sampling without replacement and without regard
  for order
• The number of ways of selection r objects from a
  set of n objects without replacement without
  regard for order is

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Probability

• Sampling with replacement and without regard for
  order
• r out of n
• Ans

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Probability

• Probability Axioms
• Let S be a sample space of a random experiment
• Axioms
• (A1) for any event A,
• (A2) P(S) 1
• (A3)
• It follows easily that (n finite)
• if Ai are
  mutually exclusive

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Probability

• (A3) For any countable sequence of events
  A1,A2,...,An,,that are mutually exclusive.?
• (i.e. )

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Some implications of the Axioms

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Principles of inclusion exclusion

• If A1,A2,,An are any events then

Proof by induction on the number of events n
(recitation) Related result in combinatoric S
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Conditional Probability

• Probability that event A has occurred given that
  B has occurred , notation P(AB)
• Def

Generalization
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Independent Events

• Def Events A B are said to be independent iff
• In words, given that B occurred, given no
  information about the probability that A
  occurred.
• Note If P(AB)P(A),that
• i.e. symmetric

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Independent Events

• Alternative Def
• follows from first def
• but if P(AB)P(A)
• then P(A)P(B) P(A?B)

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Independent Events

• Ex
• Piprob. of introducing an error, assume that n
  msg arrives in error
• P(msg arrives with error)?
• Let Eievent that link I introduces an error
• P(msg arrives with error)
• assume are independent. are
  indep. ??
• gt P(msg arrives with error)

Generalized
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Some random properties

• If
• If A B are independent
• And BC are independent
• AC are independent

Ex B 1, 2, 3, 4 A 1, 3, 5
C 2, 4, 6
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Independence of a set of events

• Def A list of n events A1,A2,,An is said to be
  mutually indep. iff for each set of k (2 k n)
  distinct indices i1,i2,,Ik
• pairwise indep. mutually indep.

Note duality mutually exclusive
mutually indep
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• Ex N blocks in a file.
• n accesses, uniformly distributed.
• What is the prob. That the first block
• is accessed at least once ?
• P1st block accessed
• ways to make the n accesses Nn
• ways with 0 accesses to 1st block (N-1)n
• ways with gt0 accesses to 1st block Nn- (N-1)n
• ?

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• Let Ei event that the first block is accessed
  on i-th access
• P1st block accessed at least once

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Bayes Rule

• In general
• Let B1, , Bn be mutually exclusive and
  exhaustive events.
• Let A be any event.

Bi exhaustive
Mutually exclusive
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• Ex Drawers
• G G
• G S
• Select a drawer at random
• reach in and take a coin without looking.
• The coin is gold, what is the prob that drawer 1
  was chosen ?
• Find P(drawer1gold)?
• P(drawer1)p(drawer2)1/2
• P(golddrawer1)1
• P(golddrawer2)1/2

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Chap 2 Discrete Random Variable

• Def A random variable X on a sample space S is a
  function
• X S?R from S to the reals
• Numbers often are naturally associated with
  sample space outcomes.
• e.g. face value on a die
• But not always
• e.g what if the die had colors

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Probability mass function

• Def Px(x)P(Xx)
• is also called the density function for X
• (Cumulative) Distribution function

3/36
1/36
2
3
4
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• Ex Bruin rot
• 1/10000 of having to disease Medical test.
• Two possibilities P (positive) or N (negative)
• PPBR0.99
• PPNBR0.01
• Henry gets a test result is positive.
• PBRP

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Particular Cases

• Bernoulli

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Particular Cases

• Experiments with two outcomes, repeat some number
  of this n, There are 2n possible sequences of
  length n for outcomes of the experiments.
• Call the outcomes success and failure
• Letfor any single experiment
• P(success)p
• P(failure)1-pq
• Consider a sequence of length n
• ?????

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Particular Cases

• Consider

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Particular Cases

• Generalization

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Geometric r.v Z

• (An) Interpretation
• Number of Bernoulli trials until a success is
  obtained sample space is infinite
• All sequences of some number of 0s followed by a
  1 or fff.f s
• PZ(i)qi-1p , I gt 1

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Geometric r.v Z

• Distribution function
• Fz(t)P(Zltt)
• ?????,???

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Geometric r.v Z
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Poisson Pmf
???????
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Probability Generating Function (Z transform)

• Transformation
• Will simplify many calculations later. (non
  negative integer valued r.v.)

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Probability Generating Function (Z transform)

• An example

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Probability Generating Function (Z transform)

• ??????
• So

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Some transforms

• Bernoulli r.v.x

???????
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Some transforms

• Poisson

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Linear recurrence relation
???
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Linear recurrence relation
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Review
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Review
????
???????
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Z-transform

• Fn n0,1,2,3
• Define
• Property 1
• Property 2

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Z-transform
???
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Z-transform
??????,????
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Z-transform kleinrock Appendix I
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Z-transform
Last 4 are special cases of the following
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Z-transform
p.335
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Discrete random vectors

• Multivariate distributions

???
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Discrete random vectors

• What is the condition pmf

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Continuous r.v
???? nondecreasing
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Application to simulation

• How to generate a r.v. with desired FX(x)
• ? generate a uniformly distributed r.v. y
• Then x0FX-1(y0) got it
• Nonnegative r.v

???????
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Exponential distribution

• Service time at a resource (server)
• Distribution of time between arivals

F(x)
x
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Exponential distribution
Fx(x)
x
t
Remaining life distribution
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Exponential distribution

• Memoryless property

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Poisson distribution of exponential distribution

• Poisson distribution was for the xxxxxxx of
  events in an interval of length t
• where

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Poisson distribution of exponential distribution

• Exponential interarrival
• Poisson arrival process ? memoryless property ?
  exponential interarrival times
• What is the memoryless property important ?
• Real answer must wait but intuitively.

Arrival process
departures
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Expectation

• Def of expectation
• X a random variable
• Moments

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Expectation
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Expectation

• Expectation is a linear operation
• Variance Xi are mutually
  indep.

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Expectation

• Expectation of functions of more than one random
  variable

a
0
current request
Next request
x2
x1
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Conditional Distribution and Conditional
Expectation

• Conditional Probabilities

? Marginal pmf
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Conditional Distribution and Conditional
Expectation

• Continuous r.v.

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??????
???
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Conditional Distribution and Conditional
Expectation
????? example ??????,????
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Conditional Expectation
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Conditional Expectation
If U1 then V0 If U0 then V has same pmf as X
1EX
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Example
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Jobs arrivals

?exeuction time
Repair time
time
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Laplace Transforms
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Some properties
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Laplace transform pairs
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Why we need transforms ?

• Ex Random sum of random variables

(continuous)
z-transform
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Inversion of Laplace transforms

1. Explicit inversion formula -- we wont use
2. Inspection using table
3. Partial fraction expansion

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Random Sum of random variables
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Last-come-first-serve
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Review of Probability
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