Basic Queuing Model 1

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Unit - IV
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Source

• Probability and statistics with reliability,
  Queuing and Computer Science Applications, IInd
  Edition, authored by K. S. Trivedi, John Wiley.
• Communication Systems, IV Edition, A. B.
  Carlson and others, McGraw Hills.

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Random variable X(s)

• Random variable is a rule
• X, that assigns a real number
• to each sample point, s, of a
• sample space

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Discrete Random Variable and Probability Mass
Function

• Tossing of a coin
• Transmission of Digital signals
• Access of Cache memory by the processor

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Continuous random variable
X
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CDF and PDF

• F (x) P(Xlt x)
• Draw F (x) Vs X for experiment of tossing of two
  coins.
• What is the intuitive shape of F (x) for previous
  example of disk?
• d/dx F(x) p(x)

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Properties of PDF

• Integration of p(x) over entire range 1
• P (altXltb) F(b) F(a)
• Bays theorem is applicable

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Stochastic Process X(s,t)

• It represents a family of random variables
• Involves states, s
• (associated type of random variable )
• as well as time, t
• (either sampling is done at discrete instant of
  time or continuously the system is being
  monitored).
• Four possible types
• DSDT, DSCT, CSDT, CSCT

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Basic Queuing Model
Service discipline
Arrival process
Queue
Servers
Customer population
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• Define stochastic processes W, X, Y, Z over the
  queuing system such that
• W is DSDT
• X is DSCT
• Y is CSDT
• Z is CSCT
• Write down and show your notebook.

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• W number of jobs in the system at the time of
  departure of the kth customer.
• X(t) number of jobs in the system at time t.
• Y time that the kth customer has to wait in the
  system before receiving service.
• Z(t) cumulative service requirement of all jobs
  in the system at time t.

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Other types of stochastic processes

• Strictly stationary
• Widesense stationary
• Ergodic, etc

• Markov Process
• Birth-Death Process
• Poisson Process, etc

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Markov Process

• If future state probabilities independent of the
  past states and depends only on the present, then
  process called Markov.
• Markov property makes analysis easier, since past
  history need not be remembered.
• Markov Chain Discrete-state Markov process

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Markov Chains

• Interested in Markov chains with stationary state
  transition probabilities. i.e.,

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Birth-Death Process

• Birth-Death process A Markov chain that only
  allows transitions to neighboring states
• Represent states by integers. A process in state
  n can transition to state n-1 or n1
• E.g., of jobs in queue in a single server
  system can be represented by birth-death process
• Birth gt arrival of a job gt 1 state transition
• Death gt departure gt -1 state transition
• Batched arrivals cannot be modeled!

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• Using stochastic flow balance, the steady state
  probability of being in state n can be computed.

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Relationships between Processes
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M/M/1 Queue

• One server, one queue, FIFO service
• exponentially distributed interarrival and
  service times
• infinite population, infinite capacity
• Can model as a birth-death process

Notation pn steady-state probability of being
in state n
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• Using stochastic flow balance equations
• pn (?/?)n p0 ?np0, n0,1,,?
• ? is traffic intensity (lt1 for stability)
• Probabilities sum to one. Therefore,
• ? pn 1, n 0, 1, , ?
• p0(?0 ?1 ?2 ) 1
• p0 (1-?)
• pn ?n(1-?)
• Important performance measures follow

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• Utilization prob. of one or more jobs in system
• U 1- p0 ?
• Mean jobs in System
• En ?npn , n 0, 1, , ?
• En ?/(1-?)
• Mean response time (Littles law)
• number in system arrival rate x response time
• En ?Er
• Er (1/?)(?/(1-?)) (1/?)(1/(1-?))

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• Mean of jobs in queue
• Enq ?(n-1)pn , n 1, , ?
• Enq (?2)/(1-?)
• Can also be obtained using En Enq Ens
• Mean waiting time in queue (Littles Law)
• Number in queue arrival rate x mean waiting
  time
• Enq ?Ew
• Ew (1/?)((?2)/(1-?)) ?((1/µ)/(1-?))

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• Prob. of finding n or more jobs in system
• P( in system n) ?pj , j n,n1, , ?
• ?(1-?)?j ?n
• Waiting time and response time distributions
• Waiting times in queue exponentially distributed
• P0 lt w t 1 - ?e-?t(1-?)
• Response times exponentially distributed
• P0 lt r t 1 - e-?t(1-?)

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M/M/1 Queue Example

• Packets arrive at 100 packets/second at a router.
  The router takes 1 ms to transmit the incoming
  packets to an outgoing link. Using an M/M/1
  model, answer the following
• What is utilization?
• Probability of n packets in router?
• Mean time spent in the router?
• Probability of buffer overflow if router could
  buffer only 5 packets?
• Buffer requirement to limit packet loss to 10-6?

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• Arrival rate
• ? 100 pps
• Service rate
• ? 1/.001 1000pps
• Traffic intensity
• ? 0.1
• Mean packet residence time at router
• r (1/?)(1/(1-?))
• 1.01 ms
• Prob. of buffer overflow
• P( 6) ?6 10-12
• To limit loss to less than 10-6
• ?n 10-6
• n gt log(10-6)/log(0.1) gt 3

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M/M/c Queue

• c servers, one queue, FIFO service, exponentially
  distributed interarrival and service times
• infinite population, infinite capacity
• Model as a birth-death process with K states

State transition diagram for M/M/C Queue with C3
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• Mean arrival rate ?
• Mean service rate c?
• Traffic Intensity (avg. utilization) ? ?/(c?)
• ? lt 1 for stability
• Flow balance equations yield
• pn ((c?)n/n!)p0 n 1,,c-1
• pn ((c?)n/(c!cn-c))p0 n c
• Using law of total probability

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• Newly arrivals wait if all servers are busy,
  i.e., c or more jobs are in the system
• P( c jobs) pc pc1 pc2
• ? (c?)c/c!(1-?) p0
• ? Known as Erlangs C formula
• Mean of jobs in system
• En ?npn n 0, 1, , ?
• p0(c?)c/c!(1-?)2 c?
• c? ??/(1-?)

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• Mean of jobs in queue
• Enq ?(n-c)pn n c, , ?
• p0?(c?)c/c!(1-?)2
• ??/(1-?)
• Mean response time (Littles law)
• En ?Er
• Er 1/? ?/c?(1-?)
• Mean waiting time
• Ew Enq/ ? ??/(1-?)/? ?/c?(1-?)

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Pattern Recognition
Source Pattern Classification, II edition, R. O.
Duda et al, Wiley

• The act of taking in raw data and making an
  action based on the category of pattern
• Over the past tens of millions of years we have
  evolved highly sophisticated neural and cognitive
  systems for such tasks.
• What is a pattern?

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The modules of a PR system

• Sensing
• Segmentation and grouping
• Feature extraction (invariant feature,
  translation rotation, scale, occlusion,
  projective distortion, rate, deformation)
• Classification (decision theory, decision
  boundary, generalization)
• Post processing (error rate, risk, context,
  multiple classifier)

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The Design Cycle

• Data collection
• Feature choice
• Model choice
• Training (supervised, unsupervised,
  reinforcement)
• Evaluation (the issue of overfitting)
• Computational Complexity.