Chapter 2 Machine Interference Model

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Chapter 2Machine Interference Model

• Long Run Analysis
• Deterministic Model
• Markov Model

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Problem Description

• Group of m automatic machines
• Operator must change tools or perform minor
  repairs
• How many machines should be assigned to one
  operator?
• Performance measures
• Operator utilization ? fraction of time the
  operator is busy
• Production rate TH finished items per unit
  time
• Machine availability ? TH/G, where G is the
  gross production rate, or the production rate
  that would be achieved if each machine were
  always available
• Note In this queuing system, the machines are
  the customers!

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Long Run Analysis

• Each machine has gross production rate h
• Pn is the proportion of time that exactly n
  machines are down
• Then, given Pn,

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Eliminate some unknowns

• Suppose the mean time to repair a machine is 1/?,
  and the mean time between failures for a single
  machine is 1/?.
• avg. of repairs in (0,t ??t
• avg. of failures in (0,t
• In the long run, assuming the system is stable,

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Queuing Measures of Performance

• average of machines waiting for
  service
• average number of machines down
• average downtime duration of a machine
• average duration of waiting time for
  repair

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Littles Formula

• Observe from the previous equations
• where
• is the total number of failures per unit time
• the arrival rate of customers to the queuing
  system
• Littles formula relates mean of customers in
  system to mean time a customer spends in the
  system.

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A Deterministic Model

• Suppose each machine spends exactly 1/? time
  units working followed by exactly 1/? time units
  in repair. Then if
• and we could stagger the failure times, we would
  have no more than one machine unavailable at any
  time, so that
• (Otherwise,

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A Markov Model

• Let be the time between the (n-1)st and the
  nth failure of machine j, and
• be the time duration of the nth repair
• The time until the first failure is
• N(t) of machines down at time t follows a
  CTMC with
• S 0, 1, , m and

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Steady-State Probabilities

• satisfy the balance equations or level-crossing
  equations

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Solution