Modeling and Analysis of Manufacturing Systems Session 2 QUEUEING MODELS

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Modeling and Analysis of Manufacturing
SystemsSession 2QUEUEING MODELS

• January 2001

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SINGLE WORKSTATION

• SYSTEM STATION INPUT QUEUE
• INPUT Batches of raw materials.
• WORKSTATION one or more identically capable
  processors (servers).
• OUTPUT Completed products.
• SIMPLEST SPECIAL CASE (M/M/1)
• Batch size 1 Server size 1
• Exponential intearrival and service times
• FCFS service policy
• Service time set-up time processing time

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Single Station (contd)

• Average arrival rate ?
• Average service rate ?
• Utilization factor (expected number of items in
  process) ? ? / ?
• Expected number of items at station L Lq ?
• Expected throughput time W Wq 1/?
• Actual number of items at station n
• Probability of having n items at time t pt(n)

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Single Station (contd)

• Probability of n 0 at t
• pt?t(0) pt(0) (1 - ? ?t) pt(1) ? ?t
• Probability of n gt 0 at t
• pt?t(n) pt(n) (1 - ? ?t - ? ?t) pt(n1) ?
  ?t pt(n-1) ? ?t

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Single Station (contd)

• In rate form
• For n 0
• dpt?t(0)/dt - ? pt(0) ? pt(1)
• For n gt 0
• dpt?t(n)/dt - (? ? ) pt(n)
• ? pt(n1) ? pt(n-1)

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Single Station (contd)

• At steady state pt?t(n) pt(n) p(n)
• For n 0
• ? p(0) ? p(1)
• For n gt 0
• (? ? ) p(n) ? p(n1) ? p(n-1)

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Single Station (contd)

• Steady state probabilities
• For n 0
• p(1) ??????? p(0)
• For n gt 0
• p(n1) (? ? )/? p(n) -
• ??????? p(n-1)

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Single Station (contd)

• Steady state probabilities (contd)
• p(n) ???????n p(0)
• Constraint
• ? p(n) 1

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Single Station (contd)

• Combining
• p(0) ???????
• Also
• p(n) ?????????n
• Expected number of items in system
• L ? n p(n) ? /???????

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Single Station (contd)

• Expected throughput time
• W 1/? ???????
• Littles Law
• L ?W
• See summary in Table 11.1, p. 366
• See Example 11.1

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Single Station (contd)

• Poisson arrivals, general FCFS service
• M/G/1
• E(S) expectation for service time (1/?)
• E(T) expectation for throughput time T
• E(N) expectation for number of jobs N
• See Example 11.2, p. 367

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Single Station (contd)

• How about other that FCFS policy?
• If multiple parts with different priorities are
  being processed then priority service may have to
  be instituted
• See Sec. 11.2.3 and Example 11.3, p. 369

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Networks of Workstations

• Consider M workstations with jobs moving
  between workstation pairs following a routing
  scheme.
• If an external arrival process generates jobs
  that enter the network anytime, we have an open
  network.
• If the number of jobs in the network is
  maintained constant we have a closed network.

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Facts about Networks

• The sum of independent Poisson random variables
  is Poisson.
• If arrival rate is Poisson, the time interval
  between arrivals is Exponential.
• If service time is Exponential , the output rate
  is Poisson.

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Facts about Networks (contd)

• The interdeparture time from an M/M/c system with
  infinite queue capacity is Exponential.
• If a Poisson process of rate ? is split into
  multiple processes with probability pi, the
  individual streams become Poisson with arrival
  rates equal to ? pi

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Open Networks

• Illustration of Facts
• See Example 11.4, p. 372
• Poisson Arrivals and FCFS policy
• Parts are taken from Warehouse for Kitting
• Kits are sent to Assembly station(s)
• Finished parts are sent to Inspection/Packing
• See Fig. 11.2, p. 373

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Open Networks (contd)

• Kitting-Assembly-Inspect/Pack Problem
• Kitting queue has always 1 hr worth of work
• Kitting rate 10 kits/hr
• Assembly rate 12 parts/hr
• Inspection/Pack rate 15 parts/hr
• Assume all times are Exponential.
• Serial System with Random Processing Times.

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Kitting-Assembly-Inspect/Pack

• Output rate from Kitting is Poisson.
• Arrival time into Assembly is Exponential.
• Output from Assembly is Poisson.
• Arrival time into Inspect/Pack is Exponential.
• State of system described by number of jobs at
  Assembly and Inspect/Pack (n1, n2)

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Kitting-Assembly-Inspect/Pack

• States and transitions diagram (Fig. 11.3)
• Steady-state balance equations (Eqn. 11.13, p.
  373)
• Product Form Solution
• p(n1,n2) (1 - ?1) ?1n1 (1 - ?2) ?2n2
• Recall for single workstation
• p(n) ?????????n

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Important Note

• The product form solution allows the analysis of
  the M-station network by first analyzing the M
  individual stations separatedly and then
  combining the results.
• See Example 11.5

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Jacksons Generalization

• M workstations with cj servers each.
• External arrivals are Poisson with rate ?j
• FCFS
• Service times are Exponential w/mean 1/?j
• Job at station j transfers to k with probability
  pjk
• Queue sizes are unlimited.

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Jackson (contd)

• Effective arrival rate External arrivals
  Internal arrivals
• ?j ?j ?k ?k pkj
• Note this is a system of linear algebraic
  equations for the various ?j
• Utilization factors must then be computed using
  the Effective arrival rates.

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Jackson (contd)

• The state of system is given by the vector
• n (n1, n2, n3, ..., nM)
• The probability of the system being in a state n
  is p(n) .

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Procedure for Open Networks

• 1.- Solve for the effective arrival rates in all
  workstations (Eqn. 11.15)
• 2.- Analyze each station independently using
  Table 11.1.
• 3.- Aggregate results across stations to obtain
  performance measures.
• See Example 11.6, p. 377, Ex. 11.7, p. 378

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Closed Networks

• Sometimes it may be convenient not to introduce
  new jobs into the system but until a unit is
  completed and delivered.
• This maintains the number of jobs in the system
  at a constant level N .
• In this case WIP becomes a control parameter not
  an output statistic.

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Closed Networks

• As N increases, both peoduction rate and
  throughput increase.
• Production rate is limited by lowest service rate
  station.
• Worsktations are not independent now.
• Set of possible states is such that
• ?? nj N

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Mean Value Analysis

• Assume P part types (? njp Np ??Np N)
• Mean service time for part p on station j
  1/?jp
• Throughput time of part p at j
• Wjp 1/?jp ((Np-1)/Np) Ljp/ ?jp
• ? Ljr/ ?jp

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MVA (contd)

• Throughput rates
• Xp Np/(? vjp Wjp)
• Number of visits of part p to station j vjp
• Queue lengths
• Ljp Xp vjp Wjp

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MVA (contd)

• Iterative Solution Procedure
• 1.- Guess the values of Ljp
• 2.- Obtain Wjp
• 3.- Compute Xp
• 4.- Compute improved values of Ljp
• 5.- Repeat until satisfied.
• See Example 11.0, pp. 388-392

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Product Form Solutions forClosed Networks

• Probability of selecting part of type p to
  enter the system next dp
• Station visit count vj ? vjp dp
• Total work required at station j
• ?j ? vjp dp ??jp
• Service rate at j
• 1 ??jp ?j / vj

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Product Form Solutions forClosed Networks
(contd)

• Rate station j serves customers under n
• rj(n) min(nj,cj) ?j
• Probability of job leaving station j for k pjk
• Steady state equation (Eqn 11.32, p. 394)
• p(n) ? rj(n) ????p(njk) pjk rj(njk)
• See Example 11.10, p. 394-

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Product Form Solutions forClosed Networks
(contd)

• The solution to the balance equations is
• p(n) G-1 (N) (f1f2f3 ...fM)
• Where, if nj lt cj
• fj(nj) ?j nj/nj!
• And if nj gt cj
• fj(nj) ?j nj/(cj! cjnj-cj)
• And
• G-1 (N) ? (f1f2f3 ...fM)

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Hybrid Systems

• See Sec. 11.5