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Modeling and Analysis of Manufacturing Systems Session 2 QUEUEING MODELS

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... to k with probability pjk. Queue sizes are unlimited. Jackson ... Probability of job leaving station j for k pjk. Steady state equation (Eqn 11.32, p. 394) ... – PowerPoint PPT presentation

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Title: Modeling and Analysis of Manufacturing Systems Session 2 QUEUEING MODELS


1
Modeling and Analysis of Manufacturing
SystemsSession 2QUEUEING MODELS
  • January 2001

2
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

3
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)

4
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

5
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)

6
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)

7
Single Station (contd)
  • Steady state probabilities
  • For n 0
  • p(1) ??????? p(0)
  • For n gt 0
  • p(n1) (? ? )/? p(n) -
  • ??????? p(n-1)

8
Single Station (contd)
  • Steady state probabilities (contd)
  • p(n) ???????n p(0)
  • Constraint
  • ? p(n) 1

9
Single Station (contd)
  • Combining
  • p(0) ???????
  • Also
  • p(n) ?????????n
  • Expected number of items in system
  • L ? n p(n) ? /???????

10
Single Station (contd)
  • Expected throughput time
  • W 1/? ???????
  • Littles Law
  • L ?W
  • See summary in Table 11.1, p. 366
  • See Example 11.1

11
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

12
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

13
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.

14
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.

15
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

16
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

17
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.

18
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)

19
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

20
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

21
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.

22
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.

23
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) .

24
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

25
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.

26
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

27
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

28
MVA (contd)
  • Throughput rates
  • Xp Np/(? vjp Wjp)
  • Number of visits of part p to station j vjp
  • Queue lengths
  • Ljp Xp vjp Wjp

29
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

30
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

31
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-

32
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)

33
Hybrid Systems
  • See Sec. 11.5
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