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Modeling and Analysis of Manufacturing Systems Session 2 QUEUEING MODELS

Modeling and Analysis of Manufacturing Systems Session 2 QUEUEING MODELS. January 2001. SINGLE WORKSTATION. SYSTEM: STATION + INPUT QUEUE INPUT: Batches of raw materials. WORKSTATION: one or more identically capable processors (servers). OUTPUT: Completed products.

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Modeling and Analysis of Manufacturing Systems Session 2 QUEUEING MODELS

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  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 (cont’d) • 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 (cont’d) • Probability of n = 0 at t pt+t(0) = pt(0) (1 - t) + pt(1) t • Probability of n > 0 at t pt+t(n) = pt(n) (1 - t - t) + pt(n+1) t + pt(n-1) t

  5. Single Station (cont’d) • In rate form: • For n = 0 dpt+t(0)/dt = -  pt(0) +  pt(1) • For n > 0 dpt+t(n)/dt = - ( +  ) pt(n) +  pt(n+1) +  pt(n-1)

  6. Single Station (cont’d) • At steady state pt+t(n) = pt(n) = p(n) : • For n = 0  p(0) =  p(1) • For n > 0 ( +  ) p(n) =  p(n+1) +  p(n-1)

  7. Single Station (cont’d) • Steady state probabilities: • For n = 0 p(1) =  p(0) • For n > 0 p(n+1) = [( +  )/] p(n) -  p(n-1)

  8. Single Station (cont’d) • Steady state probabilities (cont’d): p(n) = n p(0) • Constraint:  p(n) = 1

  9. Single Station (cont’d) • Combining: p(0) =  • Also: p(n) = n • Expected number of items in system L =  n p(n) =  /

  10. Single Station (cont’d) • Expected throughput time: W = 1/ • Little’s Law: L = W • See summary in Table 11.1, p. 366 • See Example 11.1

  11. Single Station (cont’d) • 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 (cont’d) • 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 (cont’d) • 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 (cont’d) • 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. Jackson’s 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 (cont’d) • 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 (cont’d) • 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 (cont’d) • Throughput rates Xp = Np/( vjp Wjp) • Number of visits of part p to station j = vjp • Queue lengths Ljp = Xp vjp Wjp

  29. MVA (cont’d) • 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 (cont’d) • 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 (cont’d) • The solution to the balance equations is p(n) = G-1 (N) (f1*f2*f3 ...fM) • Where, if nj < cj fj(nj) = jnj/nj! • And if nj > cj fj(nj) = jnj/(cj! cjnj-cj) • And G-1 (N) =  (f1*f2*f3 ...fM)

  33. Hybrid Systems • See Sec. 11.5

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