1. Modeling and Analysis of Manufacturing Systems Session 2 QUEUEING 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 = L q +  Expected throughput time: W = W q + 1/  Actual number of items at station: n Probability of having n items at time t: p t (n)

4. Single Station (cont’d) Probability of n = 0 at t p t+  t (0) = p t (0) (1 -  t) + p t (1)   t Probability of n > 0 at t p t+  t (n) = p t (n) (1 -  t -   t) + p t (n+1)   t + p t (n-1)  t

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

6. Single Station (cont’d) At steady state p t+  t (n) = p t (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 p i, the individual streams become Poisson with arrival rates equal to p i

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 )  1 n1 (1 -  2 )  2 n2 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 c j 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 p jk Queue sizes are unlimited.

22. Jackson (cont’d) Effective arrival rate = External arrivals + Internal arrivals j ’ = j +  k k ’ p kj 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,..., n M ) 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 X p = N p /(  v jp W jp ) Number of visits of part p to station j = v jp Queue lengths L jp = X p v jp W jp

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 for Closed Networks Probability of selecting part of type p to enter the system next d p Station visit count v j =  v jp d p Total work required at station j  j =  v jp d p  jp Service rate at j 1  jp =  j / v j

31. Product Form Solutions for Closed Networks (cont’d) Rate station j serves customers under n r j (n) = min(n j,c j )  j Probability of job leaving station j for k p jk Steady state equation (Eqn 11.32, p. 394) p( n )  r j (n) =  p( n jk ) p jk r j ( n jk ) See Example 11.10, p. 394-

32. Product Form Solutions for Closed Networks (cont’d) The solution to the balance equations is p( n ) = G -1 (N) (f1*f2*f3...fM) Where, if nj < cj f j (n j ) =  j nj /nj! And if nj > cj f j (n j ) =  j nj /(cj! cj nj-cj ) And G -1 (N) =  (f1*f2*f3...fM)

33. Hybrid Systems See Sec. 11.5