Queueing Model for an Assemble-to-Order Manufacturing System- A Matrix Geometric Solution Approach

1. Queueing Model for an Assemble-to-Order Manufacturing System- A Matrix Geometric Solution Approach Sachin Jayaswal Department of Management Sicences University of Waterloo Beth Jewkes Department of Management Sciences University of Waterloo

2. Outline • Motivation • Model Description • Literature Review • Analysis • Future Directions

3. Motivation • Get a better understanding of Assemble-to-Order (ATO) production systems • Develop a queuing model for a two stage ATO production system and evaluate the following measures of performance: • Distribution of semi-finished goods inventory • Distribution of order fulfillment time

4. Model Description λ Demand Arrival

5. Model Description Notations: • λ : demand rate (Poisson arrivals) • μj : service rate at stage j, j=1, 2 (exponential service times) • B1: base stock level at stage 1 (parameter) • N1: queue occupancy at stage 1 • N2: queue occupancy at stage 2 • I1 : semi-finished goods inventory after stage 1. I1 = [B1 – N1]+ • BO :Number of units backordered at stage 1. BO = [N1 – B1]+

6. Model Description If B1 = 0, the system is MTO and operates like an ordinary tandem queue: • The process describing the departure of units from each stage is Poisson with rate λ • Individual queues behave as if they are operated independently. In equilibrium, N1 and N2 are independent

7. Model Description • For the ATO with B1 > 0: • Arrival process to stage 2 is no longer poisson. • There is a positive dependence between the arrival of input units from stage 1 to stage 2. Times between successive arrivals to stage 2 are correlated.

8. Related Literature • Buzacott et al. (1992) observe that C.V. of inter-arrival times at stage 2 is between 0.8 and 1 and, therefore, recommend using an M/M/1 approximation for stage-2 queue. Lee and Zipkin (1992) also assume M/M/1 approximation for stage 2. (BPS-LZ approximation) • Buzacott et al. (1992) further improve upon this approximation by modeling the congestion at stage 2 as GI/M/1 queue. (BPS approximation) • Gupta and Benjaafar (2004) use BPS-LZ approximation to compare alternative MTS and MTO systems

9. Solution – Matrix Geometric Method State space representation 1 • Consider a finite queue before stage 2 with size k • State description: {N = (N1, N2) : N1 ≥ 0; 0 ≤ N2 ≤ k+1}

10. Infinitesimal Generator Q = This is a special case of a level dependent QBD

11. Solution… State space representation 2 • Consider a finite queue before stage 1 with size k • State description: {N = (N2, N1) : N2 ≥ 0; 0 ≤ N1 ≤ k+1}

12. Infinitesimal Generator Q is a level independent QBD process and hence can be solved using standard Matrix-Geometric Method Q =

13. An Exact Solution • The above methods are not truly exact as one of the queues is truncated • We next present an exact solution for the doubly infinite problem, using censoring (Grassmann & Standford (2000); Standford, Horn & Latouche (2005)) • State description: {N = (N2, N1) : N1 ≥ 0, N2 ≥ 0}

14. Censoring Infinitesimal Generator Q =

15. Censoring Transition Matrix: ; P =

16. P(1) = where Censoring Censoring all states above level 1 gives the following transition matrix: Censoring level 1 gives: P(0) infinite only in one dimension However, P(0) may no longer be QBD

17. Censoring R matrix: R = R matrix possesses asymptotically block Toeplitz form

18. Censoring P(0) = P(0) is also asymptotically of block Toeplitz form Hence, one can use GI/G/1 type Markov chains to study P(0)

19. Censoring GI/G/1 type Markov chain is of the form: P =

20. Censoring To make P(0) conform to GI/G/1 type Markov chain, we choose B0 to be sufficiently large to contain those elements not within a suitable tolerance of their asymptotic forms P(0) =

21. Censoring Transition matrix with all states beyond level n censored (Grassmann & Standford, 2000)

22. Censoring

23. ; ; t determined using generating function (Grassmann & Standford (2000)) Solution to level-0 probabilities Non-normalized probabilities αj for censored process Normalized probabilities for censored process

24. Solution Stationary vector at positive levels • Performance measures • EK2: Expected no. of units stage 2 still needs to produce to meet the pending demands. EK2 = E(N2+BO) • EI: Expected no. of work-in-process units. EI = E(I1+N2)

25. Initial Results λ=1; μ1 =1.25; μ2 =2

26. Future Directions • To construct an optimization model using the performance measures obtained • To compare the results obtained with the approximations suggested in the literature

27. Thank You !