On Successive Lumping of Large Scale Systems

Transcription

1 On Successive Lumping of Large Scale Systems Laurens Smit Rutgers University Ph.D. Dissertation supervised by Michael Katehakis, Rutgers University and Flora Spieksma, Leiden University April 18, 2014

2 Outline I present solutions to Many O.R. Models (e.g., Queueing, Inventory) that are notoriously hard to solve. The basic models Successive Lumping DES Quasi-skip-free (QSF) processes RES QSF processes Solving Queueing Models Computational Benefit

3 Priority Queue Two types of customers arrive to a system, and enter two di erent queues Queue 1 has priority over queue 2 Customers arrive in di erent group sizes There is a single server Queue 1 Server Queue 2

4 Priority Queue Two types of customers arrive to a system, and enter two di erent queues Queue 1 has priority over queue 2 Customers arrive in di erent group sizes There is a single server Queue 1 Server Queue 2

5 Longest Queue Model Customers arrive in two queues with di erent arrival rates A single server serves the longest queue Queue bu ers are finite or infinite Queue 1 Server Queue 2

6 Longest Queue Model Customers arrive in two queues with di erent arrival rates A single server serves the longest queue Queue bu ers are finite or infinite Queue 1 Queue 2 Server

7 An inventory model with lost sales One stream of customers arrives New products arrive in batches of variable sizes (random yield) New batches of products can arrive at any time Arrival rate can depend on the inventory level (dynamic prices) Stock Customer

8 An inventory model with lost sales One stream of customers arrives New products arrive in batches of variable sizes (random yield) New batches of products can arrive at any time Arrival rate can depend on the inventory level (dynamic prices) Stock Customer

9 An inventory model with lost sales One stream of customers arrives New products arrive in batches of variable sizes (random yield) New batches of products can arrive at any time Arrival rate can depend on the inventory level (dynamic prices) Stock Customer

10 An inventory model with lost sales One stream of customers arrives New products arrive in batches of variable sizes (random yield) New batches of products can arrive at any time Arrival rate can depend on the inventory level (dynamic prices) Stock Customer

11 An inventory model with lost sales One stream of customers arrives New products arrive in batches of variable sizes (random yield) New batches of products can arrive at any time Arrival rate can depend on the inventory level (dynamic prices) Stock Customer

12 An inventory model with lost sales One stream of customers arrives New products arrive in batches of variable sizes (random yield) New batches of products can arrive at any time Arrival rate can depend on the inventory level (dynamic prices) Stock Customer

13 An inventory model with lost sales One stream of customers arrives New products arrive in batches of variable sizes (random yield) New batches of products can arrive at any time Arrival rate can depend on the inventory level (dynamic prices) Stock

14 An inventory model with lost sales One stream of customers arrives New products arrive in batches of variable sizes (random yield) New batches of products can arrive at any time Arrival rate can depend on the inventory level (dynamic prices) Stock Customer

15 An inventory model with lost sales One stream of customers arrives New products arrive in batches of variable sizes (random yield) New batches of products can arrive at any time Arrival rate can depend on the inventory level (dynamic prices) Stock Customer

16 An inventory model with lost sales One stream of customers arrives New products arrive in batches of variable sizes (random yield) New batches of products can arrive at any time Arrival rate can depend on the inventory level (dynamic prices) Stock

17 Successive Lumping idea [Katehakis and Smit, 2012] The Successive Lumping method is an exact solution procedure to calculate the steady state distribution of a Markov Chain. There is a partition such that one subset of the state space has an entrance state. We represent this subset as one state. The new process contains a subset with an entrance state. This process can be repeated.

18 Successive Lumping D 0 D 1 D 2 D 3 0, 1 1, 1 2, 1 3, 1 3, 2 0, 2 1, 2 2, 2 3, 3

19 Successive Lumping D 0 0, 1 0, 2

20 Successive Lumping D 0 0, 1 0, 2

21 Successive Lumping 0 0, 1 0, 2

22 Successive Lumping 1 D 2 D 3 1, 1 2, 1 3, 1 1, 0 3, 2 1, 2 2, 2 3, 3

23 Successive Lumping D0 D1 D2 D3 0, 1 1, 1 2, 1 3, 1 3, 2 0, 2 1, 2 2, 2 3, 3 0 0, 1 1 D2 D3 1, 1 2, 1 3, 1 1, 0 3, 2 0, 2 1, 2 2, 2 3, 3 Feinberg(1987)

24 Successively Lumpable Steps D0 D1 D2 D3 0, 1 1, 1 2, 1 3, 1 3, 2 0, 2 1, 2 2, 2 3, , 1 1, 1 2, 1 3, 1 1, 0 2, 0 3, 0 3, 2 0, 2 1, 2 2, 2 3, 3 Figure Successively Lumpable Markov Process X(t) splitted up in sub processes. (m, j) = m (m, j) MY k=m+1 (k, 0), 8(m, j) 2X. k

25 Maximal SL Markov Chain SL Property Violating Transition Probabilities 0, 1 1, 1 2, 1 3, 1 3, 2 0, 2 1, 2 2, 2 3, 3

26 Successively Lumpable with respect to D 0 D 0 0 D 0 1 D 0 2 D 0 4 0, 1 1, 1 2, 1 3, 1 3, 2 D 0 3 0, 2 1, 2 2, 2 3, 3

27 Upper Bound (m, j) = m (m, j) MY k=m+1 (k, 0), 8(m, j) 2X. k For elements in D m the following holds when cutting o at level L m: LY (m, j) apple m (m, j) (k, 0). k k=m+1

28 Multiple Successive Lumping L 1 1 L 1 0 L 2 0 L 2 1 1, 1, 1 1, 0, 1 2, 0, 1 2, 1, 1 1, 1, 2 2, 1, 2 1, 1, 3 1, 0, 2 2, 0, 2 2, 1, 3 Katehakis & Smit (2012)

29 Multiple Successive Lumping D 1 1 D 1 0 D 2 0 D 2 1 1, 1, 1 1, 0, 1 2, 0, 1 2, 1, 1 1, 1, 2 2, 1, 2 1, 1, 3 1, 0, 2 2, 0, 2 2, 1, 3 Katehakis & Smit (2012)

30 Multiple Successive Lumping L 1 1 L 1 0 L 2 0 L 2 1 1, 1, 1 1, 0, 1 2, 0, 1 2, 1, 1 1, 1, 2 2, 1, 2 1, 1, 3 1, 0, 2 2, 0, 2 2, 1, , 0, 0 2, 0, 0 (n, m, j) = (n) n n m (j) Mn Y k=m+1 n nk (0) for all (n, m, j) 2X. (n, m, j) apple n n m (j) Nn Y k=m+1 n nk (0) for all (n, m, j) 2X, N n apple M n. Feinberg (1987) for (n) and Katehakis & Smit (2012) for remainder of the above Eq.

31 Basic Notation for QSF Processes A quasi-skip-free process is a Markov process with states that can be specified by tuples of the form (mi) where m represents the current level of the state i (i =1,...,`m) represents a state within the level L m (m =0, ±1, ±2,...). In addition, their probability transition law does not permit transitions to a state with level more than two units away from the current state s level in one direction (downwards).

32 Basic Notation for QSF Processes - Continued Their infinitesimal generator (transition rate matrix) has the form: D m 1 W m 1 U m 1,m U m 1,m+1 U m 1,m+2 Q = 0 D m W m U mm+1 U mm D m+1 W m+1 U m+1,m in the above specification of Q we use the letters D, W, and U todescribe down, within, and up transition rates, respectively. For a QSF process X(t) and for a fixed m 2{M 1,...,M 2 },astate (m, "(L m)) 2 L m is an entrance state for L e m if and only if the following is true for all (m +1,i) 2 L m+1 : d(mj m +1,i)=0, if (mj) 6= (m "(L m)).

33 DES QSF Processes The identity matrix I m of dimension `m `m. The (row) vectors of dimension `m: i) The vector 0 m identically equal to 0; ii) The vector 1 m identically equal to 1; iii) The vector m with 1 as its first coordinate and 0 elsewhere. The matrix U e mn = P M 2 k=n U mk 1 0 k m of dimension `m `m. The elements of U e mn will be denoted by ũ( mi):= P M 2 P`k k=m+1 j=1 u(kj mi). Define the matrices A m and B m : 2 6 A m = 4 eu M 1 m+1 + U M 1 m. eu m 1 m+1 + U m 1 m B m = U mm+1 + W m

34 The steady state distribution via the rate matrices R k m These results are described in Katehakis, Smit and Spieksma (2013). For a DES-QSF process there exist a matrix set {R k m} k,m such that: Theorem [K - S - S] m = e m k R k m. Since all subsets have an entrance state from above, the process is successively lumpable. A possible solution matrix R 1 m can be calculated exactly as a function of U k,m,u k,m+1 (k<m) and W m : R 1 m = A m (B m ) 1. All other R k m can be calculated recursively by: Theorem[K - S - S] h i Rm k = Ĩ m k Rm 1 (k 1) Rm k 1.

35 The construction of the rate matrix R The Successive Lumping method: Works for non-homogeneous processes. Gives an upper bound for the steady state probabilities when M 2 is infinite. Provides a limiting sequence when M 1 is infinite. Allows the size of levels ` to be infinite.

36 Discussion For a subclass of QBD processes when M 1 is finite, a similar result was derived with di erent methods in: Introduction to matrix analytic methods in stochastic modeling, by Latouche and Ramaswami (1999) [L - R].

37 Priority Queue Customers arrive with rate for each queue in batches of size 1 (for simplicity, not necessary) Queue 1 has priority over queue 2 There is an exponential single server with rate µ In state (n, j) there are n customers in queue 2 and j in queue 1. Queue 1 Server Queue 2

38 Explicit Solutions for the Priority Queue Model ,` 1,` 2,` 1 µ 1 µ 1 µ 0, 1 2 1, 1 2 2, µ 1 µ 1 µ 0, 0 2 1, 0 2 2, 0 2 µ µ µ Transition diagram for the priority model

39 Explicit Solutions for the Priority Queue Model - Continued The down, up and within transition rate matrices are: 2 D = W = 6 4 µ where d =2 + µ ,U = d 0. µ d.. 0 µ d

40 Explicit Solutions for the Priority Queue Model - Continued R = ( + µ) µ d 0. µ d.. 0 µ d We constructed an iterative method (Katehakis, Smit and Spieksma (2013b)) to do the inversion above in O(`2) steps. When ` is infinite our method gives a tight approximation. 1.

41 RES QSF Processes For a QSF process X(t) and for all m 2{M 1,...,M 2 },thestate (M 2, "(L m)) 2 L M2 is an entrance state for e L m if the following is true for all states (n, i) 2 L e m 1: u(k, j n, i) =0, if (k, j) 6= (M 2, "(L m)). In addition to the notation previously defined we introduce: ed m = D m 1 0 k m.

42 The steady state distribution via the rate matrices R k m These results are described in Katehakis, Smit and Spieksma (2013). For a RES-QSF process there exist a set {R m } m such that: Theorem [K - S - S] m = m+1 R m. Since all subsets have an entrance state from below, the process is successively lumpable. Amatrixset{R m } m that is a solution to the equation above can be calculated exactly as is shown below: Theorem[K - S - S] R m := D m+1 (W m ) 1. In addition, to find the steady state distribution, one can normalize using that: M2 (W M2 + e D M2 )=0 M2.

43 Restart Model The system contains 3 machines. Machine i break down with exponential rate µ i Machines are repaired with rate when all have broken down. The states are described as a vector x where x i = 0 (1) if the i-th machine is broken (working).

44 Explicit Solutions for the Restart Model µ1 µ 3 µ 2 0, 0, 1 0, 1, 1 µ 3 µ1 0, 0, 0 µ 2 0, 1, 0 1, 0, 1 µ 2 1, 1, 1 µ 3 µ1 µ 3 µ 1 1, 0, 0 µ 2 1, 1, 0 Transition diagram for the restart model

45 Explicit Solutions for the Restart Model - Continued The down, up and within transition rate matrices are: In this example of a restart process, when modeled as a QSF process, M 1 =0and M 2 =3. We derive that D m,w m and U m,m 2 have the from given below: 2 D 2 = 4 and W 0 =, U 0,3 =, U 1,3 =0 0 1, U 2,3 =0 0 2, 2 µ µ D 1 = 4 µ 2 5, W 1 = 4 0 µ 2 0 5, µ µ 1 µ 2 µ 3 0 µ 1 0 µ 3 0 µ 1 µ , W 2 = 4 µ 2 µ µ 1 µ µ 1 µ 2 D 3 = µ 1 µ 2 µ 3, W 3 = (µ 1 + µ 2 + µ 3 ). 3 5,

46 Explicit Solutions for the Restart Model - Continued R 0 = D 1 (W 0 ) 1 =1/ 2 R 1 = D 2 (W 1 ) 1 = µ 3 µ 2 µ 1 3 5, µ 2 /µ 3 µ 3 /µ 2 0 µ 1 /µ 3 0 µ 3 /µ 1 0 µ 1 /µ 2 µ 2 /µ 1 R 2 = D 3 (W 2 ) 1 = µ1 /(µ 2 + µ 3 ) µ 2 /(µ 1 + µ 3 ) µ 3 /(µ 1 + µ 2 ). 3 5,

47 Lattice Path Counting AknownalternativeforcalculatingtheratematrixisLattice Path Counting, introduced in v. Leeuwaarden, Squillante and Winands (2009). Works only for QBDs Uses a probabilistic interpretation of the rate matrix Uses hypergeometric functions For interior points: only transitions to nearest neighbor process are allowed No transition to west, north-west and south-west for interior points

48 Complexities in the homogenous Priority Queue M T = 10 5 T = 10 6 T = Red: Complexity of Brute Force Method Blue: Complexity of LPC to steady state distribution Green: Complexity of Successive Lumping [K - S - S] for specific models

49 Complexities in the non-homogenous Priority Queue M T = 10 5 T = 10 6 T = Red: Complexity of Brute Force Method Blue: Complexity of LPC to steady state distribution Green: Complexity of Successive Lumping [K - S - S] for specific models

50 Thank you!

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