Positive Harris Recurrence and Diffusion Scale Analysis of a Push Pull Queueing Network. Haifa Statistics Seminar May 5, 2008

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1 Positive Harris Recurrence and Diffusion Scale Analysis of a Push Pull Queueing Network Yoni Nazarathy Gideon Weiss Haifa Statistics Seminar May 5,

2 Outline 1 Preview of Results 2 Introduction Queueing Networks Push Pull 3 Analysis and Results Stochastic Model Fluid Analysis Markov Chain Setting Diffusion Scale Analysis 4 Open Questions 2

3 Outline 1 Preview of Results 2 Introduction Queueing Networks Push Pull 3 Analysis and Results Stochastic Model Fluid Analysis Markov Chain Setting Diffusion Scale Analysis 4 Open Questions 3

4 Preview of Results The Push Pull Network Server 1 Server 2 Q 2 µ 1 µ 2 ν 2 ν 4 µ 4 µ 3 Q 4 THEOREM 1: Fluid stability. THEOREM 2: Positive Harris recurrence. THEOREM 3: Diffusion approximation. Insights Full Utilization and stable network general processing times. Diffusion scale covariance between outputs negative. Diffusions of outputs same for all stable policies. 4

5 Outline 1 Preview of Results 2 Introduction Queueing Networks Push Pull 3 Analysis and Results Stochastic Model Fluid Analysis Markov Chain Setting Diffusion Scale Analysis 4 Open Questions 5

6 Outline 1 Preview of Results 2 Introduction Queueing Networks Push Pull 3 Analysis and Results Stochastic Model Fluid Analysis Markov Chain Setting Diffusion Scale Analysis 4 Open Questions 6

7 Queueing Networks Demonstration with the Job Shop Simulator Single class networks Jackson Networks. Generalized Jackson Networks. Multi-class networks several job classes per node Choose a policy. Analysis is hard, policy dependent. Optimal exact solutions even harder. Optimal approximate solutions sometimes possible. 7

8 Stability Depends on Policy KSRS Example Q 1 Server 1 Server 2 Q 2 α 1 µ 1 µ 2 µ 4 µ 3 α 2 Q 4 Q 3 Kumar-Seidman, Rybko-Stoylar (90 s) Offered loads: ρ 1 = α 1 µ 1 + α 2 µ 4, ρ 2 = α 1 µ 2 + α 2 µ 3 Necessary Condition for stability: ρ 1, ρ 2 < 1. Condition is not sufficient KSRS example. 8

9 KSRS in Balanced Heavy Traffic Q 1 Server 1 Server 2 Q 2 α 1 µ 1 µ 2 µ 4 µ 3 α 2 Q 4 Q 3 Some "good" policies exist (e.g Max-Pressure [Dai & Lin]). Yet as α i = ρ i 1 = congestion. 9

10 KSRS in Balanced Heavy Traffic Max-Pressure, µ 1 = µ 3 = 1.25 µ 2 = µ 4 = 1 α i ρ =.9 ρ =.99 ρ = 1.0 ρ =

11 Outline 1 Preview of Results 2 Introduction Queueing Networks Push Pull 3 Analysis and Results Stochastic Model Fluid Analysis Markov Chain Setting Diffusion Scale Analysis 4 Open Questions 11

12 The Push Pull Network Server 1 Server 2 Q 2 µ 1 µ 2 ν 2 ν 4 µ 4 µ 3 Q 4 No input stream infinite supply of work. Steps 1 and 3 push, Steps 2 and 4 pull. Infinite supply allows full utilization. Two queues Q 2 (t), Q 4 (t), no congestion. 12

13 Rates for Full Utilization Server 1 Server 2 Q 2 µ 1 µ 2 ν 2 ν 4 µ 4 µ 3 Q 4 θ i allocation to i. Require: Q 2, Q 4 not congested. θ 1 = 1 θ 4, θ 2 = 1 θ 3 µ 1 θ 1 = µ 2 θ 2, µ 3 θ 3 = µ 4 θ 4 ν 1 = ν 2 = µ 1µ 2 (µ 3 µ 4 ) µ 1 µ 3 µ 2 µ 4, ν 3 = ν 4 = µ 3µ 4 (µ 1 µ 2 ) µ 1 µ 3 µ 2 µ 4. 13

14 Case 1 Inherently Stable Case 1 Parameters: µ 1 < µ 2, µ 3 < µ 4. Policy: Pull Priority. Exponential Processing Times Kopzon & Weiss (2001) Q 4 µ 3 µ 4 µ 3 µ 4 Push Server 1 Server Pull 4 3 µ 3 µ 4 µ 2 µ 2 µ 2 µ 1 µ 1 µ 1 Q 2 14

15 Case 2 Inherently Unstable Case 2 Parameters: µ 1 > µ 2, µ 3 > µ 4. Policy: Threshold push when other server s queue low. Exponential Processing Times Kopzon & Weiss (2007) Q 4 µ 3 µ 2 µ 2 µ 4 µ 2 µ 1 µ 1 µ 3 µ 4 µ 4 µ 3 µ 3 µ 2 µ 1 µ 2 µ 1 µ 2 µ 1 µ 2 µ 1 µ 4 µ 2 µ 1 µ 4 µ 2 µ 2 µ 2 µ 2 µ 2 µ 2 µ 1 Pull Push Server 1 Server µ 3 µ 4 µ 3 µ 4 µ 3 µ 4 µ 3 µ 4 µ 3 µ 4 µ 3 µ 3 µ 4 µ 3 µ 4 µ 3 µ 4 µ 3 µ 4 µ 3 µ 4 µ 1 µ 1 µ 1 µ 1 µ 1 µ 1 15

16 For Case 2, Use Linear Threshold Policy Why not fixed thresholds? Threshold is distribution dependent: ( ) ( ) µ 1 s2 µ4 ( µ3 ) s4 ( µ2 ) > 1, > µ 2 µ 3 µ 4 µ 1 1. Fluid approximation how? Linear thresholds Choose κ 1 > µ 3 µ 1, κ 2 > µ 1 µ 3, Q 4 Q 4 =κ 1 Q 2 µ 3 Q 2 = µ 1 Q 4 Q 2 = κ 2 Q 4 Server 1 Server Q 2 16

17 Outline 1 Preview of Results 2 Introduction Queueing Networks Push Pull 3 Analysis and Results Stochastic Model Fluid Analysis Markov Chain Setting Diffusion Scale Analysis 4 Open Questions 17

18 Outline of Analysis and Results Server 1 Server 2 Q 2 µ 1 µ 2 ν 2 ν 4 µ 4 µ 3 Q 4 THEOREM 1: Fluid stability. THEOREM 2: Positive Harris recurrence. THEOREM 3: Diffusion approximation. 18

19 Outline 1 Preview of Results 2 Introduction Queueing Networks Push Pull 3 Analysis and Results Stochastic Model Fluid Analysis Markov Chain Setting Diffusion Scale Analysis 4 Open Questions 19

20 A Multi-Class Queueing Network Model Random processing time sequences ξ i = {ξ j i, j = 1, 2,...}, S i(t) = departure in processing t. Allocation processes: T i (t) T i (0) = 0, T i ( ), T i (t) T i (s) t s Lipschitz, absolutely continuous, derivative exists a.e. Outputs and queue lengths Outputs: D i (t) = S i (T i (t)). Queue Lengths: Q 2 (t), Q 4 (t) 0. Q 2 (t) = Q 2 (0) + D 1 (t) D 2 (t), Q 4 (t) = Q 4 (0) + D 3 (t) D 4 (t) 20

21 Policies Head of the line. Preemptive resume. Servers never idle T 1 (t) + T 4 (t) = t, T 2 (t) + T 3 (t) = t. Case 1 Pull priority t 0 Q 4 (s)dt 1 (s) = 0, t 0 Q 2 (s)dt 3 (s) = 0. Case 2 Linear thresholds t 0 1{0 < Q 4(s) < κ 1 Q 2 (s)}dt 1 (s) = 0 t 0 1{0 Q 2(s) 1 κ 1 Q 4 (s)}dt 4 (s) = 0 t 0 1{0 Q 4(s) 1 κ 2 Q 2 (s)}dt 2 (s) = 0 t 0 1{0 < Q 2(s) < κ 2 Q 4 (s)}dt 3 (s) = 0 21

22 Assumptions A1 Rates (fluid scaling) lim n n j=1 ξj i n = 1 µ i a.s. A2 Renewal processing (positive Harris recurrence) (a) ξ i, i = 1, 2, 3, 4, are mutually independent i.i.d. Technical Assumptions: (b) Spread-out and unbounded ξ i for i = 1, 3, or (b ) Compacts are petite. A3 Second moments (diffusion scaling) µ 2 i Var(ξ1 i ) = c 2 i <. 22

23 Outline 1 Preview of Results 2 Introduction Queueing Networks Push Pull 3 Analysis and Results Stochastic Model Fluid Analysis Markov Chain Setting Diffusion Scale Analysis 4 Open Questions 23

24 Fluid Scaled Processes Assume rates exist (A1) lim n n j=1 ξj i n = 1 µ i a.s. A sequence of processes parameterized n = 1, 2,... Y n (t) = (Q n (t), T n (t)) all sharing the same ξ i (ω). Q n (0): A sequence of initial queue lengths. Fluid Scalings: Definition Ȳ n (t, ω) = Y n (nt,ω) n. Ȳ (t) = ( Q(t), T (t)) is a fluid limit of the network: for some sample path ω and some r, Ȳ r (, ω) Ȳ ( ), u.o.c. 24

25 Fluid Limit Model and Fluid Model Equations Theorem Fluid limits exist. They form the fluid limit model. They satisfy: Fluid Model Equations Q i (t) = Q i (0) + µ i 1 Ti 1 (t) µ i Ti (t) 0, i = 2, 4. T i (0) = 0, Ti, Ti (t) T i (s) t s, i = 1, 2, 3, 4. and additional equations which depend on the policy. 25

26 Policy Related Fluid Model Equations Full utilization T 1 (t) + T 4 (t) = t, T2 (t) + T 3 (t) = t. Case 1 Pull priority policy t 0 Q 4 (s)d T 1 (s) = 0, Case 2 Linear threshold policy t 0 Q 2 (s)d T 3 (s) = 0. t 0 1{0 < Q 4 (s) < κ 1 Q2 (s)}d T 1 (s) = 0 t 0 1{0 Q 2 (s) 1 κ 1 Q4 (s)}d T 4 (s) = 0 t 0 1{0 Q 4 (s) 1 κ 2 Q2 (s)}d T 2 (s) = 0 t 0 1{0 < Q 2 (s) < κ 2 Q4 (s)}d T 3 (s) = 0 26

27 Fluid Stability Result Definitions ( Q, T ) that satisfies the fluid model equations is a fluid solution. A fluid model is stable if there exists a δ > 0 such that for every fluid solution with Q(0) = 1, Q(t) = 0 for any t δ. THEOREM 1 Fluid Stability of Push Pull Network The fluid model of the push pull network, for Case 1 with pull priority and for Case 2 with linear threshold policy, is stable. Corollary Fluid Approximation For fixed Q(0), Y (nt)/n converges as n u.o.c. a.s. to a fluid limit Ȳ (t) with: Ti (t) = θ i t, Di (t) = ν i t, Qi (t) = 0. 27

28 Lyapounov Proof for Fluid Stability Lemma Let f be absolutely continuous, f (t) 0, with derivative ḟ (t) at regular points. Assume: (i) f(t) = 0 implies ḟ (t) = 0. (ii) There exists ɛ > 0 for which at regular t: If f (t) > 0 then ḟ (t) ɛ. Then: f (t) = 0 for all t > f (0)/ɛ, furthermore f ( ) is non-increasing so once it reaches 0 it stays there. Lyapunov Function g( Q 2, Q 4 ) 0, and g( Q 2, Q 4 ) = 0 if and only if Q 2 = Q 4 = 0. Q = Q 2 + Q 4 = 1 then g( Q 2, Q 4 ) B. Let f (t) = g( Q 2 (t), Q 4 (t)), if f satisfies the assumptions of the Lemma, then the network fluid model is stable, with δ = B/ɛ. 28

29 Lyapounov Proof for Fluid Stability Case 1 Simple Full Proof for Illustration Lyapounov function f (t) = Q 2 (t) + Q 4 (t). Note: f (t) 0 and f (t) = 0 if and only if Q(t) = 0 and Q(0) = 1 then f (0) is bounded. Take ɛ = min{µ 2 µ 1, µ 4 µ 3. Now bound ḟ (t): Q 2 (t), Q4 (t) > 0: T 1 = T3 = 0, T 2 = T4 = 1, Q i (t) = µ i f (t) = (µ 2 + µ 4 ). Q 2 (t) > 0, Q 4 (t) = 0: T 3 = 0, T 2 = 1, f (t) = µ 1 T1 (t) µ 2 µ 4 T4 (t) = = µ 1 µ 2 (µ 1 + µ 4 ) T4 (t) (µ 2 µ 1 ) Q 2 (t) = 0, Q 4 (t) > 0, Similarly, f (t) (µ 4 µ 3 ). 29

30 Lyapounov Proof for Fluid Stability Case 2 Same Technique Different Lyapounov Function Piecewise linear Lyapounov function (1 + β) Q 2 (t) (κ 2 β) Q 4 (t) if Q 2 (t) κ 2 Q4 (t), f (t) = (1 + β) Q 4 (t) (κ 1 β) Q 2 (t) if Q 4 (t) κ 1 Q2 (t), β( Q 2 (t) + Q 4 (t)) otherwise. where β = 1 2 min{κ 1 µ 3 µ µ 3 µ 1 µ 3, κ 2 µ µ }. 1 µ 3 We again obtain ḟ ɛ and f (t) 0 and f (t) = 0 if and only if Q(t) = 0, and if Q(0) = 1 then f (0) is bounded by some finite value B. 30

31 Lyapounov Proof for Fluid Stability Case 2 same technique different Lyapounov function

32 Outline 1 Preview of Results 2 Introduction Queueing Networks Push Pull 3 Analysis and Results Stochastic Model Fluid Analysis Markov Chain Setting Diffusion Scale Analysis 4 Open Questions 32

33 Markovian Network State Process Assume Renewal processing (A2) ξ i, i = 1, 2, 3, 4 are mutually independent i.i.d. A Markovian State Define the network state process, X(t) = (Q(t), U(t)). (U 1 (t), U 3 (t)), (U 2 (t), U 4 (t)) residual processing times. State space: S = Z 2 + R 2 + R 2 +. Evolution of X(t) between arrivals and departures is deterministic, X(t) is piecewise deterministic. Proposition X = {X(t), t 0} is a strong Markov process on S. 33

34 Positive Harris Recurrence (PHR) For x S, B B(S) P t (x, B) = P x (X(t) B) = P{X(t) B X(0) = x}. π σ-finite on (S, B(S)) is invariant if for all t π(b) = P t (x, B) π(dx), B B(S). S τ A = inf{t 0 : X(t) A}. X is Harris recurrent if for measure ν, A B(S) with ν(a) > 0 implies P x (τ A < ) = 1 for all x S. Harris Recurrent = invariant measure π. When π is a probability X is positive Harris recurrent. PHR implies ergodicity: for all x S, and all f 0 1 t lim f (X(s))ds = f (x) π(dx) t t 0 S P x a.s. 34

35 Main Stability Result PHR THEOREM 2: Positive Harris Recurrence Under Assumptions (A1), (A2a) and (A2b ), the network state process X is PHR for Case 1 under the pull priority policy and for Case 2 under the linear threshold policy. Furthermore, for Case 1 we may substitute Assumptions (A2b ) with (A2b). Proof We use the framework of Dai 95. Dai has shown that for a MCQN (with external arrival streams), fluid model stability implies PHR. With infinite virtual queues we need to modify the proof. 35

36 Dai s Framework Dai shows that PHR follows from two statements: (i) Convergence of process scaled by initial state lim x 1 x E x X(δ x ) = 0, someδ > 0 (ii) Every compact set is petite (A2b ) The argument that (i) and (ii) imply PHR needs no modification. Dai s main result is that fluid model stability implies (i). This also needs no modification. Hence, by our Theorem on Fluid Stability, under (A2a,A2b ), PHR follows. 36

37 Petite Sets and Compact Sets A S is petite if there exists a probability distribution a on (0, ) and a nontrivial measure ν on (S, B(S)), such that for all x A 0 P t (x, B)a(dt) ν(b), for all B B(S). Petiteness of A may be interpreted as the property that all sets B are "equally accessible" from any x A. Compact sets in S are closed sets with bounded Q i, U i. In fluid scaling they represent the point 0. Petiteness is a property of the Markov Process, and therefore depends on the policy. It is not straightforward to check if compact sets are petite. 37

38 The Technical Assumptions (A2b ) and (A2b) Our technical assumptions are (A2b ) Every compact set is petite (A2b) Spread out unbounded processing times: For the push activities (Infinite Virtual Queues) Proposition P(ξ 1 i x) > 0 for all x > 0, k i 0 > 0, q i( ) 0 with 0 q i(x)dx > 0 : P(ξ 1 i ξ k i 0 i dx) q i (x)dx, i = 1, 3. Under pull priority policy, (A2b) implies (A2b ) We were unable to prove a similar result for the linear threshold policy 38

39 Outline 1 Preview of Results 2 Introduction Queueing Networks Push Pull 3 Analysis and Results Stochastic Model Fluid Analysis Markov Chain Setting Diffusion Scale Analysis 4 Open Questions 39

40 Diffusion Scaling Relations Finite n Diffusion Scalings where (fluid scalings and limits), Ŝi n (t) = S i (nt) S i (nt) n, n ˆT i (t) = T i (nt) T i (nt) n, ˆD i n (t) = D i (nt) D i (nt) n, ˆQn i (t) = Q i (nt) n. S i n (t) = S i (nt) S n i (t) = µ i t T i n (t) = T i (nt) T n i (t) = θ i t D i n (t) = D i (nt) D n i (t) = θ i µ i t, a.s. (u.o.c) n Decomposition of output variability ˆD n i (t) = D i (nt) D i (nt) n = S i (n T i n (t)) S i (n T i n (t)) n + S i (n T i n (t)) n D i (nt) n = Ŝn i ( T i n T (t)) + µ i (nt) T i (nt) Ti (nt) i n + µ i n D i (nt) n = Ŝn i ( T n i (t)) + µ i ˆT n i (t) + θ i µ i nt θi µ i nt = Ŝ n i ( T n i (t)) + µ i ˆT n i (t) 40

41 Diffusion Scaling Relations Finite n Relations between diffusion scalings ˆD n 1 (t) = Ŝn 1 ( T n 1 (t)) + µ 1 ˆT n 1 (t) ˆD n 2 (t) = Ŝn 2 ( T n 2 (t)) + µ 2 ˆT n 2 (t) ˆD n 3 (t) = Ŝn 3 ( T n 3 (t)) + µ 3 ˆT n 3 (t) ˆD n 4 (t) = Ŝn 4 ( T n 4 (t)) + µ 4 ˆT n 4 (t), { ˆQn 2 (t) = ˆD n 1 (t) ˆD n 4 (t) ˆQ n 4 (t) = ˆD n 3 (t) ˆD n 4 (t) }, { ˆT n 2 (t) = ˆT 3 n(t) } ˆT 4 n(t) = ˆT 1 n(t) "Limit Ready" Representation where, 1 A = µ 1 µ 3 µ 2 µ 4 ˆD n 2 (t) ˆD n 4 (t) ˆT n 2 (t) ˆT n 4 (t) = A Ŝ n 1 ( T n 1 (t)) Ŝ n 2 ( T n 2 (t)) Ŝ n 3 ( T n 3 (t)) Ŝ n 4 ( T n 4 (t)) µ 2 µ 4 µ 1 µ 3 µ 1 µ 2 µ 1 µ 2 µ 3 µ 4 µ 3 µ 4 µ 2 µ 4 µ 1 µ 3 µ 4 µ 4 µ 1 µ 1 µ 3 µ 3 µ 2 µ 2 + B [ ˆQn 2 (t) ˆQ n 4 (t) ], B = 1 µ 1 µ 3 µ 2 µ 4 µ 2 µ 4 µ 1 µ 2 µ 3 µ 4 µ 2 µ 4 µ 4 µ 1 µ 3 µ 2. 41

42 Diffusion Result THEOREM 3: Diffusion Limit Under assumption (A1) (A3), as n, ( ˆD n (t), ˆT n (t), ˆQ n (t)) converges weakly to a 10 dimensional driftless Brownian motion. Furthermore: ˆD 1 n(t) ˆD 2 n(t) = ˆQ 2 n (t) 0, ˆD 4 n(t) ˆD 3 n(t) = ˆQ 4 n (t) 0. Some Elements of the Covariance Matrix Var(ˆD 2 (1)) = µ 1 µ 2 (µ 1 µ 3 µ 2 µ 4 ) 3 {µ 1µ 2 µ 3 µ 4 (c c2 4 )(µ 1 µ 2 ) + (µ 2 1 µ2 3 c2 2 + µ2 2 µ2 4 c2 1 )(µ 3 µ 4 )} Cov(ˆD 2 (1), ˆD 4 (1)) = µ 1µ 2 µ 3 µ 4 (µ 1 µ 3 µ 2 µ 4 ) 3 {(µ 1µ 3 c µ 2µ 4 c 2 3 )(µ 1 µ 2 ) + µ 1 µ 3 c µ 2µ 4 c 2 1 )(µ 3 µ 4 )} 42

43 Highly Negative Correlation Between Outputs Covariance for symmetric push pull Take c 2 i = c 2, equal, µ 2 = µ 4 = 1, µ 1 = µ 3 = λ. Plot ρ λ = Cov(ˆD 2 (1),ˆD 4 (1)) Var(ˆD2 (1))Var(ˆD 4 (1))

44 Insensitivity to Specific Policy "Inputs" to Diffusion Proof ˆQ n i (t) 0 (weakly). T n (t) θt (u.o.c.). Assumptions (A3) Second Moment. Conclusion Asymptotic variance rate of outputs from fully utilizing stable push pull network does not depend on specific policy. 44

45 Outline 1 Preview of Results 2 Introduction Queueing Networks Push Pull 3 Analysis and Results Stochastic Model Fluid Analysis Markov Chain Setting Diffusion Scale Analysis 4 Open Questions 45

46 Open Questions and Future Work Petiteness of compacts other policies. Anomalies in simulation and numerical approximations. Push pull in heavy traffic State Space Collapse? Behavior of MCQN with infinite virtual queues. 46

47 Thank You 47