OPEN MULTICLASS HL QUEUEING NETWORKS: PROGRESS AND SURPRISES OF THE PAST 15 YEARS. w 1. v 2. v 3. Ruth J. Williams University of California, San Diego

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1 OPEN MULTICLASS HL QUEUEING NETWORKS: PROGRESS AND SURPRISES OF THE PAST 15 YEARS v 2 w3 v1 w 2 w 1 v 3 Ruth J. Williams University of California, San Diego 1

2 PERSPECTIVE MQN SPN Sufficient conditions for e.g., parallel server system, HL stability and diffusion packet switch approximations Non- e.g., LIFO, Processor Sharing e.g., Internet congestion HL (single station, control / bandwidth sharing PS: network stability) model 2

3 Answers for HL MQN STABILITY Subcritical fluid models PERFORMANCE ANALYSIS (in heavy traffic) Reflecting diffusions and state space collapse via critical fluid models 3

4 Scaling Fluid scale: r W () t = W( rt)/ r Diffusion scale: ˆ r () ( 2 )/ r ( ) W t = W r t r = W rt Performance processes do not require centering for the heavy traffic diffusion approximation. Diffusion scale is obtained by considering large times in fluid scale. 4

5 Assumptions HL: jobs within a buffer are stored in the order in which they arrived and service is always given to the job at the head-of-the-line. Also, the discipline is non-idling. Primitive arrival, service and routing processes are assumed to satisfy functional central limit theorems. 5

6 Outline SIMPLE MULTICLASS EXAMPLE OPEN MULTICLASS HL NETWORK (CONJECTURES) HISTORY UP THROUGH EARLY 1990 s OPEN MULTICLASS HL NETWORK (SETUP) FLUID MODELS AND STABILITY REFLECTING BROWNIAN MOTIONS HEAVY TRAFFIC LIMIT THEOREM VIA SSC FURTHER DEVELOPMENTS 6

7 SIMPLE MULTICLASS EXAMPLE 7

8 Multiclass FIFO Station λ 1 m 1 λ I m I Renewal arrivals to class i at rate λ i i.i.d. service times for class i, mean m i Service discipline: FIFO across all classes 8

9 Performance Processes λ 1 m 1 λ I m I Queuelength for class i: Workload: W Q i Idletime: Y 9

10 Stability λ 1 m 1 λ I Traffic Intensity m I I ρ = λ m 1 i= 1 i i Stability iff ρ <

11 Stability λ 1 m 1 λ I Traffic Intensity m I I ρ = λ m 1 i= 1 i i Stability iff Heavy traffic ρ < 1 1 ρ

12 Simulation of a Multiclass FIFO queue (Poisson arrivals, exponential service times) λ 1 = 0.05 m 1 = 1 λ 2 = 0.3 m 2 = 0.5 λ 3 = m 3 = 1 ρ =

13 Simulation of a Multiclass FIFO queue 13

14 Stability λ 1 m 1 λ I Traffic Intensity m I I ρ = λ m 1 i= 1 i i Stability iff ρ < 1 1 Heavy traffic ρ (assume ρ = 1 1 for simplicity)

15 Heavy Traffic Diffusion Approximation Wˆ t = W r t r Yˆ t = Y r t r r 2 r 2 () ( )/, () ( )/, ˆ r ( ) ( 2 i i ) /, 1,, Q t = Q r t r i = I Theorem (Whitt 71) ˆ r ˆr ˆ r * * * ( W, Y, Q ) ( W, Y, Q ) * where W is a one-dimensional reflecting Brownian * * * motion with local time Y and Q = λw (state space collapse). W * () t t W Y * () t = X () t + Y () t * * * ( t X t * ) = sup { ( s): 0 s } * X = Brownian motion 15

16 OPEN MULTICLASS HL NETWORK (CONJECTURES) 16

17 Open Multiclass HL Queueing Network First order parameters m α λ P λ = α + P λ ρ = λm, k = 1,..., k i i i k K 17

18 Natural Conjectures Stability: Network is stable provided ρ < 1 for each k = 1,, K k Heavy traffic diffusion approximation: If k 1, k 1,,, then ˆ r ˆr ˆ r * * where Q = ΔW for some IxK lifting matrix ρ = K * * * ( W, Y, Q ) ( W, Y, Q ) (that depends on the HL service discipline), and * * * W = X + RY is a reflecting Brownian motion (RBM) in the K-dimensional orthant. Δ 18

19 HISTORY 19

20 Affirmative Answers (Refs. are for diffusion approximations through early 1990s) SINGLE CLASS (FIFO): Single station: Borovkov ( 67), Iglehart-Whitt ( 70) Acyclic network: Iglehart-Whitt ( 70), Tandem queue: Harrison ( 78) Network: Reiman ( 84), Chen-Mandelbaum ( 91) MULTICLASS: Single station, priorities: Whitt ( 71), Harrison ( 73) Network, priorities: Johnson ( 83, SP), Peterson ( 91, feedforward) Single station, feedback, round robin & FIFO: Reiman ( 88), Dai- Kurtz ( 95) Rely on continuous mapping construction of RBM and do not cover multiclass networks with general feedback. 20

21 Counterexamples (two-stations, deterministic routing) STABILITY Kumar and Seidman ( 90): dynamic policy. Lu and Kumar ( 91): static priorities, deterministic interarrival and service times. Rybko and Stolyar ( 92): static priorities, exponential interarrival and service times. (See also Botvitch and Zamyatin ( 92)) Seidman ( 94): FIFO, deterministic interarrival and service times. Bramson ('94): FIFO, exponential interarrival and service times. DIFFUSION APPROXIMATION Dai-Wang ( 93): FIFO, exponential interarrival and service times. 21

22 Dai-Wang 93 FIFO Counterexample α m1 1 m m 2 3 m 2 5 m 4 Poisson arrivals (rate ) i.i.d. exponential service times with class means Traffic intensities α Proposed workload approximation: R m =,,,, ρ = α( m + m + m ) = α ρ2 = α( m3+ m4) = α =

23 OPEN MULTICLASS HL NETWORK (SETUP) 23

24 Open Multiclass HL Queueing Network V E A D Φ 24

25 Open Multiclass HL Queueing Network Network Structure K single server stations I customer classes (buffers) C constituency matrix (KxI ) C = 1 ( if class i served at station k) ki All buffers have infinite capacity HL: FIFO service within each class and non-idling (e.g., FIFO, static priorities, HLPS) 25

26 Open Multiclass HL Queueing Network Stochastic Primitives (E,V, Φ) Ei () t = # of exogenous class i arrivals in [0,t] Vi ( n ) = cumulative service time for first n jobs processed from class i ( Si () t = # class i jobs completed after t units of service is given to class i) j () n = Φi # of the first n departures from class j that are routed next to class i 26

27 Open Multiclass HL Network Performance Processes and Model Equations V E A 1 D K Φ Ai Q() t = Q (0) + A() t D() t (queuelength) i ( t) = E () t +Φ ( D( t) ) D () t = S ( T() t ) i i i i i Wk() t = Vi( Qi(0) + Ai( t) ) t + Yk( t) (workload) i k Y () t t T ( t : k = i k i i ) (idletime can increase only when W is 0) i i k 27

28 Open Multiclass HL Network Performance Processes and Model Equations V E A 1 D K Φ At ( ) = Et () +Φ ( Dt ( )) D( t) = S( T( t) ) Qt () = Q(0) + At () D() t Wt () = CV ( Q( 0) + A( t) ) 1t + Y( t) Yt () = 1t CT() t + additional equations depending on service discipline, e.g., FIFO: D( t+ W ( t)) D( t) = Q () t whe n i k i k i i 28

29 STABILITY AND FLUID MODELS 29

30 Fluid Model for HL Network (formal FLLN approximation) m α A D 1 K P At () = αt+ P D(), M = diag m = 1 ' t ( ), D() t M T() t T() t = total time allocated to class i by tim e t i 30

31 Fluid Model for HL Network (formal FLLN approximation) m α A D 1 K P At () = αt+ P D(), M = diag m = 1 ' t ( ), D() t M T() t T() t = total time allocated to class i by tim e t i W() t= W (0) + mat () t + Y(), t Q t = Q(0 ) + At () D () t k k i i k i i i i i k Yk( t) = t Ti( t) is non-decreasing, increases only when Wk = 0 i k ( ) 31

32 Fluid Model for HL Network (formal FLLN approximation) m α A D 1 K P At () = αt+ P D(), M = diag m = 1 ' t ( ), D() t M T() t T() t = total time allocated to class i by tim e t i Wk() t = Wk(0) + miai() t t + Yk(), t Qi t = Qi (0) + Ai() t Di() t i k Yk( t) = t Ti( t) is non-decreasing, increases only when Wk = 0 i k + additional conditions depending on service discipline, e.g., FIFO: D ( t + W ( t)) D ( t) = Q ( t) w hen i k i k i i ( ) 32

33 Stability via Fluid Models Definition: A fluid model is (uniformly) stable if there is such that for all fluid model solutions Qt ( ) = 0 for all t t Q( 0). 0 t 0 > 0 33

34 Stability via Fluid Models Definition: A fluid model is (uniformly) stable if there is such that for all fluid model solutions Qt ( ) = 0 for all t t Q( 0). 0 t 0 > 0 Theorem* (Dai 95): Fix an open multiclass HL queueing network and consider an associated fluid model. Under mild conditions**, if the fluid model is stable, then a Markov process describing the multiclass network is positive Harris recurrent. * See also Stolyar ( 95) **includes unboundedness and spread out assumptions on interarrival times 34

35 Examples Since 1995, many authors have used the fluid model approach to obtain sufficient conditions for stability of open multiclass HL networks (e.g., Bertsimas, Bramson, H. Chen, Dai, Foss, Hasenbein, Meyn, Stolyar, Weiss,...) Bramson 96: FIFO Kelly-type and HLPPS networks (Kelly type: mean service times are station dependent) used subcritical fluid models to establish stability when ρ < 1 for all k. k 35

36 Examples Since 1995, many authors have used the fluid model approach to obtain sufficient conditions for stability of open multiclass HL networks (e.g., Bertsimas, Bramson, H. Chen, Dai, Foss, Hasenbein, Meyn, Stolyar, Weiss,...) Bramson 96: FIFO Kelly-type and HLPPS networks (Kelly type: mean service times are station dependent) used subcritical fluid models to establish stability when ρ < 1 for all k. k t established asymptotic behavior (as ) of critical fluid models ( ρ for all k) --- uniform k = 1 convergence to invariant manifold. 36

37 SEMIMARTINGALE REFLECTING BROWNIAN MOTIONS (SRBMs) 37

38 SRBM DATA State space: K R+ Brownian statistics: drift θ, covariance matrix Reflection matrix: R v,..., v 1 = K ( ) Γ w3 v1 v 2 w 2 w 1 v 3 38

39 SRBM DEFINITION (w/starting point ) x 0 A continuous K-dimensional process (i) W = X + RY K (ii) W has paths in R+ (iii) for k=1,...,k, Y (0) = 0, k is continuous, such that non-decreasing, and it can increase only when W k = (iv) X is a ( θ, Γ) BM s.t. X (0) = x0, { X () t θt, t 0} is a martingale relative to the filtration generated by ( W, X, Y ) Y k W 0 39

40 Necessary Condition for Existence Defn: R is completely-s iff for each principal submatrix R of R there is y > 0: Ry > 0 v 2 w3 v1 w 2 R v,..., v 1 = K ( ) w 1 v 3 40

41 Existence and Uniqueness in Law Theorem (Reiman-W 88, Taylor-W 93) There is an SRBM W starting from each point x0 in R K + iff R is completely-s. In this case, each such SRBM is unique in law and these laws define a continuous strong Markov process. 41

42 Oscillation Inequality Assume that R is completely-s. There is a constant C>0 such that whenever δ > 0, 0 t1 < t2 <, and w, x, y are r.c.l.l. satisfying (i) wt () = x() t + Ry() t for t [ t1, t2] K (ii) w lives in R+ (iii) for k=1,...,k, yk ( t1) 0, yk is continuous, non-decreasing, and can increase only when wk < δ, Then Osc( w,[ t, t ]) + Osc( y,[ t, t ]) C( Osc( x,[ t, t ]) + δ ) Cts case: Bernard-El Kharroubi 91, discts case: W 98 42

43 Analysis of multidimensional SRBMs Sufficient conditions for positive recurrence Dupuis-W 94, Chen 96, Budhiraja-Dupuis 99, El Kharroubi-Ben Tahar- Yaacoubi 00 Stationary distribution Characterization: Harrison-W 87, Dai-Harrison 92, Dai-Kurtz 98 Analytic solutions -two-dimensions: Foddy 84, Trefethen-W 86, Harrison 06 -product form: Harrison-W 87 Numerical methods: Dai-Harrison 91, 92, Shen-Chen-Dai-Dai 02, Schwerer 01 Large deviations Majewski 98, 00, Avram-Dai-Hasenbein 01, Dupuis-Ramanan 02, 43

44 RBMs in piecewise smooth domains Motivation capacitated queues (Dai-Dai `99) single station-polling (Coffman-Puhalskii-Reiman `95) dynamic HLPS (Ramanan-Reiman `03) bandwidth sharing (Kelly-W 04) Input queued packet switch (Shah-Wischik `06) Sufficient conditions for existence and uniqueness Dupuis-Ishii `93 (piecewise smooth domains, reflected diffusions, strong solutions) Dai-W `95 (polyhedral domains, SRBMs, weak solutions) Dupuis-Ramanan `99, Ramanan `06 (strong solutions, extended Skorokhod problem) Mazumdar et al., (variable reflection directions) Invariance Principle (oscillation inequality) Kang-W `06 (piecewise smooth domains) 44

45 HEAVY TRAFFIC LIMIT THEOREM VIA STATE SPACE COLLAPSE 45

46 Heavily Loaded Multiclass HL Network E A D Stochastic Model V 1 K Φ Start system empty Assume ρ = 1 k k Ai ( t) = E () t +Φ ( D( t) ) D () t = S ( T() t ) i Q() t = A() t D() t i i i Wk() t = Vi ( Ai()) t t + Yk () t i k i + additional equations depending on service discipline, e.g., FIFO: D ( t + W ( t)) D ( t) = Q () t whe n i k i k i i i i i 46

47 State Space Collapse Definition: Multiplicative state space collapse (MSSC) holds if there is a IxK matrix Δ such that for each T 0 : ˆ r ˆ r Q () i ΔW () i T max( ˆ r W () i T,1) r in probability as. 0 47

48 State Space Collapse Definition: Multiplicative state space collapse (MSSC) holds if there is a IxK matrix Δ such that for each T 0 : ˆ r ˆ r Q () i ΔW () i T max( ˆ r W () i T,1) r in probability as. 0 Theorem (Bramson 98) MSSC holds if critical fluid model solutions converge uniformly to the invariant manifold. In particular, MSSC holds for FIFO Kelly type and HLPPS networks. 48

49 Sufficient Conditions for HT Limit Theorem Theorem (W 98) Assume standard heavy traffic assumptions and (i) multiplicative state space collapse, (ii) the reflection matrix R is completely-s. Then where W* is an SRBM with pushing process Y* and Q* = W*. Δ Wˆ Yˆ Qˆ W Y Q r r r * * * (,, ) (,, ) Examples: FIFO Kelly type and HLPPS networks; FBFS, LBFS reentrant lines; some static priority networks (see e.g., Bramson 98, W 98, Bramson-Dai 01) 49

50 Dai-Wang-Wang 92 example A multiclass FIFO network of Kelly type Renewal arrivals (rate ), i.i.d. service times for each class Assume Traffic intensities α m1 = m2 = m3 = m4 = m5 = m6 = m ρ1 = ρ2 = ρ3 = 2αm Reflection matrix for SRBM approximation to workload process No continuous mapping constr. for SRBM R = 1 Λ

51 FURTHER DEVELOPMENTS 51

52 Some Related Work on Diffusion Approximations for Stochastic Processing Networks HT limits that are not SRBMs (& have no state space collapse) Single station-polling: Coffman-Puhalskii-Reiman 95 Dynamic HLPS: Ramanan and Reiman 03 SRBMs in piecewise smooth (non-polyhedral domains): - conjectured to arise from Internet congestion control and input queued packet switch (Kelly-W 04, Shah-Wischik 06) Non-HL service disciplines (Markovian state descriptor is typically infinite dimensional) LIFO preemptive resume: Single station: Limic 00, 01 Processor sharing: Single station (Gromoll-Puha-W 01, Puha-W 03, Gromoll 03); network (stability: Bramson 04) EDF: Single station (Doytchinov-Lehoczky-Shreve 01), acyclic network (Kruk-Lehoczky-Shreve-Yeung 03), network (stability: Bramson 01) 52