Markovian N-Server Queues (Birth & Death Models)

Transcription

1 Markovian -Server Queues (Birth & Death Moels) - Busy Perio Arrivals Poisson (λ) ; Services exp(µ) (E(S) = /µ) Servers statistically ientical, serving FCFS. Offere loa R = λ E(S) = λ/µ Erlangs Q(t) = number in system (serve + queue) at time t. W (k) = queueing time of k-th arrival. Steay State: Q( ), W ( ), when exists. non-iling [Q(t) ] + = queue-length, an [Q(t) ] = number of ile servers

2 Examples: M/M//K, Abanonment, Balking M/M/ : λ k λ, µ k = k µ, k. Steay state π k = e R R k /k! Poisson (R) M/M//: λ k λ, µ k = k µ, k. Steay state π k = Rk k! / n= R n n! Erlang-B P {Blocke}. = E, = π, by PASTA M/M/: λ k λ, µ k = (k ) µ, k. Steay state ρ = λ µ <, servers utilization. Erlang-C P {Wait > }. = E 2, = k π k W ( ) W ( ) > = exp ( mean = µ( ρ) ) 2

3 Restriction to a Set via Time-Change X Markov, X( ) π τ A (t) = t {X(u) A} u time in A t a t a X A (t) = X ( τ A (t)) X restricte to A: Markov X A ( ) X( ) X( ) A Example: M/M// = M/M/ restricte to {,,..., }. E, = P r{x R = }/P r{x R }, X R Poisson (R) 3

4 Example: M/M/ (Erlang-C) Q Q + Q(t) = number in system at time t Q = Q restricte to {,,..., } : M/M/-/- ; λ, µ. Q + = Q restricte to {, +,...} : M/M/ ; λ, µ. Evolution of Q: Alternates between M/M/ (Q + ) an P (Wait > ) = E 2, = M/M/-/- (Q ). T, T, + T,, by PASTA, where ( after conitioning on the first step ): T, = µ π + () = µ( ρ) Hence, in which T, = E 2, = [ E, = R! λ π ( ) =. λ E, + T, T, / k= R k k! ] = ; ρ = λ µ [ + ρ ], ρ E, ; R = ρ = λ µ 4

5 M/M// (Erlang-B) with Many Servers: Assume: µ fixe, while λ as. Recall: R = R = λ /µ, ρ = ρ = R /. Erlang-B: E, = P r{x R = }/P r{x R }, where X R = Poisson (R). R large X R R + Z R, where Z = (, ), suggesting R + β R, < β <. Then E, = P r{ < X R }/P r{x R } P r { } β R < Z β /P r{z β} R ϕ(β)/φ(β) ϕ(β) φ(β) = = ϕ( β) φ( β) = h( β), h = hazar rate. Algebra: µ fixe,. QED: R + β R, for some β, < β < λ µ βµ ρ β, namely lim ( ρ ) = β. Theorem: (Jagerman, 974) QED lim E, = h( β). 5

6 M/M/ (Erlang-C) with Many Servers: Q Q + Q() = : all servers busy, no queue. Recall [ E 2, = + T ], = T, [ + ρ ]. ρ E, Here T, = λ E, µ h( β)/ /µ h( β) which applies as ( ρ ) β, < β <. Also T, = µ( ρ ) /µ β Hence, which applies as above, but for < β <. E 2, [ + β ], assuming β >. h( β) QED: R + β R for some β, < β < λ µ βµ ρ β, namely lim ( ρ ) = β. Theorem (Halfin-Whitt, 98) QED lim E 2, = [ + β h( β)]. 6

7 QED M/M/: Steay-State Theorem (Halfin-Whitt, 98) Consier a sequence equivalent M/M/, =,2,.... Then the following are lim P {W ( ) > } = α, for some < α < ; lim ( ρ ) = β, for some < β < ; λ = µ µβ + o ( ), i.e., R + β R, in which case α = α(β) = [ + β ] Halfin-Whitt function h( β) W ( ) Ŵ ( ), Ŵ ( ) Ŵ ( ) > = exp(µβ). Moreover (Queue must be orer ) ( [Q ( ) ] +, ) W ( ) where ˆQ + ( ) = µŵ ( ) = exp(β). ( ˆQ + ( ), Ŵ ( ) ) Proof: Let ( ) be Poisson(λ ). Then ivie by in an take [W ( )] = [Q ( ) ] + (Haji+ewell, 97). 7

8 QED M/M/ : Process View Framework : Sequence of M/M/ systems, inexe by =, 2,... Q = {Q (t), t } number in system V = {V (t), t } virtual waiting time, uner FCFS Parameters λ, µ µ Offere loa R = λ µ = λ /µ Traffic intensity ρ = R / Each M/M/ is a Birth & Death process, which is ergoic iff ρ <, in which case ρ = servers utilization. QED Scaling : λ = µ βµ, namely ρ = β Approximations of Process an Stationary Distribution, as : ˆQ (t) = [Q (t) ], t ˆV (t) = V (t). 8

9 umber in System, Centere an Rescale: ˆQ (t) (µ =, β =.5) 2 = 3 = time time 2 = 2.5 = time time Thanks to G. Shaikhet for the simulations an insight. 9

10 Approximating Queueing an Waiting Q = {Q (t), t } : Q (t) = number in system at t. ˆQ = { ˆQ (t), t } : stochastic process obtaine by centering an rescaling: ˆQ = Q ˆQ ( ) : stationary istribution of ˆQ ˆQ = { ˆQ(t), t } : process efine by: ˆQ (t) ˆQ(t). ˆQ (t) t ˆQ ( ) ˆQ(t) t ˆQ( ) Approximating (Virtual) Waiting Time ˆV = V ˆV = [ ] + µ ˆQ (Puhalskii, 994)

11 Diffusion Processes in IR X = {X t, t } Markov process with continuous sample paths Kolmogorov/Feller/Dynkin: characterize (Strong Markov, Continuous) by Drift function µ t (x) (infinitesimal mean) E [X t+ɛ X t X t = x] = µ t (x)ɛ + o(ɛ), ɛ ; Diffusion function σ t (x) (infinitesimal variance) Var [X t+ɛ X t X t = x] = σ t (x)ɛ + o(ɛ), ɛ ; Bounary behavior: inaccessible; absorbing, sticky, reflecting Time-homogeneous: µ t (x) µ(x), σ t (x) σ(x). Examples:. µ(x) µ ; σ(x) σ > Brownian Motion: BM (µ, σ 2 ) µ = σ = Stanar BM (SBM) 5 BM, generate as (t) t, where (t) Poisson() process 4 Brownian Motion time

12 Diffusions: Examples (Continue) 2. µ(x) = a bx, σ(x) σ Orenstein-Uhlenbeck (OU) Stationary istribution b > : ( a b, σ2 2b ) 3. Reflecte BM (µ, σ 2 ), reflecte at. (Harrison s book) { Xt = µt + σb t + Y t, X X, Y, XY = Stationary istribution µ < : exp(2 µ /σ 2 ) 4. Reflecte OU on [, ), (Glynn an War), or generally (Dupuis) { Xt = µ t (X t )t + σ t (X t )B t + Y t ; X given X, Y, XY = Ito: Characterize by solutions to stochastic ifferential equations X t = µ t (X t )t + σ t (X t )B t, t ; X given. (Reference: Karlin an Taylor, 2n Course ; Karatzas an Shreve.) Formalizes the infinitesimal escription : X t+ɛ X t X t = x µ t (x) ɛ + σ t (x) (B t+ɛ B t ) where B t+ɛ B t (, ɛ), inepenent of {B u, u t}. (Equivalently, B = {B t, t } is SBM.) 2

13 QED M/M/ : Diffusion Approximation Q = {Q (t), t } Birth an Death with λ = µ µβ birth rate, constant µ (x) = µ (x ) eath rate at state x ˆQ (t) = [Q (t) ] Scale queue length of customers (+) or servers ( ) Diffusion: Var [ ˆQ (t + ɛ) ˆQ (t) ˆQ (t) = x ] = 2µ ɛ + o(ɛ) Drift: E [ ˆQ (t + ɛ) + ˆQ (t) ˆQ (t) = x ] = [ E Q (t + ɛ) Q (t) Q (t) = + x ] = [ λ µ ( + x )] ɛ + o(ɛ) = {µ µβ [( µ + x ) ]} ɛ + o(ɛ) = = µβ ɛ + o(ɛ) x (BM) = µ(β + x) ɛ + o(ɛ) x (OU) } = µ(β x ) ɛ+o(ɛ) Expect ˆQ ˆQ iffusion: µ(x) = µ(β x ), σ(x) = 2µ Proof: Apply Stone (963), as in Halfin-Whitt (98). 3

14 QED M/M/ : Diffusion Approximation Theorem (Halfin-Whitt). Consier a sequence M/M/, =,2,.... Assume QED(β). Define ˆQ (t) = [Q (t) ], t <, ˆV (t) = V (t), t <. If ( ˆQ (), ˆV () ) ( ˆQ(), ˆV () ), then ( ( ) ˆQ, ˆV ˆQ, ) µ ˆQ + (Functional CLT), where parameters ˆQ is a iffusion process starting at ˆQ(), with infinitesimal µ(x) = { µβ x µ(x + β) x <, σ2 (x) = 2µ ; an steay-state istribution ˆQ( ) given by P { ˆQ( ) } = α(β), P { ˆQ( ) > x ˆQ( ) } = e xβ P { ˆQ( ) x ˆQ( ) } = φ(x + β)/φ(β) (exp RBM) (normal OU) 4

15 E-Driven M/M/ : Approximations Recall: : staffing gave rise to λ µ µγ lim ( ρ ) = γ, < γ <. Applying the usual QED scaling: ˆQ (t) = (Q (t) ), results in a Halfin-Whitt iffusion limit with β = : o stationary istribution for ˆQ + (t) = lim (Q (t) ) +. To ientify a more informative scaling, recall: ( W ( ) W ( ) > = exp mean = Also, Hence, lim P {W ( ) > } =. µ( ρ ) ). W ( ) Ŵ ( ) = exp(µγ). An (by Haji & ewell) [Q ( ) ] + ˆQ( ) + = µŵ ( ) = exp(γ). Expect a non-egenerate behavior of Q (t) = [Q (t) ] Q(t), t 5

16 Process View: Simulations of Q (t) E riven (with Q ()=),= E Driven (with Q ()=/2), = umber in system, centere an rescale.5.5 umber in system, centere an rescale time time Proposition: Let lim ( ρ ) = γ, < γ <. If [Q () ] Q(), then Q (t) Q(t) = { Q(), if Q(), Q()e µt, if Q(). Proof: Diffusion: Var [ Q (t + ɛ) Q (t) Q (t) = x ] = + o(ɛ) Drift: E [ Q (t + ɛ) + Q (t) Q (t) = x ] = } = ɛ + o(ɛ) x = µx ɛ + o(ɛ). = µx ɛ + o(ɛ) x Conclue Q Q : µ(x) = µx, σ(x). 6

17 Discovere that Q (t), t < egenerates, as. But Q ( ) = [Q ( ) ] + exp(γ). What s going on? Two explanations (assuming Q(), non-ranom, for simplicity):. For k (with λ = µ γµ ), T k+c,k = θ() T k+,k = θ() θ(), as. Hence, Q (t) takes close to infinite time to move a one-unit istance, suggesting that, in the limit, Q(t) freezes. 2. Consier Q + : the restriction of the Q to {, +,..}. Q + = M/M/ with λ + = µ γµ, µ + = µ. = M/M/ with λ = λ+ µ, µ = µ+ µ, accelerate by. Formally, Q + (t) = Q+ (t) Q() + is a flui limit, since Q + (t) = Q+ (t)( with λ+, µ + ) = Q+ (t)( with λ, µ) The limit is eterministic, hence a egenerate stationary istribution. To get a iffusion limit, accelerate: ˆQ + (t) = Q+ (t)( with λ+, µ + ) = Q+ ( 2 t)( with λ, µ) 7

18 Simulations of ˆQ (t) = [Q (t) ] 5 Starting at, =2, µ=, γ=.5 5 Starting at, =2, µ=, γ=.5 number in system, centere an rescale number in system, centere an rescale t time (<t<3.5) t time (<t<3.5) Theorem: Let lim ( ρ ) = γ, < γ <. Define ˆQ (t) = [Q (t) ]. Then Proof: ˆQ (t) ˆQ(t) = RBM( γµ, 2µ). Diffusion: σ (x) σ(x) = (2 x )µ Drift: µ (x) µ(x) = { µγ x + x Via ranom time-change: ˆQ(t) = RBM ( γµ, 2µ). (RBM : Since the infinitesimal rift is + at x.) ote: ˆQ( ) = exp(γ), establishing convergence over t. ote: V (t) ˆV (t) = ˆQ(t), t : no scaling neee. µ (Check at t =, using Haji & ewell). 8