Markov processes and queueing networkslelarge/X15/cours/slides3.pdf · )Convergence of Laplace...

Markov processes and queueing networks

Laurent Massoulie

September 22, 2015

Laurent Massoulie Markov processes and queueing networks

Outline

Poisson processes

Markov jump processes

Some queueing networks

The Poisson distribution(Simeon-Denis Poisson, 1781-1840)

{e−λ λ

}n∈N As prevalent as Gaussian distribution

Law of rare events (a.k.a. law of small numbers)pn,i ≥ 0 such that limn→∞ supi pn,i = 0, limn→∞

∑i pn,i = λ > 0

Then Xn =∑

i Zn,i with Zn,i : independent Bernoulli(pn,i ) verifies

XnD→ Poisson(λ)

{e−λ λ

∑i pn,i = λ > 0

Then Xn =∑

XnD→ Poisson(λ)

{e−λ λ

∑i pn,i = λ > 0

Then Xn =∑

XnD→ Poisson(λ)

Point process on R+

Definition

Point process on R+:

Collection of random times {Tn}n>0 with 0 < T1 < T2 . . .

Alternative descriptionCollection {Nt}t∈R+ with Nt :=

∑n>0 ITn∈[0,t]

Yet another descriptionCollection {N(C )} for all measurable C ⊂ R+ where

N(C ) :=∑n>0

ITn∈C

Poisson process on R+

Definition

Point process such that for all s0 = 0 < s1 < s2 < . . . < sn,

1 Increments {Nsi − Nsi−1}1≤i≤n independent

2 Law of Nt+s − Ns only depends on t

3 for some λ > 0, Nt ∼ Poisson(λt)

In fact, (3) follows from (1)–(2)

λ is called the intensity of the process

Definition

A first construction

Given dλne i.i.d. numbers Un,i , uniform on [0, n], let

N(n)t :=

∑ni=1 IUn,i≤t

Then for any k ∈ N, s0 = 0 < s1 < s2 < . . . < sn,

{N(n)si − N

(n)si−1}1≤i≤k

D→ ⊗1≤i≤kPoisson(λ(si − si−1))

Proof: Multinomial distribution of {N(n)si − N

(n)si−1}1≤i≤k

⇒ Convergence of Laplace transform

E exp(−∑k

i=1 αi (N(n)si − N

(n)si−1)) for all αk

1 ∈ Rk+

Suggests Poisson processes exist and are limits of this construction

N(n)t :=

∑ni=1 IUn,i≤t

{N(n)si − N

(n)si−1}1≤i≤k

E exp(−∑k

1 ∈ Rk+

N(n)t :=

∑ni=1 IUn,i≤t

{N(n)si − N

(n)si−1}1≤i≤k

E exp(−∑k

1 ∈ Rk+

A second characterization

Proposition

For Poisson process {Tn}n>0 of intensity λ, its interarrival timesτi = Ti+1 − Ti , where T0 = 0, verify{τn}n≥0 i.i.d. with common distribution Exp(λ)

Density of Exp(λ): λe−λxIx>0

Key property: Exponential random variable τ is memoryless,i.e. ∀t > 0, P(τ − t ∈ ·|τ > t) = P(τ ∈ ·)

Proposition

A third characterization

Proposition

Process with i.i.d., Exp(λ) interarrival times {τi}i≥0 can beconstructed on [0, t] by1) Drawing Nt ∼ Poisson(λt)2) Putting Nt points U1, . . . ,UNt on [0, t] where Ui : i.i.d. uniformon [0, t]

Proof.

Establish identity for all n ∈ N, φ : Rn+ → R:

E[φ(τ0, τ0 + τ1, . . . , τ0 + . . .+ τn−1)INt=n] = · · ·

e−λt (λt)n

n! × n!∫

(0,t]n φ(s1, s2, . . . , sn)Is1<s2<...<sn

∏ni=1 dsi

= P(Poisson(λt) = n)×E[φ(S1, . . . ,Sn)]

where Sn1 : sorted version of i.i.d. variables uniform on [0, t]

A third characterization

Proposition

Process with i.i.d., Exp(λ) interarrival times {τi}i≥0 can beconstructed on [0, t] by1) Drawing Nt ∼ Poisson(λt)2) Putting Nt points U1, . . . ,UNt on [0, t] where Ui : i.i.d. uniformon [0, t]

Proof.

Establish identity for all n ∈ N, φ : Rn+ → R:

E[φ(τ0, τ0 + τ1, . . . , τ0 + . . .+ τn−1)INt=n] = · · ·

e−λt (λt)n

n! × n!∫

(0,t]n φ(s1, s2, . . . , sn)Is1<s2<...<sn

∏ni=1 dsi

= P(Poisson(λt) = n)×E[φ(S1, . . . ,Sn)]

where Sn1 : sorted version of i.i.d. variables uniform on [0, t]

Laplace transform of Poisson processes

Definition

The Laplace transform of point process N ↔ {Tn}n>0 is thefunctional whose evaluation at f : R+ → R+ is

LN(f ) := E exp(−N(f )) = E(exp(−∑n>0

f (Tn))).

Proposition: Knowledge of LN(f ) on sufficiently rich class offunctions f : R+ → R+ (e.g. piecewise continuous with compactsupport) characterizes law of point process N.

Laplace transform of Poisson process with intensity λ:

LN(f ) = exp(−∫R+

λ(1− e−f (x))dx)

(i) Previous construction yields expression for LN(f )(ii) For f =

∑i αiICi

⇒N(Ci ) ∼ Poisson(λ∫Cidx), with

independence for disjoint Ci . Hence existence of Poisson process...

Definition

LN(f ) := E exp(−N(f )) = E(exp(−∑n>0

f (Tn))).

λ(1− e−f (x))dx)

∑i αiICi

Definition

LN(f ) := E exp(−N(f )) = E(exp(−∑n>0

f (Tn))).

λ(1− e−f (x))dx)

∑i αiICi

Definition

LN(f ) := E exp(−N(f )) = E(exp(−∑n>0

f (Tn))).

λ(1− e−f (x))dx)

∑i αiICi

independence for disjoint Ci . Hence existence of Poisson process...Laurent Massoulie Markov processes and queueing networks

Poisson process with general space and intensity

Definition

For λ : Rd → R+ locally integrable function, N ↔ {Tn}n>0 pointprocess on Rd is Poisson with intensity function λ if and only iffor measurable, disjoint Ci ⊂ Rd , i = 1, . . . , n,N(Ci ) independent, ∼ Poisson(

∫Ciλ(x)dx)

Proposition

Such a process exists and admits Laplace transform

LN(f ) = exp

(−∫Rd

λ(x)(1− e−f (x))dx

Poisson process with general space and intensity

Definition

For λ : Rd → R+ locally integrable function, N ↔ {Tn}n>0 pointprocess on Rd is Poisson with intensity function λ if and only iffor measurable, disjoint Ci ⊂ Rd , i = 1, . . . , n,N(Ci ) independent, ∼ Poisson(

∫Ciλ(x)dx)

Proposition

Such a process exists and admits Laplace transform

LN(f ) = exp

(−∫Rd

λ(x)(1− e−f (x))dx

Further properties

Strong Markov property: Poisson process N with intensityλ, stopping time T (i.e. ∀t ≥ 0, {T ≤ t} ∈ σ(Ns , s ≤ t))then on {T < +∞}, {NT+t − NT}t≥0: Poisson withintensity λ and independent of {Ns}s≤T

Superposition: For independent Poisson processes Ni withintensities λi , i = 1, . . . , n then N =

∑i Ni : Poisson with

intensity∑

Thinning: For Poisson process N ↔ {Tn}n>0 with intensityλ, {Zn}n>0 independent of N, i.i.d., valued in [k],processes Ni : Ni (C ) =

∑n>0 ITn∈CIZn=i are independent,

Poisson with intensities λi = λP(Zn = i)

Further properties

Strong Markov property: Poisson process N with intensityλ, stopping time T (i.e. ∀t ≥ 0, {T ≤ t} ∈ σ(Ns , s ≤ t))then on {T < +∞}, {NT+t − NT}t≥0: Poisson withintensity λ and independent of {Ns}s≤TSuperposition: For independent Poisson processes Ni withintensities λi , i = 1, . . . , n then N =

intensity∑

Further properties

Strong Markov property: Poisson process N with intensityλ, stopping time T (i.e. ∀t ≥ 0, {T ≤ t} ∈ σ(Ns , s ≤ t))then on {T < +∞}, {NT+t − NT}t≥0: Poisson withintensity λ and independent of {Ns}s≤TSuperposition: For independent Poisson processes Ni withintensities λi , i = 1, . . . , n then N =

intensity∑

Process {Xt}t∈R+ with values in E , countable or finite, is

Markov ifP(Xtn = xn|Xtn−1 = xn−1, . . .Xt1 = x1) = P(Xtn = xn|Xtn−1 = xn−1),tn1 ∈ Rn

+, t1 < · · · < tn, xn1 ∈ En

Homogeneous if P(Xt+s = y |Xs = x) =: pxy (t) independent of s,s, t ∈ R+, x , y ∈ E

⇒ Semi-group property pxy (t + s) =∑

z∈E pxz(t)pzy (s),or P(t + s) = P(t)P(s) with P(t) = {pxy (t)}x ,y∈E

Definition

{Xt}t∈R+ is a pure jump Markov process if in addition(i) It spends with probability 1 a strictly positive time in each state(ii) Trajectories t → Xt are right-continuous

+, t1 < · · · < tn, xn1 ∈ En

Definition

+, t1 < · · · < tn, xn1 ∈ En

Definition

Markov jump processes: examples

Poisson process {Nt}t∈R+ : then Markov jump process withpxy (t) = P(Poisson(λt) = y − x)

Single-server queue, FIFO (“First-in-first-out”) discipline,arrival times: N Poisson (λ), service times: i.i.d. Exp(µ)independent of NXt= number of customers present at time t: Markov jumpprocess by Memoryless property of Exponential distribution +Markov property of Poisson process(the M/M/1/∞ queue)

Infinite server queue with Poisson arrivals and Exponentialservice times: customer arrived at Tn stays in system tillTn + σn, where σn: service timeXt= number of customers present at time t:Markov jump process (the M/M/∞/∞ queue)

Structure of Markov jump processes

Infinitesimal Generator

∀x , y , y 6= x ∈ E , limits qx := limt→01−pxx (t)

t , qxy = limt→0pxy (t)

texist in R+ and satisfy

∑y 6=x qxy = qx

qxy : Jump rate from x to yQ := {qxy}x ,y∈E where qxx = −qx : Infinitesimal Generator ofprocess {Xt}t∈R+

Formally: Q = limh→01h [P(h)− I ] where I : identity matrix

Sequence {Yn}n∈N of visited states: Markov chain with transitionmatrix pxy = Ix 6=y

Conditionally on {Yn}n∈N, sojourn times {τn}n∈N in successivestates Yn: independent, with distributions Exp(qYn)

Infinitesimal Generator

∀x , y , y 6= x ∈ E , limits qx := limt→01−pxx (t)

t , qxy = limt→0pxy (t)

texist in R+ and satisfy

∑y 6=x qxy = qx

qxy : Jump rate from x to yQ := {qxy}x ,y∈E where qxx = −qx : Infinitesimal Generator ofprocess {Xt}t∈R+

Formally: Q = limh→01h [P(h)− I ] where I : identity matrix

Sequence {Yn}n∈N of visited states: Markov chain with transitionmatrix pxy = Ix 6=y

Conditionally on {Yn}n∈N, sojourn times {τn}n∈N in successivestates Yn: independent, with distributions Exp(qYn)

Examples

for Poisson process (λ): only non-zero jump rateqx ,x+1 = λ = qx , x ∈ N

For FIFO M/M/1/∞ queue, non-zero rates: qx ,x+1 = λ,qx ,x−1 = µIx>0, x ∈ N hence qx = λ+ µIx>0

For M/M/∞/∞ queue, non-zero rates: qx ,x+1 = λ,qx ,x−1 = µx , x ∈ N hence qx = λ+ µx

Examples

Structure of Markov jump processes (continued)

Let Tn :=∑n−1

k=0 τk : time of n-th jump.

If T∞ = +∞ almost surely: trajectory determined on R+, hencegenerator Q determines law of process {Xt}t∈R+

Process is called explosive if instead T∞ < +∞ with positiveprobability. Then process not completely characterized by generator

Sufficient conditions for non-explosiveness:

supx∈E qx < +∞Recurrence of induced chain {Yn}n∈NFor Birth and Death processes (i.e. E = N, only non-zerorates: βn = qn,n+1, birth rate; δn = qn,n−1, death rate),non-explosiveness holds if∑

βn + δn= +∞

Let Tn :=∑n−1

βn + δn= +∞

Let Tn :=∑n−1

βn + δn= +∞

Kolmogorov’s forward and backward equations

Formal differentiation of P(t + h) = P(t)P(h) = P(h)P(t) yields

ddtP(t) = P(t)Q Kolmogorov’s forward equationddt pxy (t) =

∑z∈E pxz(t)qzy

ddtP(t) = QP(t) Kolmogorov’s backward equationddt pxy (t) =

∑z∈E qxzpzy (t)

Follow directly from Q = limh→01h [P(h)− I ] for finite E , in which

case P(t) = exp(tQ), t ≥ 0

Hold more generally–in particular for non-explosive processes–witha more involved proof (justifying exchange of summation anddifferentiation)

Kolmogorov’s forward and backward equations

Formal differentiation of P(t + h) = P(t)P(h) = P(h)P(t) yields

ddtP(t) = P(t)Q Kolmogorov’s forward equationddt pxy (t) =

∑z∈E pxz(t)qzy

ddtP(t) = QP(t) Kolmogorov’s backward equationddt pxy (t) =

∑z∈E qxzpzy (t)

Follow directly from Q = limh→01h [P(h)− I ] for finite E , in which

case P(t) = exp(tQ), t ≥ 0

Hold more generally–in particular for non-explosive processes–witha more involved proof (justifying exchange of summation anddifferentiation)

Stationary distributions and measures

Definition

Measure {πx}x∈E is stationary if it satisfies πTQ = 0, orequivalently the global balance equations

∀x ∈ E , πx∑

y 6=x qxy =∑

y 6=x πyqyxflow out of x flow into x

Kolmogorov’s equations suggest that, if X0 ∼ π for stationary πthen Xt ∼ π for all t ≥ 0

Example: stationarity for birth and death processes

π0β0 = π1δ1,πx(βx + δx) = πx−1βx−1 + πx+1δx+1, x ≥ 1

Definition

∀x ∈ E , πx∑

y 6=x qxy =∑

Kolmogorov’s equations suggest that, if X0 ∼ π for stationary πthen Xt ∼ π for all t ≥ 0 ,

Definition

∀x ∈ E , πx∑

y 6=x qxy =∑

Kolmogorov’s equations suggest that, if X0 ∼ π for stationary πthen Xt ∼ π for all t ≥ 0 ,

Irreducibility, recurrence, invariance

Definition

Process {Xt}t∈R+ is irreducible (respectively, irreduciblerecurrent) if induced chain {Yn}n∈N is.

State x is positive recurrent if Ex(Rx) < +∞, where

Rx = inf{t > τ0 : Xt = x}.

Measure π is invariant for process {Xt}t∈R+ if for all t > 0,πTP(t) = πT , i.e.

∀x ∈ E ,∑y∈E

πypyx(t) = πx .

Limit theorems 1

Theorem

For irreducible recurrent {Xt}t∈R+ , ∃ invariant measure π, uniqueup to some scalar factor. It can be defined as, for any x ∈ E :

∀y ∈ E , πy = Ex

∫ Rx

0IXt=ydt,

or alternatively with Tx := inf{n > 0 : Yn = x},

∀y ∈ E , πy =1

Tx∑n=1

IYn=y .

Corollaries

{πy} invariant for {Yn}n∈N ⇔ {πy/qy} invariant for{Xt}t∈R+ .For irreducible recurrent {Xt}t∈R+ , either all or no statex ∈ E is positive recurrent.

Limit theorems 1

Theorem

For irreducible recurrent {Xt}t∈R+ , ∃ invariant measure π, uniqueup to some scalar factor. It can be defined as, for any x ∈ E :

∀y ∈ E , πy = Ex

∫ Rx

0IXt=ydt,

or alternatively with Tx := inf{n > 0 : Yn = x},

∀y ∈ E , πy =1

Tx∑n=1

IYn=y .

Corollaries

{πy} invariant for {Yn}n∈N ⇔ {πy/qy} invariant for{Xt}t∈R+ .For irreducible recurrent {Xt}t∈R+ , either all or no statex ∈ E is positive recurrent.

Limit theorems 2

Theorem

{Xt}t∈R+ is ergodic (i.e. irreducible, positive recurrent) iff it isirreducible, non-explosive and such that ∃π satisfying globalbalance equations.Then π is also the unique invariant probability distribution.

Theorem

For ergodic {Xt}t∈R+ with stationary distribution π, any initialdistribution for X0 and π-integrable f ,

almost surely limt→∞

0f (Xs)ds =

∑x∈E

πx f (x) (ergodic theorem)

and in distribution XtD→ π as t →∞.

Limit theorems 2

Theorem

{Xt}t∈R+ is ergodic (i.e. irreducible, positive recurrent) iff it isirreducible, non-explosive and such that ∃π satisfying globalbalance equations.Then π is also the unique invariant probability distribution.

Theorem

For ergodic {Xt}t∈R+ with stationary distribution π, any initialdistribution for X0 and π-integrable f ,

almost surely limt→∞

0f (Xs)ds =

∑x∈E

πx f (x) (ergodic theorem)

and in distribution XtD→ π as t →∞.

Limit theorems 3

Theorem

For irreducible, non-ergodic {Xt}t∈R+ , any initial distribution forX0, then for all x ∈ E ,

limt→∞

P(Xt = x) = 0.

Time reversal and reversibility

For stationary ergodic {Xt}t∈R with stationary distribution π,time-reversed process Xt = X−t :Markov with transition rates qxy =

πyqyxπx

Definition

Stationary ergodic {Xt}t∈R with stationary distribution πreversible iff distributed as time-reversal {Xt}t∈R, i.e.

∀x 6= y ∈ E , πxqxy = πyqyx ,flow from x to y flow from y to x

detailed balance equations.

Detailed balance, i.e. reversibility for π implies global balance for π.Example: for birth and death processes, detailed balance alwaysholds for stationary measure.

πyqyxπx

Definition

πyqyxπx

Definition

Reversibility and truncation

Proposition

Let generator Q on E admit reversible measure π. Then for subsetF ⊂ E , truncated generator Q:

Qxy = Qxy , x 6= y ∈ F ,

Qxx = −∑

y 6=x Qxy , x ∈ F

admits {πx}x∈F as reversible measure.

Erlang’s model of telephone network

Call types s ∈ S: type-s calls arrive at instants of Poisson(λs) process, last (if accepted) for duration Exponential (µs)

type-s calls require one circuit (unit of capacity) per link ` ∈ s

Link ` has capacity C` circuits

Stationary probability distribution:

πx =1

∏s∈S

ρxssxs !

I∑s3` xs≤C`

where: ρs = λs/µs , Z : normalizing constant.

Basis for dimensioning studies of telephone networks (prediction ofcall rejection probabilities)More recent application: performance analysis of peer-to-peersystems for video streaming.

πx =1

∏s∈S

ρxssxs !

I∑s3` xs≤C`

πx =1

∏s∈S

ρxssxs !

I∑s3` xs≤C`

Jackson networks

Stations i ∈ I receive external arrivals at Poisson rate λi

Station i when processing xi customers completes service atrate µiφi (xi ) (e.g.: φi (x) = min(xi , ni ): queue with ni serversand service times Exponential (µi ))

After completing service at station i , customer joins station jwith probability pij , j ∈ I , and leaves system with probability1−

∑j∈I pij

Matrix P = (pij): sub-stochastic, such that ∃(I − P)−1

Traffic equations

∀i ∈ I , λi = λi +∑j∈I

λjpji

or λ = (I − PT )−1λ

Jackson networks

Stations i ∈ I receive external arrivals at Poisson rate λi

Station i when processing xi customers completes service atrate µiφi (xi ) (e.g.: φi (x) = min(xi , ni ): queue with ni serversand service times Exponential (µi ))

After completing service at station i , customer joins station jwith probability pij , j ∈ I , and leaves system with probability1−

∑j∈I pij

Matrix P = (pij): sub-stochastic, such that ∃(I − P)−1

Traffic equations

∀i ∈ I , λi = λi +∑j∈I

λjpji

or λ = (I − PT )−1λ

Jackson networks (continued)

Stationary measure:

πx =∏i∈I

ρxii∏xim=1 φi (m)

where ρi = λi/µi , and λi : solutions of traffic equations

Application: process ergodic when π has finite mass. e.g. forφi (x) = min(x , ni ), ergodicity iff ∀i ∈ I , ρi < ni .Proof: verify partial balance equations for all x ∈ NI :

∀i ∈ I ,πx [∑

j 6=i qx ,x−ei+ej + qx ,x−ei ] =∑

j 6=i πx−ei+ejqx−ei+ej ,x + πx−eiqx−ei ,x ,

πx∑

i∈I qx ,x+ei =∑

i∈I πx+eiqx+ei ,x ,

which imply global balance equations

πx [∑

i∈I (qx ,x−ei + qx ,x+ei +∑

j 6=i qx ,x−ei+ej )] =∑i∈I (πx−eiqx−ei ,x + πx+eiqx+ei ,x +

∑j 6=i πx−ei+ejqx−ei+ej ,x)

Stationary measure:

πx =∏i∈I

where ρi = λi/µi , and λi : solutions of traffic equationsApplication: process ergodic when π has finite mass. e.g. forφi (x) = min(x , ni ), ergodicity iff ∀i ∈ I , ρi < ni .

Proof: verify partial balance equations for all x ∈ NI :

∀i ∈ I ,πx [∑

πx∑

i∈I qx ,x+ei =∑

πx [∑

Stationary measure:

πx =∏i∈I

where ρi = λi/µi , and λi : solutions of traffic equationsApplication: process ergodic when π has finite mass. e.g. forφi (x) = min(x , ni ), ergodicity iff ∀i ∈ I , ρi < ni .Proof: verify partial balance equations for all x ∈ NI :

∀i ∈ I ,πx [∑

πx∑

i∈I qx ,x+ei =∑

πx [∑

Takeaway messages

Poisson process a fundamental continuous-time process,adequate model for aggregate of infrequent independentevents

Markov jump processes:i) generator Q characterizes distribution if not explosiveii) Balance equation characterizes invariant distribution ifirreducible non-explosiveiii) Limit theorems: stationary distribution reflects long-termperformance

Exactly solvable models include reversible processes, plusseveral other important classes (e.g. Jackson networks)

Markov processes and queueing networkslelarge/X15/cours/slides3.pdf · )Convergence of Laplace...

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