Large deviations for Cox processes and Cox/G/ queues

Ayalvadi Ganesh

University of Bristol

Joint work with Justin Dean and Edward Crane

Motivation: biochemical reaction networks

• Central dogma of molecular biology: DNA makes RNA makes proteins

• Protein synthesis is a stochastic process• 𝑁1 𝑡 : number of RNA molecules in cell at time 𝑡

• 𝑁2 𝑡 : number of protein molecules in cell at time 𝑡

• possibly several interacting molecular species

• Questions of biological interest• Can we characterise fluctuations in molecule numbers?

• What are the regulatory processes governing these fluctuations?

Mathematical models of reaction networks

• Mass-action kinetics: differential equations, no stochasticity.

• Markovian model of dynamics: 𝑛1 ⟶ 𝑛1 + 1 at rate 𝜆1,

⟶ 𝑛1 − 1 at rate 𝑛1𝜇1.

𝑛2 ⟶ 𝑛2 + 1 at rate 𝑛1𝜆2,

⟶ 𝑛2 − 1 at rate 𝑛2𝜇2.

• In fact, this is two interacting 𝑀 𝑀 ∞ queues.

Queuing Model

• Arrival process into second queue is a Cox process.

• Motivates the study of 𝐶𝑜𝑥/𝐺/ queues

𝑁1 𝑡

𝑁2 𝑡

Point process representation of infinite-server queues

s t

Queueing problem

• Describe queue length process over a compact interval, say 0,1

• Asymptotic regime: Sequence of queues, indexed by 𝑛 ∈ ℕ• Arrivals form Cox process, with directing measure Λ𝑛

• Service times iid with distribution F and finite mean

• 𝑄𝑛 ∙ : queue length process

• 𝐿𝑛 ∙ : measure with density 𝑄𝑛

• Suppose Λ𝑛/𝑛 satisfy an LDP. Then, do 𝐿𝑛/𝑛 do so as well?

Large deviations in two slides

• 𝑋𝑛 , 𝑛 ∈ ℕ, sequence of random variables taking values in some ‘nice’ topological space.

• We say they satisfy a large deviation principle (LDP) if𝑃 𝑋𝑛 ∈ 𝐴 ≈ 𝑒𝑥𝑝 −𝑛 𝑖𝑛𝑓𝑥∈𝐴 𝐼(𝑥)

• More precisely, there is a lower bound for open sets and an upper bound for closed sets

• 𝐼 ⋅ is called the rate function governing the LDP. It is called a good rate function if it has compact level sets, i.e.,

𝑥: 𝐼(𝑥) ≤ 𝛼 is compact for all 𝛼 ∈ ℝ

Contraction principle

• If 𝑋𝑛 satisfy an LDP with good rate function 𝐼, and 𝑓 is a continuous function, then 𝑌𝑛 = 𝑓 𝑋𝑛 satisfy an LDP with good rate function 𝐽given by

𝐽 𝑦 = 𝑖𝑛𝑓𝑥:𝑓 𝑥 =𝑦 𝐼(𝑥)

• Role of topology: It is easier to prove an LDP in a coarser topology. But a finer topology admits more continuous functions, making it easier to derive new LDPs via the contraction principle.

LDP for queue length processes

• If we can prove such an LDP, then, can recursively obtain LDPs for any number of such queues ‘in series’.

𝑁1 𝑡

𝑁2 𝑡

Remark: Topological issues

• Will be working with measure-valued random variables

• Random variables are Borel measures on an underlying topological space

• Two natural topologies on the space of measures• Weak topology: generated by bounded continuous functions on underlying

space

• Vague topology: generated by continuous functions with compact support

• Need weak topology but vague topology will be an intermediate step

Definitions and Notation

• Λ𝑛, 𝑛 ∈ ℕ : sequence of -finite random measures on ℝ

• Fix arbitrary 𝑎, 𝑏 ⊂ ℝ. Define

𝜓𝑛 𝑛𝜃 = log 𝐸 exp 𝜃Λ𝑛 𝑎, 𝑏

• Dependence of 𝜓𝑛 on 𝑎, 𝑏 has been suppressed in the notation.

• 𝑄𝑛 ⋅ : queue length process in infinite-server queue with iid service times, and Cox process arrivals with directing measure Λ𝑛

• 𝐿𝑛 ⋅ : measure on ℝ with density 𝑄𝑛

Assumptions

• Λ𝑛, 𝑛 ∈ ℕ are translation-invariant, with finite mean intensity for each 𝑛

• Λ𝑛/𝑛 | 𝑎,𝑏 satisfies an LDP on 𝑀𝑓 𝑎, 𝑏 equipped with the topology of weak convergence, with good rate function 𝐼 𝑎,𝑏

• 𝜓𝑛 𝑛𝜃 /𝑛 is bounded in some neighbourhood of 0, uniformly in 𝑛.

• The mean service time is finite.

Main result

Theorem (Dean, G., Crane, 2018)

• If the above assumptions are satisfied, then the sequence of random measures 𝐿𝑛/𝑛 | 𝑎,𝑏 satisfies an LDP on 𝑀𝑓 𝑎, 𝑏 equipped with the weak topology, with good rate function 𝐽 𝑎,𝑏 given by the solution of an optimisation problem.

Outline of key ingredients of proof

• Think of ∙∕ 𝐺 ∕ ∞ queue as a random map 𝑀 ℝ → 𝑀 ℝ on the space of -finite measures on ℝ.

• Decompose it into the two sources of randomness• Directing measure of Cox arrival process Empirical distribution of arrivals

• Empirical distribution of arrivals Queue occupancy process

• Establish an LDP for each, and put them together

In pictures

a b

A([a,b])

In words

• Λ𝑛 ⟼ Λ𝑛⨂𝐹|𝐴 𝑎,𝑏 : 𝑀 ℝ 𝑀𝑓 𝐴( 𝑎, 𝑏 )

• Λ𝑛⨂𝐹|𝐴 𝑎,𝑏 ⟼ Φ𝑛 : 𝑀𝑓 𝐴( 𝑎, 𝑏 ) 𝑀𝑓 𝐴( 𝑎, 𝑏 )

• Φ𝑛 ⟼ 𝐿𝑛 : 𝑀𝑓 𝐴( 𝑎, 𝑏 ) 𝑀𝑓 𝑎, 𝑏

• First and third map are deterministic, second is random.

• The last step is easy. Follows from continuity of the map, and the Contraction Principle.

Step 1: initial observation

• First, truncate the wedge.

au b

C(u,a,b)

Step 1: initial observation

• Λ𝑛⨂𝐹|[𝑢,𝑏]×ℝ+satisfies an LDP.

• Hence, by contraction, so does Λ𝑛⨂𝐹|𝐶(𝑢,𝑎,𝑏)

• By the Dawson-Gartner theorem, Λ𝑛⨂𝐹|𝐴( 𝑎,𝑏 ) satisfies an LDP on

𝑀𝑓 𝐴 𝑎, 𝑏 equipped with the projective limit topology, which is the vague topology. Not good enough!

• How do we strengthen LDP to weak topology?

Strengthening LDPs: Exponential tightness

• Need to control Λ𝑛 × 𝐹 (𝑇 ℎ )

au b

T(h)

Mass in the tail

0-1-2-3

Controlling mass in the tail

• Λ𝑛 × 𝐹 𝑇(ℎ) ≈ Λ𝑛 −1,0 𝐹 ℎ + Λ𝑛 −2,−1 𝐹 ℎ + 1 +…

• RHS is linear combination of identically distributed (by translation invariance of Λ𝑛) but not independent, random variables

• How do we bound the RHS?

Convex stochastic order

• 𝑋 ≼ 𝑌 in the convex stochastic order if 𝐸𝑓(𝑋) ≤ 𝐸𝑓 𝑌 for all convex functions 𝑓.

• Fact: Suppose 𝑋, 𝑋1, 𝑋2, … are identically distributed and the coefficients 𝑐1, 𝑐2, … ≥ 0 have finite sum 𝑐. Then:

𝑐1𝑋1 + 𝑐2𝑋2 + ⋯ ≼ 𝑐𝑋

• Use this fact to bound log-mgf of mass in tail, and hence prove exponential tightness via Markov’s inequality.

Step 2

• Want to deduce an LDP for Φ𝑛/𝑛 on 𝑀𝑓 𝐴 𝑎, 𝑏 equipped with its weak topology, from an LDP for Λ𝑛⨂𝐹 /𝑛 on the same space.

• Nothing special about the set 𝐴 𝑎, 𝑏 , so will do this in much greater generality, on Polish spaces.

LDP for Cox Processes

• 𝐸, 𝑑 : -compact Polish space

• 𝑀𝑓 𝐸 : space of finite Borel measures on 𝐸, equipped with the weak topology

• Φ𝑛 : sequence of Cox point processes on 𝐸, with directing measures Λ𝑛 ∈ 𝑀𝑓 𝐸

Theorem (Dean, G., Crane, 2018)

• If Λ𝑛/𝑛 satisfy an LDP on 𝑀𝑓 𝐸 with a good rate function 𝐼, then Φ𝑛/𝑛 do so as well, with a good rate function 𝐽

Related work

• LDP for Poisson point processes: Florens and Pham, Leonard

• LDP for Cox processes: Schreiber – somewhat different assumptions from us, and different method of proof

Sketch of proof

• Condition on Λ𝑛/𝑛 → 𝜆

• Conditional on Λ𝑛, Φ𝑛 is a Poisson process. In particular:

• Conditional on the number of points, 𝑁𝑛, their locations are iid with distribution Λ𝑛 ⋅ /Λ𝑛 𝐸 → 𝜆 ⋅ /𝜆 𝐸

• Hence, empirical measure satisfies an LDP by Sanov’s theorem, or more precisely, an extension of it by Baxter and Jain

• Combine this conditional LDP with the assumed LDP for Λ𝑛/𝑛 to obtain a joint LDP, and thence for the marginal Φ𝑛/𝑛

From conditional to joint LDPs

• Consider a sequence of random variables 𝑋𝑛, 𝑌𝑛 , 𝑛 ∈ ℕ

• Suppose 𝑋𝑛 satisfy an LDP with good rate function 𝐼

• Suppose that, conditional on 𝑋𝑛 → 𝑥, 𝑌𝑛 satisfy an LDP with good rate function 𝐽𝑥

• Q: Do 𝑋𝑛, 𝑌𝑛 satisfy a joint LDP? Does 𝑌𝑛 satisfy an LDP?

• A: Not completely straightforward. Need some sort of continuity condition. Studied by Dinwoodie and Zabell, Chaganty, Biggins

Finishing the proof

• We use version by Chaganty

• Not all required conditions are satisfied on a Polish space• but they are on a compact metric space

• Need to follow approach of proving results on compact sets 𝐾1, 𝐾2, … ↑ 𝐸, using projective limit approach, and proving exponential tightness

• This is where -compactness of 𝐸 comes in

• Finiteness of measures is crucial to proving exponential tightness

Open problems

• Have only considered queues in series. Can results be extended to general networks?

• Seems tractable, provided ‘influence’ is linear as here

• Model is basically multitype branching process with immigration

• Can we prove functional central limit theorems?

• Measure-valued description doesn’t seem to be right approach

• Need to think of measures as processes indexed by suitable classes of functions? Which ones?