Large deviations for Cox processes and Cox/G/ queuesΒ Β· Large deviations in two slides β’ π,...
Transcript of Large deviations for Cox processes and Cox/G/ queuesΒ Β· Large deviations in two slides β’ π,...
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
β’ Central dogma of molecular biology: DNA makes RNA makes proteins
β’ Protein synthesis is a stochastic processβ’ π1 π‘ : number of RNA molecules in cell at time π‘
β’ π2 π‘ : number of protein molecules in cell at time π‘
β’ possibly several interacting molecular species
β’ Questions of biological interestβ’ Can we characterise fluctuations in molecule numbers?
β’ What are the regulatory processes governing these fluctuations?
Mathematical models of reaction networks
β’ Mass-action kinetics: differential equations, no stochasticity.
β’ Markovian model of dynamics: π1 βΆ π1 + 1 at rate π1,
βΆ π1 β 1 at rate π1π1.
π2 βΆ π2 + 1 at rate π1π2,
βΆ π2 β 1 at rate π2π2.
β’ In fact, this is two interacting π π β queues.
Queuing Model
β’ Arrival process into second queue is a Cox process.
β’ Motivates the study of πΆππ₯/πΊ/ queues
π1 π‘
π2 π‘
Point process representation of infinite-server queues
s t
Queueing problem
β’ Describe queue length process over a compact interval, say 0,1
β’ Asymptotic regime: Sequence of queues, indexed by π β ββ’ Arrivals form Cox process, with directing measure Ξπ
β’ Service times iid with distribution F and finite mean
β’ ππ β : queue length process
β’ πΏπ β : measure with density ππ
β’ Suppose Ξπ/π satisfy an LDP. Then, do πΏπ/π do so as well?
Large deviations in two slides
β’ ππ , π β β, sequence of random variables taking values in some βniceβ topological space.
β’ We say they satisfy a large deviation principle (LDP) ifπ ππ β π΄ β ππ₯π βπ ππππ₯βπ΄ πΌ(π₯)
β’ More precisely, there is a lower bound for open sets and an upper bound for closed sets
β’ πΌ β is called the rate function governing the LDP. It is called a good rate function if it has compact level sets, i.e.,
π₯: πΌ(π₯) β€ πΌ is compact for all πΌ β β
Contraction principle
β’ If ππ satisfy an LDP with good rate function πΌ, and π is a continuous function, then ππ = π ππ satisfy an LDP with good rate function π½given by
π½ π¦ = ππππ₯:π π₯ =π¦ πΌ(π₯)
β’ 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
β’ If we can prove such an LDP, then, can recursively obtain LDPs for any number of such queues βin seriesβ.
π1 π‘
π2 π‘
Remark: Topological issues
β’ Will be working with measure-valued random variables
β’ Random variables are Borel measures on an underlying topological space
β’ Two natural topologies on the space of measuresβ’ Weak topology: generated by bounded continuous functions on underlying
space
β’ Vague topology: generated by continuous functions with compact support
β’ Need weak topology but vague topology will be an intermediate step
Definitions and Notation
β’ Ξπ, π β β : sequence of -finite random measures on β
β’ Fix arbitrary π, π β β. Define
ππ ππ = log πΈ exp πΞπ π, π
β’ Dependence of ππ on π, π has been suppressed in the notation.
β’ ππ β : queue length process in infinite-server queue with iid service times, and Cox process arrivals with directing measure Ξπ
β’ πΏπ β : measure on β with density ππ
Assumptions
β’ Ξπ, π β β are translation-invariant, with finite mean intensity for each π
β’ Ξπ/π | π,π satisfies an LDP on ππ π, π equipped with the topology of weak convergence, with good rate function πΌ π,π
β’ ππ ππ /π is bounded in some neighbourhood of 0, uniformly in π.
β’ The mean service time is finite.
Main result
Theorem (Dean, G., Crane, 2018)
β’ If the above assumptions are satisfied, then the sequence of random measures πΏπ/π | π,π satisfies an LDP on ππ π, π equipped with the weak topology, with good rate function π½ π,π given by the solution of an optimisation problem.
Outline of key ingredients of proof
β’ Think of ββ πΊ β β queue as a random map π β β π β on the space of -finite measures on β.
β’ Decompose it into the two sources of randomnessβ’ Directing measure of Cox arrival process Empirical distribution of arrivals
β’ Empirical distribution of arrivals Queue occupancy process
β’ Establish an LDP for each, and put them together
In pictures
a b
A([a,b])
In words
β’ Ξπ βΌ Ξπβ¨πΉ|π΄ π,π : π β ππ π΄( π, π )
β’ Ξπβ¨πΉ|π΄ π,π βΌ Ξ¦π : ππ π΄( π, π ) ππ π΄( π, π )
β’ Ξ¦π βΌ πΏπ : ππ π΄( π, π ) ππ π, π
β’ First and third map are deterministic, second is random.
β’ The last step is easy. Follows from continuity of the map, and the Contraction Principle.
Step 1: initial observation
β’ First, truncate the wedge.
au b
C(u,a,b)
Step 1: initial observation
β’ Ξπβ¨πΉ|[π’,π]Γβ+satisfies an LDP.
β’ Hence, by contraction, so does Ξπβ¨πΉ|πΆ(π’,π,π)
β’ By the Dawson-Gartner theorem, Ξπβ¨πΉ|π΄( π,π ) satisfies an LDP on
ππ π΄ π, π equipped with the projective limit topology, which is the vague topology. Not good enough!
β’ How do we strengthen LDP to weak topology?
Strengthening LDPs: Exponential tightness
β’ Need to control Ξπ Γ πΉ (π β )
au b
T(h)
Mass in the tail
0-1-2-3
Controlling mass in the tail
β’ Ξπ Γ πΉ π(β) β Ξπ β1,0 πΉ β + Ξπ β2,β1 πΉ β + 1 +β¦
β’ RHS is linear combination of identically distributed (by translation invariance of Ξπ) but not independent, random variables
β’ How do we bound the RHS?
Convex stochastic order
β’ π βΌ π in the convex stochastic order if πΈπ(π) β€ πΈπ π for all convex functions π.
β’ Fact: Suppose π, π1, π2, β¦ are identically distributed and the coefficients π1, π2, β¦ β₯ 0 have finite sum π. Then:
π1π1 + π2π2 + β― βΌ ππ
β’ Use this fact to bound log-mgf of mass in tail, and hence prove exponential tightness via Markovβs inequality.
Step 2
β’ Want to deduce an LDP for Ξ¦π/π on ππ π΄ π, π equipped with its weak topology, from an LDP for Ξπβ¨πΉ /π on the same space.
β’ Nothing special about the set π΄ π, π , so will do this in much greater generality, on Polish spaces.
LDP for Cox Processes
β’ πΈ, π : -compact Polish space
β’ ππ πΈ : space of finite Borel measures on πΈ, equipped with the weak topology
β’ Ξ¦π : sequence of Cox point processes on πΈ, with directing measures Ξπ β ππ πΈ
Theorem (Dean, G., Crane, 2018)
β’ If Ξπ/π satisfy an LDP on ππ πΈ with a good rate function πΌ, then Ξ¦π/π do so as well, with a good rate function π½
Related work
β’ LDP for Poisson point processes: Florens and Pham, Leonard
β’ LDP for Cox processes: Schreiber β somewhat different assumptions from us, and different method of proof
Sketch of proof
β’ Condition on Ξπ/π β π
β’ Conditional on Ξπ, Ξ¦π is a Poisson process. In particular:
β’ Conditional on the number of points, ππ, their locations are iid with distribution Ξπ β /Ξπ πΈ β π β /π πΈ
β’ Hence, empirical measure satisfies an LDP by Sanovβs theorem, or more precisely, an extension of it by Baxter and Jain
β’ Combine this conditional LDP with the assumed LDP for Ξπ/π to obtain a joint LDP, and thence for the marginal Ξ¦π/π
From conditional to joint LDPs
β’ Consider a sequence of random variables ππ, ππ , π β β
β’ Suppose ππ satisfy an LDP with good rate function πΌ
β’ Suppose that, conditional on ππ β π₯, ππ satisfy an LDP with good rate function π½π₯
β’ Q: Do ππ, ππ satisfy a joint LDP? Does ππ satisfy an LDP?
β’ A: Not completely straightforward. Need some sort of continuity condition. Studied by Dinwoodie and Zabell, Chaganty, Biggins
Finishing the proof
β’ We use version by Chaganty
β’ Not all required conditions are satisfied on a Polish spaceβ’ but they are on a compact metric space
β’ Need to follow approach of proving results on compact sets πΎ1, πΎ2, β¦ β πΈ, using projective limit approach, and proving exponential tightness
β’ This is where -compactness of πΈ comes in
β’ Finiteness of measures is crucial to proving exponential tightness
Open problems
β’ Have only considered queues in series. Can results be extended to general networks?
β’ Seems tractable, provided βinfluenceβ is linear as here
β’ Model is basically multitype branching process with immigration
β’ Can we prove functional central limit theorems?
β’ Measure-valued description doesnβt seem to be right approach
β’ Need to think of measures as processes indexed by suitable classes of functions? Which ones?