Networks of queues

Networks of queues • Networks of queues reversibility, output theorem, tandem networks, partial balance, product-form distribution, blocking, insensitivity, BCMP networks, mean-value analysis, Norton's theorem, sojourn times Richard J. Boucherie Stochastic Operations Research department of Applied Mathematics University of Twente

Networks of Queues: lecture 5 Nelson, sec 10.6 • Last time on NoQ … • Kelly-Whittle network • Partial balance • Time reversed process • Quasi reversibility • Queue disciplines, Symmetric queues, BCMP networks • Network of quasi reversible queues • Insensitivity: phase type distributions • Insensitivity: processor sharing queue • Insensitivity: general case – nominal life time • Insensitivity: general case – process description • Insensitivity and partial balance • Summary / Exercises

Routes, network description • Multiclass queueing network, type i=1,..,I • J queues • Customer type identifies route • Poisson arrival rate per type i=1,…,I • Route r(i,1),r(i,2),…,r(i,S(i)) • Type i at stage s in queue r(i,s) • S(c,x) set of states in which queue contains one more class c than in state x • State X(t)=(x1(t),…,xJ(t)) • Fixed number of visits; cannot use Markov routing • 1, 2, or 3 visits to queue: use 3 types

Queue disciplines • Operation of the queue j:(i) Each job requires exponential(1) amount of service.(ii) Total service effort supplied at rate φj(nj)(iii) Proportion γj(k,nj) of this effort directed to job in position k, k=1,…, nj ; when this job leaves, his service is completed, jobs in positions k+1,…, nj move to positions k,…, nj -1.(iv) When a job arrives at queue j he moves into position k with probability δj(k,nj + 1), k=1,…, nj +1; jobs previously in positions k,…, nj move to positions k+1,…, nj +1.

Queue disciplines • Operation of the queue j:(i) Each job requires exponential(1) amount of service.(ii) Total service effort supplied at rate φj(nj)(iii) Proportion γj(k,nj) of this effort directed to job in position k, (iv) job arriving at queue j moves into position k with prob. δj(k,nj + 1) • Examples: FCFSLCFSPSinfinite server queue • BCMP network

Symmetric queues • Operation of the queue j:(i) Each job requires exponential(1) amount of service.(ii) Total service effort supplied at rate φj(nj)(iii) Proportion γ j(k,nj) of this effort directed to job in position k, (iv) job arriving at queue j moves into position k with prob. δj(k,nj + 1) • Examples: IS, LCFS, PS • Symmetric queue QR (for general service requirement) • Instantaneous attention • Note: FCFS with identical service rate for all types is QR

Network of Quasi-reversible queues • Theorem : For an open network of QR queues(i) the states of individual queues are independent at fixed time(ii) an arriving customer sees the equilibrium distribution(ii’) the equibrium distribution for a queue is as it would be in isolation with arrivals forming a Poisson process.(iii) time-reversal: another open network of QR queues(iv) system is QR, so departures form Poisson process

Erlang(k,ν) meanEL = k/ τ CV=1/ √k < 1 Hyperexponential mean CV > 1 General distribution ν ν ν

With probabilityErlang(k,ν) phase type distribution mean General distribution: phase type distribution ν ν ν ν ν ν

With probabilityErlang(k,ν) phase 1 phase 2 phase 1 phase type distribution dense in class of distributions with non-negative support General distribution: phase type distribution ν ν ν ν ν ν

Markov chain that records the remaining number of phases and that restarts in phase k wp each time phase 1 is completed state k records number of remaining phases of renewal process state space S={1,2,…} transition rates q(k,k-1) = ν q(1,k) = ν Let H(k) denote equilibrium distribution, then H(k) satisfies global balance: H(k) ν = H(1) ν + H(k+1) ν, k=1,2,… or discrete renewal equation (TK VII-6) H(k) = H(1) + H(k+1), k=1,2,… solution where General distribution: phase type distribution

is distribution that satisfies discrete renewal equation H(k) = H(1) + H(k+1), k=1,2,… Proof insert H(k) into equation: show that H(k) is distribution: General distribution: phase type distribution

Poisson arrivals rate λ Service request L mean τ=1/μ State n = # calls in queue State space S = {0,1,…} Markov chain X = {X(t), t≥0} birth rateq(n,n+1)= λ death rateq(n,n-1)=μ Equilibrium distribution Processor sharing queue

equilibrium distribution solution global balance rate out of state n = rate into state n detailed balance Proof: (exponential case)

Poisson arrivals rate λ call length L mean τ=1/μ State call i has remaining phases; State space Markov chain X = {X(t), t≥0} Transition rates Equilibrium distribution H(k) is distribution of the remaining number of phases = remaining call length Processor sharing queue: phase type call length

Equilibrium distribution Proof global balance Erlang loss queue: phase type call length and use discrete renewal equation H(1)=μ/ν

Theorem 1Equilibrium distribution where moreover, equilibrium distribution of number of calls depends on call length distribution only through its mean (insensitivity property): Proof sum distribution over all possible configurations of phases Processor sharing queue: phase type call length

Network of Quasi-reversible queues • Rates • Theorem : For an open network of QR queues(i) the states of individual queues are independent at fixed time(ii) queue length distribution is insensitive to the distribution of the holding time except for its mean.

Assumptions, Notation, Partial balance • General Markov proces, states x, State space S, transition rates q(x,x’) • Assume that • this implies regularity • Global balance • Partial balance over A

Nominal life time • At time t amount of work T has to be done, T random. T worked off at rate ρ(u): work completed at time s such that • Also holds if ρ(u) is itself a random variable • T nominal lifetime • Now suppose event completion at time u has intensity ρ(u) • Equivalent to unit intensity at unit workrate, but workrate ρ(u) • Otherwise expressed:T is exponentially distributed with unit mean, and workrate is ρ(u)

Nominal sojourn time • A arbitrary subset of S of Markov process • Intensity out of A when current state is x • Completion of sojourn time in A, or • Completion of nominal sojourn time T in A, where Texponentially distributed with unit mean and worked off at rate • is random, since x(u), the state of the MC, is random • abbreviate

Semi Markov process type description Modify dynamics of Markov process within A • Nominal sojourn time within Aarbitrary distribution of unitmean worked off at rate • Before completion of the sojourn time transitions in A have intensities q(x,x’) (x,x’εA), transitions out of A are forbidden • Upon completion of sojourn time transition out of A immediate, with transition probability • If equilibrium distribution π(x) unaffected by modification, whatever distribution of nominal sojourn time, we say that π(x) is insensitive to nominal sojourn time in A • Theorem: The equilibrium distribution is insensitive to nominal sojourn time in A if and only if the Markov process shows partial balance in A.

Phase type distribution • Subsidiary Markov process • State space J, states j=1,2,… • Transition rates (0 is outside)with Σνj =1 • T(j) sojourn time in j up to departure out of J • Total sojourn time in J is T=UjT(j) • Expected sojourn times αj=ET(j) • So that • And ET=1 implies • T is passage time, any distribution with non-negative support can be appr. arbitrary closely by phase type distr.

Supplementary variables • Modify original Markov process on S by supplementing state x to (x,j) when x in A, with the following transition rules • First entry in A, at x say, adopt state (x,j) w.p. νj • Transition (x,j)(x,k) intensity • Transition (x,j)(x’,j) intensity • Transition (x,j)x’ insensity • Nominal sojourn time in A is passage time through J for auxiliary process

Partial balance for supplemented process • Equilibrium distribution modified process • Global balance

Semi Markov process type description Modify dynamics of Markov process within A • Nominal sojourn time within Aarbitrary distribution of unitmean worked off at rate • Before completion of the sojourn time transitions in A have intensities q(x,x’) (x,x’εA), transitions out of A are forbidden • Upon completion of sojourn time transition out of A immediate, with transition probability • If equilibrium distribution π(x) unaffected by modification, whatever distribution of nominal sojourn time, we say that π(x) is insensitive to nominal sojourn time in A • Theorem: The equilibrium distribution is insensitive to nominal sojourn time in A if and only if the Markov process shows partial balance in A.

Proof of Theorem • Insert distribution • Into global balance Satisfied Reduces to

Proof of Theorem • Partial balance sufficient for insensitivity • Necessity • Insensitivity implies: • Summing GB for xεA: • For J containing two states, μ1 ≠μ2: two eqn two unknown • We must have form • And thus must have • Done if possible,

Insensitivity and partial balance • LemmaSuppose distribution π(x) shows partial balance over each of the subsets Ai (i=1,2,…,r) of S, and that there is no single transition of positive probability in which more than one Ai is vacated or more than one Ai entered. Then π(x) shows partial balance over the intersection of any selection of the Ai. • Theorem: InsensitivitySuppose subsets Ai of S are such that there is no single transition of positive intensity in which more than one Ai is vacated or more than one Ai entered. Then the equilibrium distribution π(x) is insensitive to the nominal sojourn times in the Ai if and only if the Markov process shows partial balance in all the Ai.

Networks of Queues: lecture 4 Nelson, sec 10.6 • Last time on NoQ … • Kelly-Whittle network • Partial balance • Time reversed process • Quasi reversibility • Queue disciplines, Symmetric queues, BCMP networks • Network of quasi reversible queues • Insensitivity: phase type distributions • Insensitivity: processor sharing queue • Insensitivity: general case – nominal life time • Insensitivity: general case – process description • Insensitivity and partial balance • Summary / next / Exercises

Insensitivity and partial balance, Norton’s theorem • Theorem: InsensitivitySuppose subsets Ai of S are such that there is no single transition of positive intensity in which more than one Ai is vacated or more than one Ai entered. Then the equilibrium distribution π(x) is insensitive to the nominal sojourn times in the Ai if and only if the Markov process shows partial balance in all the Ai. • Norton’s theorem: state aggregation, flow equivalent servers, Nelson, sec 10.6.14-15 • Consider network of subnetworks, each subnetwork represented by auxiliary process. Then we may lump subnet into single node if and only if partial balance over the subnets

Examples • Semi Markov process • Network of symmetric queues

Summary / next / exercises: • Jackson network • Kelly Whittle network • Partial balance • Quasi reversibility • All customers identical • Quasi reversibility, customer types • BCMP networks • Insensitivity: general service times for symmetric queues • Nelson, sec 10.6 • Aggregation / decomposition • PASTA • MUSTA • Exercises: provided next time for tutorial next week.

Networks of queues