1 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

2 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

3 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 )=( x 1 ( t ),…, x J ( 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 φ j (n j ) (iii) Proportion γ j (k,n j ) of this effort directed to job in position k, k=1,…, n j ; when this job leaves, his service is completed, jobs in positions k+1,…, n j move to positions k,…, n j -1. (iv) When a job arrives at queue j he moves into position k with probability δ j (k,n j + 1), k=1,…, n j +1; jobs previously in positions k,…, n j move to positions k+1,…, n j +1.

Operation of the queue j: (i) Each job requires exponential(1) amount of service. (ii) Total service effort supplied at rate φ j (n j ) (iii) Proportion γ j (k,n j ) of this effort directed to job in position k, (iv) job arriving at queue j moves into position k with prob. δ j (k,n j + 1) Examples: FCFS LCFS PS infinite server queue BCMP network Queue disciplines

Symmetric queues Operation of the queue j: (i) Each job requires exponential(1) amount of service. (ii) Total service effort supplied at rate φ j (n j ) (iii) Proportion γ j (k,n j ) of this effort directed to job in position k, (iv) job arriving at queue j moves into position k with prob. δ j (k,n j + 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

8 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 / Exercises

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

10 General distribution: phase type distribution With probability Erlang( k, ν ) phase type distribution mean νν ν νν ν

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

12 General distribution: phase type distribution 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

13 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:

14 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 / Exercises

15 Processor sharing queue 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 rate q(n,n+1)= λ death rate q(n,n-1)= μ Equilibrium distribution

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

17 Processor sharing queue: phase type call length 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

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

19 Processor sharing queue: phase type call length Theorem 1 Equilibrium 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

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.

21 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 / Exercises

22 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

23 Nominal life time A t 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)

24 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 T exponentially distributed with unit mean and worked off at rate is random, since x(u), the state of the MC, is random abbreviate

25 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 / Exercises

26 Semi Markov process type description Modify dynamics of Markov process within A Nominal sojourn time within A arbitrary distribution of unit mean 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.

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

28 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. ν j 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

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

30 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 / Exercises

31 Semi Markov process type description Modify dynamics of Markov process within A Nominal sojourn time within A arbitrary distribution of unit mean 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.

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

33 Partial balance sufficient for insensitivity Necessity Insensitivity implies: Summing GB for xεA: For J containing two states, μ 1 ≠μ 2 : two eqn two unknown We must have form And thus must have Done if possible, Proof of Theorem

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

35 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

36 Theorem: Insensitivity Suppose subsets A i of S are such that there is no single transition of positive intensity in which more than one A i is vacated or more than one A i entered. Then the equilibrium distribution π ( x ) is insensitive to the nominal sojourn times in the A i if and only if the Markov process shows partial balance in all the A i. Norton’s theorem: state aggregation, flow equivalent servers, Nelson, sec 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 Insensitivity and partial balance, Norton’s theorem

37 Semi Markov process Network of symmetric queues Examples

38 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.