1. Flows and Networks (158052) Richard Boucherie Stochastische Operations Research -- TW wwwhome.math.utwente.nl/~boucherierj/onderwijs/158052/158052.html Introduction to theorie of flows in complex networks: both stochastic and deterministic apects Size 5 ECTS 32 hours of lectures : 16 R.J. Boucherie focusing on stochastic networks 16 W. Kern focusing on deterministic networks Common problem How to optimize resource allocation so as to maximize flow of items through the nodes of a complex network Material: handouts / downloads Exam: exercises / (take home) exam References: see website

2. Motivation and main question Motivation Production / storage system Internet http://www.warriorsofthe.net/ trailer http://www.warriorsofthe.net/ trailer Road traffic Main questions How to allocate servers / capacity to nodes or how to route jobs through the system to maximize system performance, such as throughput, sojourn time, utilization QUESTIONS ??

3. Aim: Optimal design of Jackson network Consider an open Jackson network with transition rates Assume that the service rates and arrival rates are given Let the costs per time unit for a job residing at queue j be Let the costs for routing a job from station i to station j be (i) Formulate the design problem (allocation of routing probabilities) as an optimisation problem. (ii) Provide the solution to this problem

4. Flows and network: stochastic networks Contents 1.Introduction; Markov chains 2.Birth-death processes; Poisson process, simple queue; reversibility; detailed balance 3.Output of simple queue; Tandem network; equilibrium distribution 4.Jackson networks; Partial balance 5.Sojourn time simple queue and tandem network 6.Performance measures for Jackson networks: throughput, mean sojourn time, blocking 7.Application: service rate allocation for throughput optimisation Application: optimal routing

5. Today: Introduction / motivation course Discrete-time Markov chain Continuous time Markov chain Birth-death process Example: pure birth process Example: pure death process Simple queue General birth-death process: equilibrium Reversibility, stationarity Truncation Kolmogorov’s criteria Summary / Next Exercises

6. Today: Introduction / motivation course Discrete-time Markov chain Continuous time Markov chain Birth-death process Example: pure birth process Example: pure death process Simple queue General birth-death process: equilibrium Reversibility, stationarity Truncation Kolmogorov’s criteria Summary / Next Exercises

7. Markov chain X n n=1,2,… stochastic process State space : all possible states Transition probability Markov property time-homogeneous Property

8. Markov chain : equilibrium distribution n-step transition probability Evaluate: Chapman-Kolmogorov equation n-step transition matrix Initial distribution Distribution at time n Matrix form

9. Markov chain: classification of states j reachable from i if there exists a path from i to j i and j communicate when j reachable from i and i reachable from j State i absorbing if p(i,i)=1 State i transient if there exists j such that j reachable from i and i not reachable from j Recurrent state i process returns to i infinitely often = non transient state State i periodic with period k>1 if k is smallest number such that all paths from i to i have length that is multiple of k Aperiodic state: recurrent state that is not periodic Ergodic Markov chain: alle states communicate, are recurrent and aperiodic (irreducible, aperiodic)

10. Markov chain : equilibrium distribution Assume: Markov chain ergodic Equilibrium distribution independent initial state stationary distribution normalising interpretation probability flux

11. Discrete time Markov chain: summary stochastic process X(t) countable or finite state space S Markov property time homogeneous independent t irreducible: each state in S reachable from any other state in S transition probabilities Assume ergodic (irreducible, aperiodic) global balance equations (equilibrium eqns) solution that can be normalised is equilibrium distribution if equilibrium distribution exists, then it is unique and is limiting distribution

12. Today: Introduction / motivation course Discrete-time Markov chain Continuous time Markov chain Birth-death process Example: pure birth process Example: pure death process Simple queue General birth-death process: equilibrium Reversibility, stationarity Truncation Kolmogorov’s criteria Summary / Next Exercises

13. Continuous time Markov chain stochastic process X(t) countable or finite state space S Markov property transition probability irreducible: each state in S reachable from any other state in S Chapman-Kolmogorov equation transition rates or jump rates

14. Continuous time Markov chain Chapman-Kolmogorov equation transition rates or jump rates Kolmogorov forward equations: (REGULAR) Global balance equations

15. Markov jump chain Hier tranparant met sprongketen, is nodig in bewijs verderop.

16. Continuous time Markov chain: summary stochastic process X(t) countable or finite state space S Markov property transition rates independent t irreducible: each state in S reachable from any other state in S Assume ergodic and regular global balance equations (equilibrium eqns) π is stationary distribution solution that can be normalised is equilibrium distribution if equilibrium distribution exists, then it is unique and is limiting distribution

17. Today: Introduction / motivation course Discrete-time Markov chain Continuous time Markov chain Birth-death process Example: pure birth process Example: pure death process Simple queue General birth-death process: equilibrium Reversibility, stationarity Truncation Kolmogorov’s criteria Summary / Next Exercises

18. Birth-death process State space Markov chain, transition rates Bounded state space: q(J,J+1)=0 then states space bounded above at J q(I,I-1)=0 then state space bounded below at I Kolmogorov forward equations Global balance equations

19. Example: pure birth process Exponential interarrival times, mean 1/  Arrival process is Poisson process Markov chain? Transition rates : let t0<t1<…<tn<t Kolmogorov forward equations for P(X(0)=0)=1 Solution for P(X(0)=0)=1

20. Example: pure death process Exponential holding times, mean 1/  P(X(0)=N)=1, S={0,1,…,N} Markov chain? Transition rates : let t0<t1<…<tn<t Kolmogorov forward equations for P(X(0)=N)=1 Solution for P(X(0)=N)=1

21. Simple queue Poisson arrival proces rate , single server exponential service times, mean 1/  Assume initially empty: P(X(0)=0)=1, S={0,1,2,…,} Markov chain? Transition rates :

22. Simple queue Poisson arrival proces rate , single server exponential service times, mean 1/  Kolmogorov forward equations, j>0 Global balance equations, j>0

23. Simple queue (ctd)    j j+1   Equilibrium distribution:  <  Stationary measure; summable  eq. distrib. Proof: Insert into global balance Detailed balance

24. Birth-death process State space Markov chain, transition rates Definition: Detailed balance equations Theorem: A distribution that satisfies detailed balance is a stationary distribution Theorem: Assume that then is the equilibrium distrubution of the birth-death prcess X.

25. Today: Introduction / motivation course Discrete-time Markov chain Continuous time Markov chain Birth-death process Example: pure birth process Example: pure death process Simple queue General birth-death process: equilibrium Reversibility, stationarity Truncation Kolmogorov’s criteria Summary / Next Exercises

26. Reversibility; stationarity Stationary process: A stochastic process is stationary if for all t 1,…,t n,  Theorem: If the initial distribution is a stationary distribution, then the process is stationary Reversible process: A stochastic process is reversible if for all t 1,…,t n,  NOTE: labelling of states only gives suggestion of one dimensional state space; this is not required

27. Reversibility; stationarity Lemma: A reversible process is stationary. Theorem: A stationary Markov chain is reversible if and only if there exists a collection of positive numbers π(j), j  S, summing to unity that satisfy the detailed balance equations When there exists such a collection π(j), j  S, it is the equilibrium distribution Proof

28. Lemma 1.9 / Corollary 1.10: If the transition rates of a reversible Markov process with state space S and equilibrium distribution are altered by changing q(j,k) to cq(j,k) for where c>0 then the resulting Markov process is reversible in equilibrium and has equilibrium distribution where B is the normalizing constant. If c=0 then the reversible Markov process is truncated to A and the resulting Markov process is reversible with equilibrium distribution Truncation of reversible processes 10 A S\A

29. Time reversed process X(t) reversible Markov process  X(-t) also, but Lemma 1.11: tijdshomogeneity not inherited for non- stationary process Theorem 1.12 : If X(t) is a stationary Markov process with transition rates q(j,k), and equilibrium distribution π(j), j  S, then the reversed process X(  - t) is a stationary Markov process with transition rates and the same equilibrium distribution Theorem 1.13: Kelly’s lemma Let X(t) be a stationary Markov processwith transition rates q(j,k). If we can find a collection of numbers q’(j,k) such that q’(j)=q(j), j  S, and a collection of positive numbers  (j), j  S, summing to unity, such that then q’(j,k) are the transition rates of the time- reversed process, and  (j), j  S, is the equilibrium distribution of both processes.

30. Kolmogorov’s criteria Theorem 1.8: A stationary Markov chain is reversible iff for each finite sequence of states Notice that

31. Today: Introduction / motivation course Discrete-time Markov chain Continuous time Markov chain Birth-death process Example: pure birth process Example: pure death process Simple queue General birth-death process: equilibrium Reversibility, stationarity Truncation Kolmogorov’s criteria Summary / Next Exercises

32. Summary / next: Basic queueing model; basic tools Markov chains Birth-death process Simple queue Reversibility, stationarity Truncation Kolmogorov’s criteria Next input / output simple queue Poisson proces PASTA Output simple queue Tandem netwerk Jackson network Partial balance Kelly/Whittle network

33. Exercises [R+SN] 1.1.2, 1.1.4, 1.1.5, 1.2.7, 1.2.8, 1.3.2, 1.3.3 (next time), 1.3.5, 1.3.6, 1.5.1, 1.5.2, 1.5.5, 1.6.2, 1.6.3, 1.6.4, 1.7.1, 1.7.8 (next time) [N] 10.1,6,7,8,9,10,12,13,15