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1. Queuing Networks Jean-Yves Le Boudec 1

2. Contents • The Class of Multi-Class Product Form Networks • The Elements of a Product-Form Network • The Product-Form Theorem • Computational Aspects • What this tells us 2

3. 1. Networks of Queues are Important but May Be Tough to Analyze • Queuing networks are frequently used models • The stability issue may, in general, be a hard one • Necessary condition for stability (Natural Condition) server utilization < 1 at every queue 3

4. Instability Examples • Poisson arrivals ; jobs go through stations 1,2,1,2,1 thenleave • A job arrives as type 1, thenbecomes 2, then 3 etc • Exponential, independent service times withmeanmi • Priorityscheduling • Station 1 : 5 > 3 >1 • Station 2: 2 > 4 • Q: Whatis the naturalstability condition ? • A: λ (m1 + m3+ m5) < 1 λ (m2 + m4) < 1 4

5. λ = 1m1 = m3 = m4 = 0.1 m2 = m5 = 0.6 • Utilization factors • Station 1: 0.8 • Station 2: 0.7 • Network is unstable ! • If λ (m1 + … + m5 ) < 1 network is stable; why? 5

6. Bramson’s Example 1: A Simple FIFO Network • Poisson arrivals; jobs go through stations A, B,B…,B, A then leave • Exponential, independent service times • Steps 2 and last: mean is L • Other steps: mean is S • Q: What is the natural stability condition ? • A: λ ( L + S ) < 1λ ( (J-1)S + L ) < 1 • Bramson showed: may be unstable whereas natural stability condition holds

7. Bramson’s Example 2A FIFO Network with Arbitrarily Small Utilization Factor • m queues • 2 types of customers • λ = 0.5 each type • routing as shown, … = 7 visits • FIFO • Exponential service times, with mean as shown S S L S S L S S L S S L • Utilization factor atevery station ≤ 4 λS • Network isunstable for S ≤ 0.01L≤ S8m = floor(-2 (log L )/L) 7

8. Take Home Message • The natural stability condition is necessary but may not be sufficient • We will see a class of networks where this never happens 8

9. 2. Elements of a Product Form Network • Customers have a class attribute • Customers visit stations according to Markov Routing • External arrivals, if any, are Poisson 2 Stations Class = step, J+3 classes Can youreduce the number of classes ? 9

10. Chains • Customers can switch class, but remain in the same chain ν 10

11. Chains may be open or closed • Open chain = with Poisson arrivals. Customers must eventually leave • Closed chain: no arrival, no departure; number of customers is constant • Closed network has only closed chains • Open network has only open chains • Mixed network may have both 11

12. 3 Stations 4 classes 1 open chain 1 closedchain ν 12

13. Bramson’s Example 2A FIFO Network with Arbitrarily Small Utilization Factor S S L S S L S S L S S L 2 Stations Many classes 2 open chains Network is open 13

14. 2 Stations 5 classes 1 chain Network is open Visit ratesθ11 = θ13 =θ15 =θ22 =θ24 =λ θsc = 0 otherwise 15

15. ν 16

16. Constraints on Stations • Stations must belong to a restricted catalog of stations • We first see a few examples, then give the complete catalog • Two categories: Insensitive (= Kelly-Whittle) and MSCCC • Example of Category 1 (insensitive station): Global Processor Sharing • One server • Rate of server is shared equally among all customers present • Service requirements for customers of class c are drawn iid from a distribution which depends on the class (and the station) • Example of Category 1 (insensitive station): Delay • Infinite number of servers • Service requirements for customers of class c are drawn iid from a distribution which depends on the class (and the station) • No queuing, service time = service requirement = residence time 17

17. Example of Category 2 (MSCCC station): FIFO with B servers • B servers • FIFO queueing • Service requirements for customers of class c are drawn iid from an exponential distribution, independent of the class (but may depend on the station) • Example of Category 2 (MSCCC station): MSCCC with B servers • B servers • FIFO queueing with constraints At most one customer of each class is allowed in service • Service requirements for customers of class c are drawn iid from an exponential distribution, independent of the class (but may depend on the station) 18

18. A • Say which network satisfies the hypotheses for product form B (FIFO, Exp) C (Prio, Exp) 19

19. A station of Category 1 is any station that satisfies the Kelly-Whittle property • Examples: Global or per-class PS, Global or per-class LCFSPR, Delay 20

20. Stations of Category 2 must have Exponential, class independent service requirements 21

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22. 3. The Product Form Theorem • Stationary distrib of numbers of customers has product formEach term depends only on the station 23

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24. Stability • Stability depends only on every station in isolation being stable • When service rates are constant, this is the natural condition 25

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27. Product form and independence • In an open network • Product form => independence of stations in stationary regime • No longer true in a closed or mixed network 28

28. Phase Type Distributions • Product form theorem requires service times to be • Either exponential (category 2 stations) i.e MSCCC including FIFO) • Or Phase type (category 1 stations) • Phase type distributions can approximate any distribution (for the topology of weak convergence) • Stationary Distribution depends only on mean service time(Insensitivity of category 1) • Therefore, it is reasonable to assume that the product form theorem applies if we replace a phase type distribution by any distribution (even heavy tailed) • Was done formally in some cases  • Take home message: • Stations of category 1 may have any service time distribution, class dependent • Stations of category 2 must have exponential distrib, class independent 30

29. 4. Computational AspectsStation Function • The station function, used in the Product Form theorem, is the stationary distribution of the station in isolation 31

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35. Different Stations may have same station equivalent service rate • FIFO single server, global PS and global LCFSPR with class independent mean service time have same station functions • Check this • Therefore they have the same equivalent service rate and have the same effect in a network as long as we are interested in the distribution of numbers of customers • Hence mean response times are the same • But distributions of response times may differ • Compare PS to FIFO 38

36. Algorithms for Mixed Networks • Open networks: all stations are independent; solve one station in isolation • Mixed Networks: suppress open chains (suppression theorem) • Closed networks: the problem is computing the normalizing constant; • Many methods exist, optimized for different types of very large networks • Convolution algorithms: fairly general, applies to tricky cases (MSCCC), requires storing normalizing constant (large) 39

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38. Example • N = nb customers at GateK = total population • Product Form theorem: μ ν 42

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40. Algorithms for Mixed Networks • Open networks: all stations are independent; solve one station in isolation • Mixed Networks: suppress open chains (suppression theorem) • Closed networks: the problem is computing the normalizing constant; • Many methods exist, optimized for different types of very large networks • Convolution algorithms: fairly general, applies to tricky cases (MSCCC), requires storing normalizing constant (large) • Mean Value Analysis does not require computing the normalizing constant, but does not apply (yet ?) to all cases 44

41. The Arrival Theorem and Mean Value Analysis (MVA) version 1 • MVA version 1 uses the arrival theorem in a closed network where all stations are • FIFO or Delay • or equivalent • Based on 3 equations and iteration on population: • Mean response time for a class c customer at a FIFO station (arrival theorem): • Little’s formula: • Total number of customers gives : 46

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44. MVA Version 2 • Applies to more general networks; • Uses the decomposition and complement network theorems 49

45. is equivalent to: where the service rate μ*(n4) is the throughput of 50