Queuing Networks Jean-Yves Le Boudec 1

Contents 1.The Class of Multi-Class Product Form Networks 2.The Elements of a Product-Form Network 3.The Product-Form Theorem 4.Computational Aspects 5.What this tells us 2

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

Instability Examples 4 Poisson arrivals ; jobs go through stations 1,2,1,2,1 then leave A job arrives as type 1, then becomes 2, then 3 etc Exponential, independent service times with mean m i Priority scheduling Station 1 : 5 > 3 >1 Station 2: 2 > 4 Q: What is the natural stability condition ? A: λ ( m 1 + m 3 + m 5 ) < 1 λ (m 2 + m 4 ) < 1

λ = 1 m 1 = m 3 = m 4 = 0.1 m 2 = m 5 = 0.6 Utilization factors Station 1: 0.8 Station 2: 0.7 Network is unstable ! If λ (m 1 + … + m 5 ) < 1 network is stable; why? 5

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 ) < 1 λ ( (J-1)S + L ) < 1 Bramson showed: may be unstable whereas natural stability condition holds

Bramson’s Example 2 A 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 7 LLSLLSSSSSSS Utilization factor at every station ≤ 4 λ S Network is unstable for S ≤ 0.01 L ≤ S 8 m = floor(-2 (log L )/L)

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

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 9 2 Stations Class = step, J+3 classes Can you reduce the number of classes ? 2 Stations Class = step, J+3 classes Can you reduce the number of classes ?

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

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 closed chain 3 Stations 4 classes 1 open chain 1 closed chain ν

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

Visit Rates 14

15 2 Stations 5 classes 1 chain Network is open 2 Stations 5 classes 1 chain Network is open Visit rates θ 1 1 = θ 1 3 = θ 1 5 = θ 2 2 = θ 2 4 = λ θ s c = 0 otherwise

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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

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

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

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

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

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

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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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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

Phase-Type Distributions 29

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 [8] 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

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

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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

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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Throughput Theorem 41

Example N = nb customers at Gate K = total population Product Form theorem: 42 μ ν

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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

The Arrival Theorem and Mean Value Analysis (MVA) version 1 45

The Arrival Theorem and Mean Value Analysis (MVA) version 1 46 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 :

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

50 is equivalent to: where the service rate μ*(n 4 ) is the throughput of

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5. What this tells us A rich class of networks with interesting properties Simple stability conditions Disciplines such as PS are insensitive to anything except mean service times Classes can be anything; this is a very rich modelling paradigm Only average visit rates matter 55

Example: Model of Internet 56

Product form holds if network can be modelled by a Kelly –Whittle station Requires that rate allocated to class c flows has the form Statistics of network depend only on traffic intensities Flow durations and think times may be anything and may be correlated 57

Requires that rate allocated to class c flows has the form Such an allocation is called « balanced fair » and is the only one with insensitivity property Is numerically closed to proportional fairness (TCP) 58

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