Queuing Networks

Jean-Yves Le Boudec

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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

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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

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Instability Examples

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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 mi

Priority schedulingStation 1 : 5 > 3 >1

Station 2: 2 > 4

Q: What is the natural stability condition ?

A: λ (m1 + m3 + m5 ) < 1 λ (m2 + m4) < 1

λ = 1m1 = m3 = m4 = 0.1 m2 = m5 = 0.6

Utilization factorsStation 1: 0.8

Station 2: 0.7

Network is unstable !

If λ (m1 + … + m5 ) < 1 network is stable; why?

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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

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

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L LS L LS S S S S S S

Utilization factor at every station ≤ 4 λ S

Network is unstable for S ≤ 0.01L ≤ S8

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

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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

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2 StationsClass = step, J+3 classes

Can you reduce the number of classes ?

Chains

Customers can switch class, but remain in the same chain

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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

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3 Stations4 classes1 open chain1 closed chain

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Bramson’s Example 2A FIFO Network with Arbitrarily Small Utilization Factor

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L LS L LS S S S S S S

2 StationsMany classes2 open chainsNetwork is open

Visit Rates

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2 Stations5 classes1 chainNetwork is open

Visit ratesθ1

1 = θ13 = θ1

5 = θ22 = θ2

4 = λ θs

c = 0 otherwise

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ν

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 SharingOne 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): DelayInfinite 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

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Example of Category 2 (MSCCC station): FIFO with B serversB 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 serversB 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)

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Say which network satisfies the hypotheses for product form

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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

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

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3. The Product Form Theorem

Stationary distrib of numbers of customers has product formEach term depends only on the station

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Stability

Stability depends only on every station in isolation being stable

When service rates are constant, this is the natural condition25

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Product form and independence

In an open networkProduct form => independence of stations in stationary regime

No longer true in a closed or mixed network

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Phase-Type Distributions

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Phase Type Distributions

Product form theorem requires service times to beEither 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

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4. Computational AspectsStation Function

The station function, used in the Product Form theorem, is the stationary distribution of the station in isolation

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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 differCompare PS to FIFO

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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)

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Throughput Theorem

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Example

N = nb customers at GateK = total population

Product Form theorem:

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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

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

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

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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

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is equivalent to:

where the service rate μ*(n4) is the throughput of

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5. What this tells us

A rich class of networks with interesting propertiesSimple 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

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Example: Model of Internet

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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 correlated57

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)

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Questions

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