Queuing Networks
Mean Value Analysis
Jean-Yves Le Boudec
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Goal of This Module
Learn on an example how to solve a
product-form queuing network
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Reminder : A queuing network is called
«Product Form» if…
Markov routing
One or several classes
External arrivals, if any are
Poisson
Station are either
FIFO
with one or more servers
(possibly with exclusion
constraints - MSCCC)
exponential service times,
independent of class
Or insensitive station:
delay, processor sharing,
LCFS among others
Service time is arbitrary,
with finite mean – may
depend on class
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A product-form queuing network…
…is stable when the natural stability condition holds
The stationary distribution of state and of number of customers has
product form (Theorem 8.7):
𝑃 𝑛1 , … , 𝑛𝑆 = 𝐾 𝑝1 𝑛1 … 𝑝𝑆 𝑛𝑆
Station 1
Normalizing
constant
Station S
Depends only on station and visit rates,
not on the othernetwork around
Visit rate for class C at station s
𝑝𝑠 𝑛𝑠 =
𝑛
𝑓𝑠 𝑛𝑠 𝜃1,𝑠1,𝑠
𝑛
… 𝜃𝐶,𝑠𝐶,𝑠
Station function
depends only on station in isolation
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Visit Rates
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Let us apply these results to this network
FIFO
K jobs in total
FIFO
Think time Z
B=1
server
B=1
server
𝑆2
𝑆1
FIFO
B=1
server
𝑆3
Single class; closed
Stations 1,2,3 are FIFO; station 0 is delay;
Markov routing : visit rates 𝜃0 = 1; 𝜃1 = 102; 𝜃2 = 30; 𝜃3 = 17
Product-Form ?
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Let us
apply
these
results to
this
network
Single class; closed
Stations 1,2,3 are FIFO; station 0 is delay;
Markov routing : visit rates 𝜃0 = 1; 𝜃1 = 102; 𝜃2 = 30; 𝜃3 = 17
Product-Form ?
Yes if service time at stations 1,2,3 (FIFO) are ∼ exponential
No condition for station 0
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The ProductForm
Network is always stable (because closed)
Product-form ⇒ 𝑃 𝑛1 , 𝑛2 , 𝑛3 =
1
𝜂 𝐾
𝑝1 𝑛1 𝑝2 𝑛2 𝑝3 𝑛3 𝑝0 𝐾 − 𝑛1 −
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FIFO
B=1
server
Station function 𝒇𝟏
𝑆1
Let us consider the simplest possible product-form queuing network: station
1 with Poisson arrivals
This is a product-form network, with visit rate 𝜃 = 𝜆
1
Therefore 𝑃 𝑛 = 𝑓1 𝑛 𝜆𝑛
𝜂
But this is a well-known system: M/M/1
𝑃 𝑛 = 1 − 𝜌 𝜌𝑛 with 𝜌 = 𝜆𝑆1
𝑃 𝑛 = 1 − 𝜌 𝑆1𝑛 𝜆𝑛
Compare and obtain:
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FIFO
B=1
server
Station function 𝒇𝟏
𝑆1
Let us consider the simplest possible product-form queuing network: station
1 with Poisson arrivals
This is a product-form network, with visit rate 𝜃 = 𝜆
1
Therefore 𝑃 𝑛 = 𝑓1 𝑛 𝜆𝑛
𝜂
But this is a well-known system: M/M/1
𝑃 𝑛 = 1 − 𝜌 𝜌𝑛 with 𝜌 = 𝜆𝑆1
𝑃 𝑛 = 1 − 𝜌 𝑆1𝑛 𝜆𝑛
Compare and obtain: 𝑓1 𝑛 = 𝑆1𝑛
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Station function 𝒇𝟎
Think time Z
Let us consider the simplest possible product-form queuing network: station
2 with Poisson arrivals
This is a product-form network, with visit rate 𝜃 = 𝜆
1
Therefore 𝑃 𝑛 = 𝑓2 𝑛 𝜆𝑛
𝜂
𝑓2 does not depend on the distribution of service time, but only on its mean
(insensitive station). To obtain 𝑓2 , we may thus consider the case where the
service time is exponential.
We obtain a well-known system: M/M/∞
𝑃 𝑛 =
𝑛
𝜌
𝑒 −𝜌
𝑛!
with 𝜌 = 𝜆𝑍
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Station function 𝒇𝟎
Think time Z
Let us consider the simplest possible product-form queuing network: station
2 with Poisson arrivals
This is a product-form network, with visit rate 𝜃 = 𝜆
1
Therefore 𝑃 𝑛 = 𝑓2 𝑛 𝜆𝑛
𝜂
𝑓2 does not depend on the distribution of service time, but only on its mean
(insensitive station). To obtain 𝑓2 , we may thus consider the case where the
service time is exponential.
We obtain a well-known system: M/M/∞
𝑃 𝑛 =
𝑃 𝑛 =
𝑛
𝜌
𝑒 −𝜌 with 𝜌
𝑛!
𝑛
𝑍
−𝜌
𝑒
𝜆𝑛
𝑛!
= 𝜆𝑍
Compare and obtain: 𝑓0 𝑛 =
𝑍𝑛
𝑛!
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The
ProductForm
Network is always stable (because closed)
Product-form ⇒ 𝑃 𝑛1 , 𝑛2 , 𝑛3 =
1
𝜂 𝐾
𝑝1 𝑛1 𝑝2 𝑛2 𝑝3 𝑛3 𝑝0 𝐾 − 𝑛1 −
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Mean Value
Analysis
𝑃 𝑛1 , 𝑛2 , 𝑛3
1
=
𝜂 𝐾
𝑆1 𝜃1
𝑛1
𝑆2 𝜃2
𝑛2
𝑆3 𝜃3
𝑛3
𝑍𝐾−𝑛1−𝑛2−𝑛3
𝐾 − 𝑛1 − 𝑛2 − 𝑛3 !
Assume we want to compute: throughput, mean response time at station 1
We can use direct computations but need to evaluate 𝜂 𝐾
Numerical problems for large 𝐾
Combinatorial explosion of number of states
The Mean Value Algorithms does this in a smarter way
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Arrival Theorem
The distribution of customers at an arbitrary point in time is
𝑃 𝑛1 , 𝑛2 , 𝑛3 =
The distribution of customers seen by a customer just before arriving at
station 1 (excluding herself)
𝑃0 𝑛1 , 𝑛2 , 𝑛3 =
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Arrival Theorem
The distribution of customers at an arbitrary point in time is
𝐾−𝑛1 −𝑛2 −𝑛3
1
𝑍
𝑃 𝑛1 , 𝑛2 , 𝑛3 =
𝑆1 𝜃1 𝑛1 𝑆2 𝜃2 𝑛2 𝑆3 𝜃3 𝑛3
𝜂 𝐾
𝐾 − 𝑛1 − 𝑛2 − 𝑛3 !
for 𝑛1 ≥ 0, 𝑛2 ≥ 0, 𝑛3 ≥ 0 and 𝑛1 + 𝑛2 + 𝑛3 ≤ 𝐾
(and 0 otherwise)
The distribution of customers seen by a customer just before arriving at
station 1 (excluding herself)
𝐾−1−𝑛1 −𝑛2 −𝑛3
1
𝑍
𝑃0 𝑛1 , 𝑛2 , 𝑛3 =
𝑆1 𝜃1 𝑛1 𝑆2 𝜃2 𝑛2 𝑆3 𝜃3 𝑛3
𝜂 𝐾−1
𝐾 − 1 − 𝑛1 − 𝑛2 − 𝑛3 !
for 𝑛1 ≥ 0, 𝑛2 ≥ 0, 𝑛3 ≥ 0 and 𝑛1 + 𝑛2 + 𝑛3 ≤ 𝐾 − 1
(and 0 otherwise)
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Mean Value Analysis
applied to our Network
Avoids the numerical problems due to computation of normalizing constant
Iterates on population 𝐾
Variables : 𝑅𝑖 𝐾) (response time at station i)
𝑁𝑖 𝐾) (mean number of jobs, station i)
𝜆 𝐾) (throughput at station 0)
Uses :
Arrival theorem:
𝑅𝑖 𝐾 = 1 + 𝑁𝑖 𝐾 − 1 𝑆𝑖 for 𝑖 = 1,2,3
𝑅0 𝐾 = 𝑍
Little’s formula :
𝑁0 𝐾 = 𝜆 𝐾 𝑍
𝑁𝑖 𝐾 = 𝜆 𝐾 𝜃𝑖 𝑅𝑖 𝐾)
Conservation of total number of customers
𝑁0 𝐾 + 𝑁1 𝐾 + 𝑁2 𝐾 + 𝑁3 𝐾 = 𝐾
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Mean Value Analysis
applied to our Network
Iterates on 𝐾
At every step:
set 𝜆 = 1 and compute 𝑁𝑖
Obtain 𝜆 by the
conservation of number of
customers
𝑁0 = 𝑁1 = 𝑁2 = 𝑁3 = 0;
for 𝑘 = 1: 𝐾
for 𝑖 = 1: 3
𝑁𝑖 = 𝜃𝑖 1 + 𝑁𝑖 𝑆𝑖 ;
end
𝑁0 = 𝑍;
𝐾
𝜆 = 𝑁 +𝑁 +𝑁 +𝑁 ;
0
1
2
3
𝑁0 , 𝑁1 , 𝑁2 , 𝑁3 = 𝜆 𝑁0 , 𝑁1 , 𝑁2 , 𝑁3 ;
end
return 𝜆, 𝑁0 , 𝑁1 , 𝑁2 , 𝑁3 )
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The algorithm we just used is called
Mean Value Analysis (MVA) version 1
It applies to closed product form networks where all stations are
FIFO or Delay
or equivalent (i.e. have the same function 𝑓𝑖 )
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MVA Version 2
Applies to more general networks;
Gives not only means but also full distribs
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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Bottleneck analysis
Waiting time
MVA for various 𝑆 and 𝐵
1 𝑆
=
c 𝐵
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Conclusions
Product-form queueing networks can be analyzed with
very efficient algorithms
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