[PPT] - Modelling retrial-upon-conflict systems with product-form stochastic PowerPoint Presentation

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Modelling retrial-upon-conflict systems with product-form stochastic Petri nets

Simonetta Balsamo Gian-Luca Dei Rossi Andrea Marin

Dipartimento di Scienze Ambientali, Informatica e Statistica Universit` a Ca’ Foscari, Venezia

ASMTA ’13, Gent, 8-10 July 2013

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Aim of the paper

Systems with a retrial-upon-conflict behaviour are common

Concurrent activities which may lead to a conflict
After a recovery phase the activity is tried again.
Examples: computer networks, transactional DBs, memory buses . . .

We consider a simple class of SPNs which can be used to model this behaviour

We show that it has a Product-Form solution according to

[Balsamo et al., 2012]

We show that, under stability, we haven’t any other rate constraint
We give numerical examples

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Context: building blocks

Definition (Building block (BB) [Balsamo et al., 2012]) Given an ordinary (connected) SPN S with set of transitions T and set of N places P, then S is a building block if it satisfies the following conditions:

1 For all T ∈ T then either O(T) = 0 or I(T) = 0. In the former case

we say that T ∈ TO is an output transition while in the latter we say that T ∈ TI is an input transition. Note that T = TI ∪ TO and TI ∩ TO = ∅, where TI is the set of input transitions and TO is the set of output transitions.

2 For each T ∈ TI, there exists T ′ ∈ TO such that O(T) = I(T ′) and

vice versa.

3 Two places Pi, Pj ∈ P, 1 ≤ i, j ≤ N, are connected, written

Pi ∼ Pj, if there exists a transition T ∈ T such that the components i and j of I(T) or of O(T) are non-zero. For all places Pi, Pj ∈ P in a BB, Pi ∼∗ Pj, where ∼∗ is the transitive closure of ∼.

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Context: Product-Form SPNs

Theorem (Theorem 2 of [Balsamo et al., 2012]) Consider a BB S with N places. Let ρy = λy/µy, where λy, µy are the firing rates for Ty, T ′

y ∈ T , |y| ≥ 1, respectively. If the following system of

equations has a unique solution ρi, (1 ≤ i ≤ N):

ρy =

i∈y ρi

∀y : Ty, T ′

y ∈ T ∧ |y| > 1

ρi = λi

µi

∀i : Ti, T ′

i ∈ T , 1 ≤ i ≤ N

(1) then the net’s balance equations – and hence stationary probabilities when they exist – have product-form solution: π(m1, . . . , mN) ∝

N

i=1

ρmi

i .

(2)

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The conflict model

A set of interconnecting building blocks

a Main Building Block (MBB)
a set L of l places L = {P1, . . . , Pl}
for each place Pi, an incoming transition Ti (rate λPi) and an
utgoing transition T ′

i (rate µPi)

for each C ⊆ L, |C| ≥ 2, an incoming transition TC (rate λC) and

an outgoing transition T ′

C (rate µC).

a set of Conflicting Building Blocks (CBBs)
single place
one for each pair of transitions T ′

C (input), TC (output).

total number of CBBs is l

k=2

l

k

= 2l − l − 1
firing semantics of transitions TC, with |C| ≥ 2, can be single server
r infinite servers
Total number of places: |P| = 2l − 1
Total number of transitions: |T | = 2|P| = 2l+1 − 2.

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Conflict model: Main Building Block

P1 P2 T1 T2 T1,2 T ′

1

T ′

2

T ′

1,2

P1 P2 P3 T1 T2 T3 T1,2 T1,3 T2,3 T1,2,3 T ′

1

T ′

2

T ′

3

T ′

1,2

T ′

1,3

T ′

2,3

T ′

1,2,3

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Conflict model: the complete picture

P1 P2 P1,2 T1 T2 T1,2 T ′

1

T ′

2

T ′

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Product form of conflict model

Proposition (Product-form of the conflict model)

The conflict model consists of building blocks satisfying the structural conditions of [Balsamo et al., 2012]. Moreover, in stability, it yields without any rate-constraint the following product-form solution: π(m) =

C∈2L\∅

gC(mC) where mC is the component of the joint state associated with place PC and gC(mC) =      (1 − λP

µP )( λP µP )mP

if C = {P} (1 − µC

λC

P ∈C

λP µP )( µC λC

P ∈C

λP µP )mC

if |C| ≥ 2 and TC is single server ( µC

λC

P ∈C

λP µP )mC exp (− µC λC

P ∈C

λP µP ) 1 mC!

if |C| ≥ 2 and TC is ∞ servers

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Stability of conflict model

Proposition The conflict model is stable if the following conditions hold: ∀i ∈ {1, . . . , l} λi < µi, (3) for the places of the main building block, while for the places of conflict building blocks PC whose corresponding TC is single server, we have that ∀C ⊆ L µC = µC

Pi∈C

ρPi < λC, (4) where µC identifies the throughput (reversed rate) of transition T ′

C.

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Applications: network collisions

Consider a computer network with a set L of l transmitting stations L = {s1, . . . , sl}.

packets become ready to be sent from each station si according to

an homogeneous Poisson process (param. λi)

time to transmit from si is exponentially distributed with parameter

µ∗

i

the channel is capable of transmitting with a global rate M
a collision can occur between any combination of k stations,

2 ≤ k ≤ L, with probability pk(L)

after a collision, an exponentially-distributed recovery time, with

parameter µC is performed. After that time, a new transmission is retried.

we assume µsi = µ1, λsi = λ1, ∀si ∈ L, λC = λ|C| and µC = µ|C|,

∀C ⊆ L, |C| ≥ 2

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Network collisions: parameters derivation

We can abstract the system with a Conflict Model with an infinite server firing semantics.

q = λ1

M is the probability, for a station, to be in transmitting phase

for C ⊂ L, |C| = k ≥ 2, the service rate is

µk = µ∗qk(1 − q)L−k

for µsi = µ1 we have

µ1 = µ∗

1 −

L

k=2

L − 1 k − 1

qk(1 − q)L−k
The average response time is

E[N] = l ρ1 1 − ρ1 +

l

k=2

l k

kρk

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Network collisions: numerical example (1)

5000 10000 15000 0.005 0.01 0.015 0.02 0.025 Packet Arrival Rate to the whole system, λ Average Response Time, E[R] L = 10 L = 20 L = 30

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Network collisions: numerical example (2)

5 10 15 20 25 30 35 40 45 50 0.02 0.04 0.06 0.08 0.1 0.12 Number of stations L Average Response Time, E[R]

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Applications: transactional databases

a set L of l processors L = {s1, . . . , sl}
transactions request to be processed by si according to an

homogeneous Poisson process with parameter λi

time for a transaction to be processed is exponentially distributed

with parameter µ∗

i

conflicts can occur during parallel transaction executions between

any subset C of k processors, 2 ≤ k ≤ L, with probability pk(L).

after a conflict, all the participating transactions are started again,

after a exponentially distributed recovery time.

we can model the system using a conflict model
since recovery requests are enqueued, conflict building blocks have

the ordinary firing semantics of SPNs.

computation of µi and ρi from µ∗

i is analogous to the previous

example E[N] =

l

k=1

L k

k

ρk 1 − ρk

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Numerical example: transactional databases (1)

10 20 30 40 50 60 70 80 90 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Transaction requests to each processor, λi Average Response Time, E[R] L = 10 L = 20 L = 30

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Numerical example: transactional databases (2)

5 10 15 20 25 30 35 40 45 50 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 Number of processors L Maximum admissible λi q = 0.1 q = 0.3 q = 0.5

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Conclusions

We have shown how retrial-upon-conflict systems can be modelled

by product-form SPNs

we have shown how this class of SPNs does not require assumptions
n rates, except for what is due stability, to be in product-form
we described two examples of possible applications of this class of

models, and we have derived some performance indices for them. Future works:

consider also closed SPNs (normalisation issues)
further explore the parametrisation issue.

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References

[Balsamo et al., 2012] Balsamo, S., Harrison, P. G., and Marin, A. (2012). Methodological construction of product-form stochastic Petri nets for performance evaluation. Journal of Systems and Software, 85(7):1520–1539. [Marin et al., 2012] Marin, A., Balsamo, S., and Harrison, P. G. (2012). Analysis of stochastic Petri nets with signals. Perform. Eval., 69(11):551–572.

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Thanks!

Thank you for your attention any question?

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