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
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 Modelling retrial-upon-conflict systems with product-form stochastic Petri nets 2 of 19
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 ∈ T O is an output transition while in the latter we say that T ∈ T I is an input transition . Note that T = T I ∪ T O and T I ∩ T O = ∅ , where T I is the set of input transitions and T O is the set of output transitions. 2 For each T ∈ T I , there exists T ′ ∈ T O such that O ( T ) = I ( T ′ ) and vice versa. 3 Two places P i , P j ∈ P , 1 ≤ i, j ≤ N , are connected, written P i ∼ P j , 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 P i , P j ∈ P in a BB, P i ∼ ∗ P j , where ∼ ∗ is the transitive closure of ∼ . Modelling retrial-upon-conflict systems with product-form stochastic Petri nets 3 of 19
Context: Product-Form SPNs Theorem (Theorem 2 of [Balsamo et al., 2012]) Consider a BB S with N places. Let ρ y = λ y /µ y , where λ y , µ y are the firing rates for T y , T ′ y ∈ T , | y | ≥ 1 , respectively. If the following system of equations has a unique solution ρ i , (1 ≤ i ≤ N ) : � ∀ y : T y , T ′ ρ y = � y ∈ T ∧ | y | > 1 i ∈ y ρ i (1) ρ i = λ i ∀ i : T i , T ′ i ∈ T , 1 ≤ i ≤ N µ i then the net’s balance equations – and hence stationary probabilities when they exist – have product-form solution: N � ρ m i π ( m 1 , . . . , m N ) ∝ i . (2) i =1 Modelling retrial-upon-conflict systems with product-form stochastic Petri nets 4 of 19
The conflict model A set of interconnecting building blocks • a Main Building Block (MBB) • a set L of l places L = { P 1 , . . . , P l } • for each place P i , an incoming transition T i (rate λ P i ) and an outgoing transition T ′ i (rate µ P i ) • for each C ⊆ L, | C | ≥ 2 , an incoming transition T C (rate λ C ) and an outgoing transition T ′ C (rate µ C ). • a set of Conflicting Building Blocks (CBBs) • single place • one for each pair of transitions T ′ C (input), T C (output). = 2 l − l − 1 • total number of CBBs is � l � l � k =2 k • firing semantics of transitions T C , with | C | ≥ 2 , can be single server or infinite servers • Total number of places: |P| = 2 l − 1 • Total number of transitions: |T | = 2 |P| = 2 l +1 − 2 . Modelling retrial-upon-conflict systems with product-form stochastic Petri nets 5 of 19
Conflict model: Main Building Block T 1 , 3 T 1 , 2 , 3 T 1 T 1 , 2 T 2 T 1 T 1 , 2 T 2 T 2 , 3 T 3 P 1 P 2 P 3 P 1 P 2 T ′ T ′ T ′ T ′ T ′ 1 , 2 2 , 3 T ′ 1 2 3 T ′ T ′ 1 , 2 1 2 T ′ 1 , 2 , 3 T ′ 1 , 3 Modelling retrial-upon-conflict systems with product-form stochastic Petri nets 6 of 19
Conflict model: the complete picture T 1 , 2 T 1 T 2 P 1 , 2 P 1 P 2 T ′ T ′ T ′ 1 , 2 1 2 Modelling retrial-upon-conflict systems with product-form stochastic Petri nets 7 of 19
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 ) = g C ( m C ) C ∈ 2 L \∅ where m C is the component of the joint state associated with place P C and g C ( m C ) = (1 − λ P µ P )( λ P µ P ) m P if C = { P } (1 − µ C µ P )( µ C λ P λ P µ P ) m C � � if | C | ≥ 2 and T C is single server P ∈ C P ∈ C λ C λ C µ P ) m C exp ( − µ C ( µ C λ P λ P 1 � � µ P ) if | C | ≥ 2 and T C is ∞ servers P ∈ C P ∈ C λ C λ C m C ! Modelling retrial-upon-conflict systems with product-form stochastic Petri nets 8 of 19
Stability of conflict model Proposition The conflict model is stable if the following conditions hold: ∀ i ∈ { 1 , . . . , l } λ i < µ i , (3) for the places of the main building block, while for the places of conflict building blocks P C whose corresponding T C is single server, we have that � ∀ C ⊆ L µ C = µ C ρ P i < λ C , (4) P i ∈ C where µ C identifies the throughput (reversed rate) of transition T ′ C . Modelling retrial-upon-conflict systems with product-form stochastic Petri nets 9 of 19
Applications: network collisions Consider a computer network with a set L of l transmitting stations L = { s 1 , . . . , s l } . • packets become ready to be sent from each station s i according to an homogeneous Poisson process (param. λ i ) • time to transmit from s i 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 p k ( L ) • after a collision, an exponentially-distributed recovery time, with parameter µ C is performed. After that time, a new transmission is retried. • we assume µ s i = µ 1 , λ s i = λ 1 , ∀ s i ∈ L , λ C = λ | C | and µ C = µ | C | , ∀ C ⊆ L, | C | ≥ 2 Modelling retrial-upon-conflict systems with product-form stochastic Petri nets 10 of 19
Network collisions: parameters derivation We can abstract the system with a Conflict Model with an infinite server firing semantics. • q = λ 1 M is the probability, for a station, to be in transmitting phase • for C ⊂ L, | C | = k ≥ 2 , the service rate is µ k = µ ∗ q k (1 − q ) L − k • for µ s i = µ 1 we have � L � � L − 1 � µ 1 = µ ∗ � q k (1 − q ) L − k 1 − k − 1 k =2 • The average response time is l � l � ρ 1 � E [ N ] = l + kρ k 1 − ρ 1 k k =2 Modelling retrial-upon-conflict systems with product-form stochastic Petri nets 11 of 19
Network collisions: numerical example (1) 0.025 L = 10 L = 20 L = 30 0.02 Average Response Time, E[R] 0.015 0.01 0.005 0 0 5000 10000 15000 Packet Arrival Rate to the whole system, λ Modelling retrial-upon-conflict systems with product-form stochastic Petri nets 12 of 19
Network collisions: numerical example (2) 0.12 0.1 0.08 Average Response Time, E[R] 0.06 0.04 0.02 0 0 5 10 15 20 25 30 35 40 45 50 Number of stations L Modelling retrial-upon-conflict systems with product-form stochastic Petri nets 13 of 19
Applications: transactional databases • a set L of l processors L = { s 1 , . . . , s l } • transactions request to be processed by s i according to an homogeneous Poisson process with parameter λ i • 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 p k ( 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 µ i and ρ i from µ ∗ i is analogous to the previous example l � L � ρ k � E [ N ] = k 1 − ρ k k k =1 Modelling retrial-upon-conflict systems with product-form stochastic Petri nets 14 of 19
Numerical example: transactional databases (1) 0.14 L = 10 L = 20 L = 30 0.12 Average Response Time, E[R] 0.1 0.08 0.06 0.04 0.02 0 0 10 20 30 40 50 60 70 80 90 Transaction requests to each processor, λ i Modelling retrial-upon-conflict systems with product-form stochastic Petri nets 15 of 19
Numerical example: transactional databases (2) 1 q = 0.1 q = 0.3 0.95 q = 0.5 0.9 0.85 Maximum admissible λ i 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0 5 10 15 20 25 30 35 40 45 50 Number of processors L Modelling retrial-upon-conflict systems with product-form stochastic Petri nets 16 of 19
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 on 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. Modelling retrial-upon-conflict systems with product-form stochastic Petri nets 17 of 19
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