Performance Modeling and Benchmarking of Event-Based ... - DVS

42 CHAPTER 4. PERFORMANCE ENGINEERING OF EVENT-BASED SYSTEMS
3XEOLVKHUS N
6\VWHP
[ WN [ WN
[ WN
Figure 4.4: Modeling Non-Poisson Event Publications.
We now show how the firing weights of the immediate transitions ϕ i for i = 1..N must be
set in order to achieve the desired routing behavior. We use the notation
µ : A 1 {c ′ 1 x 1 } ∧ A 2 {c ′ 2 x 2 } ∧ ... ∧ A n {c ′ n x n } → B 1 {d ′ 1 y 1 } ∧ B 2 {d ′ 2 y 2 } ∧ ... ∧ B m {d ′ m y m }
to denote a transition mode µ in which c i × x i tokens are taken from place A i for i = 1..n
and d j × y j tokens are deposited in place B j for j = 1..m. If an event modeled by token
color x t,k visits the system node corresponding to place π i , from there it can possibly be forwarded
to every node n j such that ν t,k
i,j > 0. Assuming that there are m such nodes, let us
denote the set of their id’s as ζ = {j 1 , j 2 , ..., j m } = {j : ν t,k
i,j > 0}. For every subset of these
nodes σ = {l 1 , l 2 , ..., l r } ⊆ {j 1 , j 2 , ..., j m }, we define a firing mode µ σ i of transition ϕ i as follows:
µ σ i : π i {1 ′ x t,k } → π l1 {1 ′ x t,k } ∧ π l2 {1 ′ x t,k } ∧ ... ∧ π lr {1 ′ x t,k } (4.17)
which means that a x t,k token is taken from place π i and a x t,k token is deposited in each
of the places π l1 , π l2 , ..., π lr , as shown in Figure 4.5. This corresponds to node n i forwarding an
arriving event of type e t , published by publisher p k , to nodes n l1 , n l2 , ..., n lr . In order to achieve
the desired routing behavior, the firing weight ψ(µ σ i ) of the mode is set as follows:
ψ(µ σ i ) = ∏ g∈σ
ν t,k
i,g
∏
g∈ζ\σ
(1 − ν t,k
i,g ) (4.18)
To explain this, let us consider the action of node n i forwarding an arriving event to node n lh
as an “event” in terms of probability theory. For h = 1..r, we have r events and their probabilities
are given by ν t,k
i,l h
. At the same time for each g ∈ ζ \ σ we can consider the action of node n i
not forwarding an event to node n g as an event. We have m − r such events in total and their
probabilities are given by (1 − ν t,k
i,g ). If we assume that all these events are independent, the
probability of all of them occurring at the same time would be equal to the product of their
probabilities. Thus, Eq. (4.18) can be interpreted as the probability of forwarding an arriving
event of type e t , published by publisher p k , to nodes n lh for h = 1..r and no other nodes. Even
though in reality the independence assumption might not hold, it is easy to see that by setting
the firing weights as indicated above, routing behavior with equivalent resource consumption is
enforced.
The model we presented is focused on capturing resource contention inside the system nodes,
however, it can easily be extended to also capture contention for network resources. Network
links can be modeled using queueing places that event tokens visit when they are sent from one
place to another. Figure 4.6 shows an example of how networks can be modeled. Nodes n i1
and n i2 (represented with places π i1 and π i2 ) communicate with nodes n j1 and n j2 over a network
(e.g., a LAN) modeled using queueing place χ 1 . Node n i2 communicates with node n j3 through
another network modeled using queueing place χ 2 . Depending on the size of QPN models,
different methods can be used for their analysis, from product-form solution methods [22] to
optimized simulation techniques [127].