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

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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].
4.2. PERFORMANCE MODELING PATTERN 43 M L L [ WN [ WN [ WN L L [ WN Figure 4.5: Firing of Transition ϕ i in Mode µ σ i O U M M M L L M M 4.2 Performance Modeling Pattern Figure 4.6: Modeling Network Connections. In this Section we introduce performance modeling patterns (PerfMP) for QPNs. To the best of our knowledge, no such patterns have been published yet for QPNs or for EBS. Overall, we define eleven patterns, which we illustrate in Table 4.1. Our goal is to provide solutions for common application scenarios in EBS. Some of the patterns target specific scenarios of EBS while others provide solutions to general modeling issues, e.g., thread pooling. In detail we reflect the following aspects in our QPN patterns: • Asynchronous communication • Pull-based vs. push-based communication • Point-to-point vs. one-to-many communication • Resource management, e.g., the number of events a consumer can process in parallel • Time controlled behavior, e.g., connection times • Load balancing Several of our patterns can be combined or modified to reflect a certain application behavior in our QPN models. In the following we have a closer look at our patterns. However, before we discuss them in detail, we introduce our template that we use to describe the patterns in a structured way. Pattern Template Our pattern is composed of four parts: 1. Characteristics: In this part the main aspects of a pattern are summarized in keywords. 2. Example: Providing a sample scenario for the pattern. 3. Description: A detailed description of the pattern including motivation and underlying ideas.
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