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22 CHAPTER 2. BACKGROUND Other extensions of ordinary PNs allow timing aspects to be integrated into the net description [26, 32]. In particular, generalized stochastic PNs (GSPNs) attach an exponentially distributed firing delay (or firing time) to each transition, which specifies the time the transition waits after being enabled before it fires. Two types of transitions are defined: immediate (no firing delay) and timed (exponentially distributed firing delay). If several immediate transitions are enabled at the same time, the next transition to fire is chosen based on firing weights (probabilities) assigned to each of the transitions. Timed transitions fire after a random exponentially distributed firing delay. The firing of immediate transitions always has priority over that of timed transitions. GSPNs can be formally defined as [26, 32]: Definition 3 A generalized Stochastic PN (GSPN) is a 4-tuple GSP N = (P N, T 1 , T 2 , W ) where: 1. P N = (P, T, I − , I + , M 0 ) is the underlying ordinary PN, 2. T 1 ⊆ T is the set of timed transitions, T 1 ≠ ∅, 3. T 2 ⊂ T is the set of immediate transitions, T 1 ∩ T 2 = ∅, T 1 ∪ T 2 = T , 4. W = (w 1 , ..., w |T | ) is an array whose entry w i ∈ R + is a rate of a negative exponential distribution specifying the firing delay, if t i ∈ T 1 or is a firing weight specifying the relative firing frequency, if t i ∈ T 2 . Combining definitions 2 and 3 leads to Colored GSPNs (CGSPNs) [26]: Definition 4 A colored GSPN (CGSPN) is a 4-tuple CGSP N = (CP N, T 1 , T 2 , W ) where: 1. CP N = (P, T, C, I − , I + , M 0 ) is the underlying CPN, 2. T 1 ⊆ T is the set of timed transitions, T 1 ≠ ∅, 3. T 2 ⊂ T is the set of immediate transitions, T 1 ∩ T 2 = ∅, T 1 ∪ T 2 = T , 4. W = (w 1 , ..., w |T | ) is an array with w i ∈ [C(t i ) ↦→ R + ] such that ∀c ∈ C(t i ) : w i (c) ∈ R + is a rate of a negative exponential distribution specifying the firing delay due to color c, if t i ∈ T 1 or is a firing weight specifying the relative firing frequency due to c, if t i ∈ T 2 . CGSPNs have proven to be a very powerful modeling formalism. However, they do not provide any means for direct representation of queueing disciplines. To overcome this disadvantage, queueing Petri nets (QPN) were introduced based on CGSPNs with so-called queueing places. Such a queueing place consists of two components, a queue and a token depository (see Figure 2.7). The depository stores tokens which have completed their service at the queue. Only tokens stored in the depository are available for output transitions. QPNs introduce two types of queueing places: 1. Timed queueing place: The behavior of a timed queueing place is as follows: (a) A token is fired by an input transition into a queueing place. (b) The token is added to the queue according to the scheduling strategy of the queue. (c) After the token has completed its service at the queue, it is moved to the depository and available for output transitions. 2. Immediate queueing place: Immediate queueing places are used to model pure scheduling aspects. Incoming tokens are served immediately and moved to the depository. Scheduling in such places has priority over scheduling/service in timed queueing places and firing of timed transitions.
2.3. INTRODUCTION TO QUEUEING PETRI NETS 23 Queue Depository Nested QPN Transition Token o o o o o o o o o Ordinary Place Queueing Place Subnet Place Figure 2.7: QPN Notation Apart from this, QPNs behaves similar to CGSPN. Formally QPNs are defined as follows: Definition 5 A Queueing PN (QPN) is an 8-tuple QP N = (P, T, C, I − , I + , M 0 , Q, W ) where: 1. CP N = (P, T, C, I − , I + , M 0 ) is the underlying Colored PN 2. Q = ( ˜Q 1 , ˜Q 2 , (q 1 , ..., q |P | )) where • ˜Q 1 ⊆ P is the set of timed queueing places, • ˜Q 2 ⊆ P is the set of immediate queueing places, ˜Q1 ∩ ˜Q 2 = ∅ and • q i denotes the description of a queue taking all colors of C(p i ) into consideration, if p i is a queueing place or equals the keyword ‘null’, if p i is an ordinary place. 3. W = ( ˜W 1 , ˜W2 , (w 1 , ..., w |T | )) where • ˜W 1 ⊆ T is the set of timed transitions, • ˜W 2 ⊆ T is the set of immediate transitions, ˜W1 ∩ ˜W 2 = ∅, ˜W1 ∪ ˜W 2 = T and • w i ∈ [C(t i ) ↦−→ R + ] such that ∀c ∈ C(t i ) : w i (c) ∈ R + is interpreted as a rate of a negative exponential distribution specifying the firing delay due to color c, if t i ∈ ˜W 1 or a firing weight specifying the relative firing frequency due to color c, if t i ∈ ˜W 2 . Example 1 (QPN [26]) Figure 2.8 shows an example of a QPN model of a central server system with memory constraints based on [26]. Place p 2 represents several terminals, where users start jobs (modeled with tokens of color ‘o’) after a certain thinking time. These jobs request service at the CPU (represented by a G/C/1/PS queue, where C stands for Coxian distribution) and two disk subsystems (represented by G/C/1/FCFS queues). To enter the system each job has to allocate a certain amount of memory. The amount of memory needed by each job is assumed to be the same, which is represented by a token of color ‘m’ on place p 1 . According to Definition 5, we have the following: QP N = (P, T, C, I − , I + , M 0 , Q, W ) where • CP N = (P, T, C, I − , I + , M 0 ) is the underlying Colored PN as depicted in Figure 2.8, • Q = ( ˜Q 1 , ˜Q 2 , (null, G/C/∞/IS, G/C/1/PS, null, G/C/1/FCFS, G/C/1/FCFS)), ˜Q 1 = {p 2 , p 3 , p 5 , p 6 }, ˜Q 2 = ∅, • W = ( ˜W 1 , ˜W2 , (w 1 , ..., w |T | )), where ˜W 1 = ∅, ˜W2 = T and ∀c ∈ C(t i ) : w i (c) := 1, so that all transition firings are equally likely.
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