On the relations between BCMP Queueing Networks and Product Form Solution. Stochastic Petri Nets. G. Balbo. [1]. , S. C. Bruell. [2]. , M. Sereno. [1]. [1].
On the relations between BCMP Queueing Networks and Product Form Solution Stochastic Petri Nets G. Balbo , S. C. Bruell , M. Sereno Dipartimento di Informatica, Universit`a di Torino, 10149 Torino, Italy Computer Science Department, University of Iowa, Iowa City, USA
Abstract In this paper we show that multi-class BCMP queueing networks can be represented by means of Product Form Solution Stochastic Petri Nets (PF-SPNs). Since the first time PF-SPNs were proposed, one of their main drawbacks concerned the fact that there was no clear relation between PFSPNs and PF-QNs, in particular the well-known multi-class BCMP-QNs. It is important to note that the existence of the product form solution is not a property of the formalism, but of the underlying Markovian process. Hence, the fact that PF-SPNs (and also PF-GSPNs) cannot directly represent BCMP-QNs is one of the weaknesses of the SPN productform results. The results presented in this paper overcome this weakness. In particular, we show that starting from a detailed GSPN representation of each of the BCMP station types, it is possible to derive equivalent compact PF-SPN representations for each station type.
1 Introduction Stochastic Petri Nets (SPNs) are a powerful tool for modeling and evaluating the performance of systems involving concurrency, nondeterminism, and synchronization. They are equivalent to continuous-time Markov chains [1] and their steady-state analysis can thus be expressed as the solution of a linear system of equilibrium equations, one for each possible marking in the corresponding state space. The major problem in the computation of performance measures for SPNs is that the size of their reachability set increases exponentially both with the number of tokens in the initial marking and with the number of places in the net. In trying to overcome this problem, a class of SPNs has been discovered [12, 13] that is characterized by the fact that the stationary probability distribution of these nets can be factored into a product of terms, one term per place in the net. Nets possessing this property are called ProductForm Stochastic Petri Nets (PF-SPNs) and are easily iden-
tified by the structural criteria proposed by Henderson et al [9, 12, 13]. In a previous paper we showed that a class of Generalized Stochastic Petri Nets (GSPNs) also possess a productform solution [3]. We started with a GSPN model that obeys the same structural criteria used to identify PF-SPNs. By introducing some additional restrictions and by employing a series of transformation steps, that convert the original GSPN into an equivalent (intermediate) GSPN and then to an equivalent PF-SPN, we established our desired objective. In this paper we show that another class of GSPNs also possess a product-form solution despite the fact that they do not obey the structural criteria used to identify PF-SPNs. The GSPNs of this new class are those that represent multiclass BCMP queueing networks [5]. It is clearly not surprising that this well-established class of PF-QNs should be representable by appropriately constructed GSPN models. However, it is not readily apparent whether they can be represented as PF-SPNs that meet the structural conditions of Henderson et al. The thrust of our paper will be to show how to convert detailed GSPN representations of PF-QNs into PF-GSPNs. The resulting PF-GSPNs will be remarkably compact. To better appreciate the results presented in this paper, we must stress two key motivating concepts.
¯ In the performance literature the comparison between different formalisms is a classical theme. In particular, many papers have been devoted to the comparison between SPNs (and their variations) and QNs. These papers describe means of mapping one formalism into the other (see, for instance, [4, 19] and [2]). In these papers, multi-class QNs (or BCMP-QNs) are often represented by using colored SPNs, i.e., the QN classes are represented by the colored Petri net classes, but little attention is devoted to explicit description of the queueing discipline that is, instead easily addressed with a detailed representation of each BCMP-QN station. ¯ Since the first time PF-SPNs ([12, 14]) were proposed, one of the common (and probably obvious) questions
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concerned the relation between PF-SPNs and PF-QNs, in particular the well-known BCMP-QNs. It is important to note that PF is not a property of the formalism, but of the underlying Markovian process. Hence, the fact that PF-SPNs (and also PF-GSPNs) could not represent multi-class BCMP-QNs is conceivable.1 The results in this paper definitely remedy the latter point. In particular, we show that starting from a detailed GSPN representation of each of the BCMP station types, it is possible to derive equivalent, remarkably compact PF-SPN representations for the same structures. The equivalent compact SPN representation is obtained by employing the property presented for PF-QNs in [16]. In a paper that follows much the same themes as ours, Bause and Buchholz have shown that PF-SPNs and PF-QNs can be incorporated into the general framework of Queueing Petri Nets (QPNs) [6]. They were then able to prove that QPNs that combined PF-QNs and PF-SPNs also possess a product-form solution. Where our work differs is that we provide explicit GSPN net-level representations for each of the BCMP station types; from these we derive their compact equivalents; and then we show that they can be embedded into PF-GSPN models. The balance of this paper is outlined as follows. Section 2 briefly reviews pertinent notation and definitions used in describing Generalized Stochastic Petri Nets. Section 3 reviews the structural conditions that are required for a Stochastic Petri Net to have a product-form solution. Section 4 quickly summarizes the class of product-form multiclass BCMP networks. We show in Section 5 the detailed GSPN representation of each BCMP station type when the queueing network has only one class of customers. This section also describes equivalent GSPN structures that can be used to replace the detailed representations. A simple example completes the section. Section 6 introduces our detailed representation of each of the multi-class BCMP station types and our compact equivalents. The important point is that the compact equivalent representation obeys the structural conditions of PF-GSPNs. We can thus use these structures as building blocks in the construction of PF-GSPN representations of multi-class PF-QNs. An example will illustrate the technique. Finally, Section 7 provides some concluding remarks.
2 Definitions and Notations In this section we review the basic concepts and notation that we use throughout this paper. More comprehensive presentations of Petri net concepts can be found, for instance, 1 BCMP-QNs can obviously be represented by GSPNs, but the explicit representations of the different queueing policies and service time distributions result into SPNs - or GSPNs - that do not satisfy the structural criteria for PF-SPNs - or PF-GSPNs.
in [17, 18]. A presentation of concepts related to Stochastic Petri Nets can be found in [1, 4]. A marked stochastic Petri net can be defined as a -tuple ¼ , where and are disjoint sets of places and transitions, is the input function, is the output function, and is the inhibition function. A net is said to be ordinary if , , and , i.e., all the arcs have is the set of firing rates for the expoweight equal to . nentially distributed transition firing times, and ¼ is the initial marking. , its pre-set, post-set, and inFor a given transition hibition set are given by , , re , and Æ spectively. In the same manner we can define the preset and postset of a given place. For any transition , using the weighted flow relation, we can then define the input vec tor ½ ¾ , the out put vector ½ ¾ , and the inhibition vector ½ ¾ . From the weighted flow relation, we can with entries also define the incidence matrix is enabled in a marking . A transition iff , and . Being enabled, may , occur (or fire) yielding a new marking ¼
and this is denoted by . The set of all the mark
¼
ings reachable from ¼ is called the reachability set, and is denoted by RS ¼ . A T-semiflow is a vector of nonnegative integers such ¼. A P-semiflow is a vector of nonnegative inthat tegers such that ¼. The set (respectively, ) is called the support of (respectively, the support of ). A semiflow (Tor P-semiflow) is called minimal if no semiflow ¼ exists whose support is a proper subset of the support of . When all the places are covered by -semiflows the net is said to be bounded, i.e., for any initial marking ¼ the , reachability set is finite. In this case, for any place it is possible to derive a bound for the number of tokens that it may contain (see [10] for details).
Brief Review of GSPN Models GSPNs are obtained by introducing a temporal specification in the class of Petri nets with priorities. Time is associated with transitions and two types of temporal specifications are possible. For transitions with priority zero, delays are exponentially distributed random variables; such transitions are consequently referred to as timed. For transitions with priority , delays are deterministically zero; such transitions are referred to as nimmediate. In this paper we only consider two priority lev-
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els; hence, transitions are simply referred to as timed and immediate. Formally, a GSPN is an -tuple:
È Ì ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¼
(1)
where the first five components of the -tuple are the same as those defined for the SPNs. is a function that specifies the priority levels associated with the transitions of the net; is a function that specifies the stochastic component of a GSPN model. In particular, it maps transitions into real positive numbers. The quantity is called the “rate” of transition if is timed, and the “weight” of transition if is immediate. The initial marking of the GSPN is denoted by ¼ . Finally, a marked place is a place that contains (at least) one token drawn as a black dot. A distribution of tokens over the places of the net identifies one possible marking. A transition is enabled in a given marking iff , if , and if no other transition with priority higher than that of exists such that and 2 . Markings that only enable timed transitions are said to be tangible, whereas markings that enable at least one immediate transition are said to be vanishing. When a vanishing marking is entered, the weights of the enabled immediate transitions are used to probabilistically select the (immediate) transition to fire. The time spent in any vanishing marking is deterministically equal to zero. When a tangible marking is entered, the rates of the transitions are used to probabilistically select one timed transition to fire. From the values of the transition rates (weights) it is possible to compute the probability that a given enabled timed (immediate) transition, say , fires in a tangible (vanishing) marking :
p
(2)
enabled in
3 Product-Form Results for SPNs In this section we review the basic concepts of the class of Stochastic Petri Nets that have a Product-Form Solution. The PFS for SPN criterion considered here is that proposed by Coleman, Henderson, Lucic, and Taylor; more comprehensive presentations of the results related to this topic can be found in the references [7, 9, 12, 13]. The key for identifying those SPNs with a PFS is to consider the input and output bags of the transitions to be states 2 In Petri nets with priorities the concept of concession has been introduced (e.g., see [1]) to capture the fact that tokens in the input and inhibitor sets of a transition satisfy the usual firing conditions, while a transition is enabled only if it has concession and no other transitions of higher priorities have concession in the same marking.
of a Markov Chain. This Markov Chain has been called the routing process [12]. We now review the basic definitions underlying the PF-SPN criteria. denote the minimal -semiflows Let ½ ¾ found from the incidence matrix. The following definitions and assumptions are essential to the analysis that will be briefly presented in this section.
Definition 1 (Closed set of transitions) (From [7]) For ¼ ¼ , let be the set of input and output bags ¼ for transitions in ; formally, ¼ is represented by the following expression:
¼
¾Ì ¼
The subset of transitions ¼ is said to be closed if, for any ¼ ¼
, there exist such that and ; i.e., each output bag is also an input bag for some transition in ¼ , and vice-versa each input bag is also an output bag.
When Definition 1 applies to the support of a -semiflow ( ), we can state that such a -semiflow is a closed support -semiflow. Definition 1 represents a criterion for identifying the closed support -semiflows of an SPN.
Definition 2 (Structural Constraints) (From [7]) An SPN ¼ is said to be closed iff
such that , there exists a minimal -semiflow and is a closed set of transitions.
3.1 Routing Process and Closed T-semiflows Definition 2 says that an SPN is closed if all its transitions are covered by closed support minimal -semiflows. Among the minimal closed support T-semiflows we can identify a relation that can be used to derive the PFS. In the following we denote by cd the set of closed support minimal T-semiflows of a net. Definition 3 [Freely related T-semiflows] Let ¼ be a closed SPN and ¼ , ¼¼ be two different minimal closed support T-semiflows of ¼ . and ¼¼ are said to be freely related, denoted as ¼ ¼¼ , if there exist ¼ ¼ and ¼¼ ¼¼ such that ¼ ¼¼ . The relation £ is the transitive closure of .
It is easy to see that the relation £ yields a partitioning of the set of minimal closed support -semiflows of into equivalence classes that we denote by .3 Since (by definition) any transition cannot be part of the supports
3
Ü
.
Ü
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Ü
is a min. cl. support
-semiflow and
Ü Ü
The term is an element of a decomposable transition probability matrix which represents the probability that if transition fires in a given marking , the next marking is ¼ , where transition becomes enabled and is defined in the following manner
if if
if
and
(3)
and
¾Ì
p5
t1
t6
p2
p6
t2
t4
t7
p3
p4
p7
t3
t5
t8 (a)
The global balance equations for the routing process
p1
are
i(t 1) i(t 2)=i(t 4) i(t 3) i(t 5) i(t 6) i(t 7) i(t 8) a=
i(t 8)
i(t 7)
i(t 3)
Using Definition 3 we can denote the routing process as a Markov chain whose state space and whose transition rates are , with
i(t 5)
is a min. cl.
and
i(t 2)=i(t4)
-sem.
i(t 3)
¾
exists only when the transition rates of the SPN are related in specific manners. That is, there are “pathological” cases where an SPN has a PFS only for some values of its transition rates. The results presented in [11] define a sub-class of PFSPNs that admit PFS, whatever the rates of the transitions of the SPN. For this sub-class of PF-SPNs Definition 2 is a necessary and sufficient condition for the existence of the PFS. In this manner when Definition 2 holds, and the SPN satisfies the additional conditions defined in [11] the SPN has a PFS, whatever the rates of its transition rates. On the other hand, if the SPN does not satysfy the conditions defined in [11], we can use the results of [9] to check if the SPN is one that has a PFS only for some values of its transition rates.
1 0 0 0 0 0
0 a 0 0 0 0
0 β 0 0 0 0
0 0 0 0 0 0
0 0 0 0 1 0
0 0 0 0 0 1
0
0
1
0
0
i(t 1)
of closed -semiflows that belong to different members of the partition , then the relation induces a partition among the transitions of the SPN. We denote by the corresponding equivalence classes, i.e.,
0 0 1 1 0 0 0
0
µ2 µ2+µ4
β=
µ4 µ2+µ4
(b)
Figure 1. An example of PF-SPN (a) and its routing probability matrix (b)
(4)
which can be interpreted as the traffic equations for the SPN and which can be observed to be partitioned into separate systems of linear equations, one for each class. Boucherie and Sereno proved that a necessary and sufficient condition for an SPN to have a solution for the traffic equations is that Definition 2 holds [7]. This is the first step in showing that such closed SPNs possess a productform solution. In the analysis of PF-SPNs the routing process plays a crucial role. This stochastic process yields the solution of the traffic equations. The result proven in [7] establishes a relation between the routing process and the structure of the T-semiflows of the SPN. In this manner the existence of a solution for the traffic equations of an SPN can be checked by using structural information, i.e., the Tsemiflows. The existence of a solution for the traffic equations is not a sufficient condition to assert a PFS for the SPN. The results presented in [9] state the there are cases where the PFS
Example 1 The SPN of Figure 1(a) allows us to illustrate some of the peculiarities of the PF-SPN analysis. In this SPN there are three minimal -semiflows with the following supports: ½ , ¾ , and ¿ . We can see that all these -semiflows satisfy Definition 1 and since there exists a minimal closed support -semiflow that covers : the SPN satisfies Definition 2 and is thus closed. We can see that there are two classes: ½ ½ ¾ , and ¿ ¿ . The classes yield a partition of the transitions, i.e., , and . Figure 1(b) shows the transition probability matrix for the routing process, where the elements are defined according to Equation (3). It can be observed that the routing matrix is partitioned into two independent submatrices yielding a partition of the traffic equations into two separate systems of linear equations.
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4 Brief Review of BCMP Queueing Networks In their classic paper, Baskett, Chandy, Muntz, and Palacios developed a broad class of queueing networks that possess a product-form solution [5]. The queueing netstaworks they considered contain a finite number of tions. There are four different types of stations depending on the queueing (or service) discipline. The permissible service disciplines are first come first served (FCFS), processor sharing (PS), infinite servers (IS), and last come first served preemptive-resume (LCFS). All customers have the same exponential service time distribution at a FCFS center. The model permits the definition of multiple classes of customers whose characteristics (service rates and transition frequencies) may be distinct. Let there be distinct . Each class of customer may classes numbered have a distinct service time distribution at the other types of stations. The only restriction is that the service time distribution must have a rational Laplace transform. A PS station contains a single server which is simultaneously shared by customers. The net effect is that when there are customers at the station, each is receiving service at a rate of . For an IS station each customer receives its own server immediately. That is, there are always at least as many servers as customers. Hence, no queueing delays arise at such a station. Each customer class has defined for it a separate set of frequencies governing the movement of customers of that class between stations. More specifically, a customer of class which completes service at station will next require service from station with a certain transition frequency denoted by . In a closed queueing network the total number of customers remains fixed at some number . If we let denote the number of class customers, then
A state, , in such a model is a feasible distribution of customers over classes and stations. More formally, the , model is said to be in state where cus , when there are tomers of class at service center . The must satisfy four constraints for the state to be feasible (or meaningful). First, the number of customers of each class at each station and must be nonnegative (i.e., , for ). Second, only if it is possible (under the transition rules of the network) for customers of class to be at station . Third, the total number of customers in a state must be equal to , i.e.,
The
equilibrium
where
for
probability
of
is:
the
state
if station is load dependent and has a FCFS, PS, or LCFS service discipline
if station is load independent and has a FCFS, PS, or LCFS
service discipline
if station has an IS service discipline.
customer at denotes the mean service rate for a class station . At a FCFS station though, all customers must have the same mean service rate, i.e., . The are the solution of the sets of linear equations:
and
Each of the sets contains linear equations. These equations balance the flows into and the flows out of each station by each of the customer classes. denotes the total number of customers of any Also, class at service center . That is,
customers at all
Fourth, the sum of the number of class the stations must be , i.e.,
is a normalizing constant chosen so that all the sum to one.
5 GSPN Representation BCMP Networks
of
Single-Class
Ideally, we would like to have a method that allows us to translate in a natural manner a queueing network into a Petri net preserving the properties of the original model and obtaining a detailed analysis. representation that is suitable for the preliminary Petri net structural analysis. The approach
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could be that of having a catalogue of detailed representations for the service distributions and queueing disciplines for each possible queueing node (component of the queueing network) from which we extract submodels that can be easily composed to derive the desired representation of the queueing system. A model constructed in this manner could be preprocessed by a Product Form Analyzer (PFA) that, using all the criteria proposed in the literature, would decide whether the product form solution can be computed. If the model does not satisfy the Product Form Criteria, a standard solution technique, based on the state space generation, could be used. This analysis process is depicted in Figure 2. Obviously we would like to make sure that if all the queueing nodes of the model satisfy the BCMP theorem, the Product Form Solution is computed; on the other hand, if queueing nodes are composed with components that do not satisfy the BCMP theorem criteria, the standard solution technique must be employed.
tions show that FCFS, LCFS and Random Order queueing disciplines are equivalent if first order performance measures (i.e., throughput, average queue length, average waiting time) are the only analysis results we are intersted in. Using this approach we can handle relatively complex cases similar to that discussed in the next example. Service Discipline
Detailed GSPN Representation
FCFS SS semantics
LCFS SS semantics
PS Catalogue BCMP stations PN subnets
Model Builder
Product Form Analyzer
YES
Compact GSPN Representation
µ(n)=(1/n) µ SS semantics
IS semantics
Normalization Constant Mean Value Analysys
IS NO
IS semantics IS semantics
State Space Based Analysis
Figure 2. A Product Form Analyzer Architecture A straighforward representation of the four service center types permitted in BCMP networks is depicted in Figure 34 The inhibitor arcs used to represent the FCFS and LCFS queueing disciplines can be easily removed by using the complementary place approach that is quite simple in this specific case. An important point that needs to be observed is that, even in the simplest case of a closed queueing network with only three customers and two nodes, one of which has either a FCFS or LCFS queueing discipline, the resulting GSPN (that we know by construction to possess a Product Form Solution) would be refused by our hypothetical PFA because the input/output bag rule discussed in [7, 12], is not satisified. This shows that there are cases in which the product form solution depends upon behavioural properties that cannot be captured by structural analysis techniques and that additional considerations are needed. In the case of single-class models the compact representations depicted in Figure 3 are easy to derive and to justify by means of simple considerations that are based on the results presented in [16]. In particular these reduc4 For the sake of simplicity, in this paper we do not explicitly address the problem of representing the non-exponential distributions allowed by the BCMP theorem; if needed we could use GSPNs representations of Phase Type distributions as suggested, for instance, in [1].
Figure 3. Detailed and Compact GSPN Representations of the Four BCMP Service Center Types (SS = Single Server and IS = Infinite Server)
Example 2 Table 1 contains the description of a five station . Each of the single-class queueing network with four BCMP service center types is employed. In addition, a load-dependent FCFS station is included. The compact GSPN model for this PF-QN is shown in Figure 4. The immediate transitions can be easily removed by fusing them onto timed transitions as described in [3], thereby obtaining a PF-SPN. Note that this model obeys the structural conditions of Henderson et al. We evaluated the QN model using Supernet [8] and the compact GSPN using DSPNExpress [15].
6 GSPN Representation BCMP Networks
of
Multi-Class
Following the idea presented for the single-class case, Queueing Network Nodes (QNN) of multiple-class cases could be easily represented by using colored Petri nets (where each customer class is encoded by a different color) or by deriving a detailed structure where the different
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Service Discipline
station 4 0.3
Detailed GSPN Representation
µ1
Compact GSPN Representation
station 2 station 1
µ1 = µ2
FCFS
0.5
µ2 station 3
0.7
µ1
0.5
µ1
µ1
µ1
station 5
p1
LCFS class 1
µ2
Figure 4. Compact GSPN Representation of a Single-Class BCMP Network
µ2
p1
µ2
µ(.)=(1/pt) µ1
SS semantics (m.d. rate)
pt
µ2 class 2
p2 SS semantics (m.d. rate)
St. 1
Ser.D. FCFS
2
LCFS
3 4
PS FCFS
5
IS
Serv.Rate
, ,
PS
IS semantics
pt
´´½µµ
´ ¾µ
´ µ
½ ¾
p2 µ(.)=(1/p ) µ t 2
IS semantics
IS semantics
IS
Table 1. Parameters of Single-Class BCMP QN with
classes are encoded by different sequences (paths) of transitions. Accordingly, Figure 5 contains the detailed representation of each of the four service disciplines. Note that the figure illustrates only two classes (Ê ) visiting a station with a maximum queue length of four total customers. Clearly, more classes and larger queue lengths can be accommodated using simple extensions of the patterns embodied in the figure. The two methods are very much related, since the second approach corresponds to an explicit unfolding of the colored representation of the first; using either method, these models are completely defined only after the specification of the rates of the different timed transitions involved. As one could easily expect, the multiple-class models obtained by composing these blocks will be “refused” most of the time by the PFA (even if the model does possess a Product Form Solution) because the structural criteria of [7, 12] are not satisfied. Also in this case the solution can be found by deriving a compact representation for each of the service disciplines. Figure 4 contains on its right side these compact versions in which the marking dependent firing times of the timed transitions are defined using formulas that are proved in the Appendix. Note that the general compact structure incorporates a place that contains the total number of customers at the station independent of class. In addition, one place and transition are used for
IS semantics
IS semantics
Figure 5. Detailed and Compact GSPN Representations of the Four BCMP Service Center Types
each class. The place contains the number of customers of the respective class. Therefore, to extend our compact representation to handle, say, four classes, we would use the structure shown in Figure 6. It is interesting to note that the proofs of the equivalence between detailed and compact representations are based on the results developed in [16], where multiclass FCFS servers must have the same mean service time for the customers of different classes, while in the case of LCFS and PS queueing disciplines different per-class mean service times are allowed. This “subtlety” is lost in the compact representation that has the same form for the three queueing disciplines of before. This means that if different firing rates are specified for a block that is intended to represent a multi-class FCFS service center, incorrect performance figures are produced without having the possibility to structurally prevent them from being computed. Example 3 Table 2 contains the description of the same QN that we used in the previous section, but with one addi). We set the number of tional customer class (i.e., Ê
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p1
T1
p1
p4
p7
class 1
p2 T 2 class 2
t1
t4
t7
p2
p5
p8
t5
t8
p3
p6
p9
t3
t6
t9
t2
T3
pt
class 3
p4
pt
T4 class 4
p4
Figure 8. A PF-SPN that represents a nonBCMP type network
Figure 6. Representation of a Four-Class BCMP Station
customers in class 1 to be
½ and for class 2, ¾ . station 4 0.3
station 2
0.5
0.7
station 1 0.7 0.5 0.3 0.7
0.3
station 3
station 5
Figure 7. Compact GSPN Representation of a Multi-Class BCMP Network
The compact GSPN model for the PF-QN is shown in Figure 7. Table 2 also describes the throughput and mean queue lengths for the two classes at each of the five stations by class. As before, these performance indices were obtained from SuperNet and DSPNExpress.
7 Conclusions In this paper we have shown how to represent multi-class BCMP QNs by remarkably compact PF-GSPNs. We have developed compact building block structures for each of the four BCMP station types. For each station type, our compact representation incorporates one place and transition for each of the customer casses visiting the station. An additional place is used to contain the total number of customers independent of class5 . For FCFS, LCFS, and PS stations, the marking-dependent firing rate of the transition used to specify the completion rate of a class customer at station is (with the provision that at a FCFS station ½ ¾ ). Each transition uses the single-server semantics. To represent an IS server station, we do not use any marking-dependent rates and each of transitions uses infinite server semantics. Although we based our results on closed PF-QNs, the extension of our technique to open and mixed BCMP networks is staightforward (although for open networks the detailed representation of a queue is infeasible). Furthermore, the incorporation of customers switching between classes should be accommodated using the notion of freely-related -semiflows described in Section 3. In fact, it should be noted that PF-SPN models of non standard Product Form Queueing Networks can be constructed using the combinations of the compact building blocks developed in this paper and special structures that have become common in the discussion of PF-SPNs.6 An example of this possibility 5 We like to explicitly include this place in the model because, even if “implicit” (i.e., its marking can always be derived from the marking of other places), it emphasizes the fact that these transitions, one per class of the model, by having a part of their input bags in common, are not independent, but represent “replicas” of the same server. 6 Compact building blocks and special structures can be considered as elements of the catalogue that we briefly mentioned in Section 5 that can be intuitively composed in very general manners, but for which specific composition rules and composition languages should be defined.
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Station 1
Serv.Disc. FCFS
2
Service Rate
LCFS
3
PS
4
FCFS
5
IS
, ,
,
, ,
Table 2. Parameters of Multi-Class BCMP QN with is depicted in Figure 8. This PF-SPN represents a multiclass BCMP networks with two classes and three stations in each class (the left part of the SPN). The subnet composed of places and transitions represents the multiclass FCFS station. The non-BCMP feauture is the possibility of having a weak form of synchronization between customers of one of the two classes with some external resources represented by tokens in place . That is, when a customer of the class , , , is waiting for its service at station 4 (token in place ), if there is an available token in place the customer can choose (with a probability that is given by the ratio ) to go to station 5 (token in place ) or it can choose (with a probability ) the synchronization with the token in place and then in this case the path is . With the firing of one token moves in place (the one that represents the customers) and one moves in . Future work on this topic may follows different directions. Generalizations to load dependent service rates are obvious extensions of the results presented in this paper, that should be relatively easy to obtain. The negative results showing that detailed representations of queueing disciplines and non-exponential service time distributions yield non Product Form GSPN models equivalent to BCMP-Queueing Networks, suggest that the problem of automatically recognizing when a GSPN models a Product Form Queueing network is difficult to solve. It seems that to obtain this type of result a different concept of Product Form GSPN needs to be derived where the factorisation of the solution is not related to specific places but to groups of places. Obviously, the construction of models that exhibit nice solution properties by composition of well behaving submodels is an easier task that requires, however a precise definition of the composition rules and of the interfaces that such components must present with respect to the rest of the model.
and
A Appendix Our objective is to show that the detailed representation for a FCFS multi-class queue is equivalent to the proposed compact representation (e.g., Figure 5). We will closely follow the technique developed by Muntz in [16]. Hence, consider a FCFS station with Poisson arrival streams with a customer of class arriving at rate . Let the vector denote the state of the FCFS station where each of the is a customer class and also represents the class of the customer in the -th position of the queue. Using the property we obtain
We can verify the solution
where denotes the total number of customers of class in the station, by substitution into the global balance equations:
The result of the substitution provides the equality
which implies that the global balance equations are satisfied iff all the are equal. This also implies that all the states with the same number of each class of customers has the same equilibrium probability. Let be the set of states with customers of class at the FCFS station. Then where . Hence,
Proceedings of the 10th International Workshop on Petri Nets and Performance Models (PNPM’03) 1063-6714/03 $17.00 © 2003 IEEE
¾
Now, using the same approach for the compact model we can observe that the property in this case requires the following balance equation to hold:
(5)
From the BCMP theorem we can assume that
This allows to write
The latter equation when substituted in Equation (5) proves that in order for the local balance property to hold, the marking dependent transition rate must be specified in the following way
Following a similar line of analysis it is possible to prove that the property derived for the LCFS detailed model does not require that customers of different classes be served with the same rate (for the detailed model the local balance equation similar to [16] involves, for any arbitray state, always a generic arrival rate and the corresponding rate ). Because of this result, in order to sastisfy the balance equation of the compact model, the generic firing rate ´ · ½µ must have the following form
References [1] M. Ajmone Marsan, G. Balbo, G. Conte, S. Donatelli, and G. Franceschinis. Modelling with Generalized Stochastic Petri Nets. John Wiley & Sons, 1995. [2] G. Balbo, S. C. Bruell, and S. Ghanta. Combining queueing network and generalized stochastic Petri nets for the solution of complex models of system behavior. IEEE Transactions on Computers, 37(10):1251–1268, October 1988.
[3] G. Balbo, S. C. Bruell, and M. Sereno. Product Form Soultion for Generalized Stochastic Petri Nets. IEEE Transactions on Software Engineering, 28(10):915–932, 2002. [4] G. Balbo and M. Silva, editors. Performance Models for Discrete Event Systems with Synchronisation: Formalisms and Analysis Techniques. KRONOS, Zaragoza, Spain, 1998. Book of the MATCH Advanced School. [5] F. Baskett, K. M. Chandy, R. R. Muntz, and F. Palacios. Open, closed and mixed networks of queues with different classes of customers. Journal of the ACM, 22(2):248–260, April 1975. [6] F. Bause and P. Buchholz. Queueing Petri Nets with Product Form Solution. Performance Evaluation, 32:265–299, 1998. [7] R. J. Boucherie and M. Sereno. On closed support tinvariants and traffic equations. Journal of Applied Probability, (35):473–481, 1998. [8] S. Bruell, G. Balbo, S. Ghanta, and P. Afshari. A mean value analysis based package for the solution of productform queueing network models. In Proc. Intern. Conf. on Modelling Techniques and Tools for Performance Analysis, Paris, May 1984. [9] J. L. Coleman, W. Henderson, and P. G. Taylor. Product form equilibrium distributions and an algorithm for classes of batch movement queueing networks and stochastic Petri nets. Performance Evaluation, 26(3):159–180, Sept. 1996. [10] J. M. Colom, E. Teruel, and M. Silva. Logical properties of P/T systems and their analysis. In Performance Models for Discrete Event Systems with Synchronisati ons: Formalisms and Analysis Techniques, pages 185 – 232. KRONOS, Zaragoza, Spain, 1998. Book of the MATCH Advanced School. [11] S. Haddad, P. Moreaux, M. Sereno, and M. Silva. Structural characterization and qualitative properties of Product Form Stochastic Petri Nets. In 22-th Intern. Conf. on Appl. and Theory of Petri nets, Newcastle, UK, June 2001. Spinger Verlag, LNCS 2075. [12] W. Henderson, D. Lucic, and P. Taylor. A net level performance analysis of stochastic Petri nets. Journal of Australian Mathematical Soc. Ser. B, 31:176–187, 1989. [13] W. Henderson and P. Taylor. Embedded processes in stochastic Petri nets. IEEE Transactions on Software Engineering, 17:108–116, Feb. 1991. [14] A. A. Lazar and T. G. Robertazzi. Markovian Petri net protocols with product form solution. Performance Evaluation, 12:67–77, 1991. [15] C. Lindemann. Performance Modelling with Deterministic and Stochastic Petri Nets. John Wiley & Sons, 1999. [16] R. R. Muntz. Poisson Departure Processes and Queueing Networks. Technical Report RC-4145, IBM Research Report, IBM T. J. Watson Research Center, 1972. [17] T. Murata. Petri nets: properties, analysis, and applications. Proceedings of the IEEE, 77(4):541–580, April 1989. [18] M. Silva. Las Redes de Petri en la Automatica y la Informatica. Ed. AC, Madrid, Spain, 1985. In Spanish. [19] M. Vernon, J. Zahorjan, and E. D. Lazowska. A Comparison of Performance Petri Nets and Queueing Network Models. In Proc. Intern. Workshop on Modelling Techniques and Performance Evaluation, pages 181–192, Paris, France, March 1987. AFCET.
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