From Bcmp Queueing Networks to Generalized Stochastic Petri Nets: an Algorithm and an Equivalence Definition

In this paper we define an algorithm that converts a BCMP queueing network (QN) with multiple classes of customers into a generalized stochastic Petri net (GSPN). Product-form property of BCMP networks is preserved by the GSPN model. The algorithm can be embedded in a hybrid formalisms modelling tool. In fact a product-form model can be partially expressed in terms of queueing network and partially in terms of GSPN. Then the algorithm determines an equivalent GSPN representation of the whole system, which can be eventually analyzed using exact techniques (taking advantages of the product-form property) or by GSPN simulators. It is worthwhile pointing out that multiple classes QNs are considered; hence their GSPN representation is not trivial, as queueing disciplines must be represented. INTRODUCTION In this paper we consider different formalisms to model systems for performance evaluation purposes. We consider the integration of different types of models into a unique framework, in order to take advantage of the relative merit of each of the modelling formalism involved. To this aim we consider the relation between the class of queueing network models and the generalized stochastic Petri nets. Specifically, we propose an algorithm to transform a class of queueing networks into the corresponding stochastic Petri net model. Queueing networks (QNs) are useful stochastic models for the performance evaluation of systems that consist of a set of customers which compete for a set of resources. The exact analysis of such kind of models, if possible, is usually computationally unfeasible due to the high cardinality of the set of all possible system states. Some classes of QNs have been introduced which allow a closed expression for the stationary probabilities, i.e., it can be expressed as product of functions that depend on the state and on the type of each node. The main result is known as BCMP theorem (Baskett et al. 1975) and it considers open, closed and mixed QNs that consist of multiple class of customers, probabilistic routing, Poisson arrivals and four types of service centers. Generalized Stochastic Petri Nets (GSPNs) are stochastic models that represent systems with concurrency and synchronization and they are usually defined as models at a lower level of abstraction with respect to QNs. In fact the GSPN semantic is strictly defined in terms of transitions, places and rules of firing, that are the GSPN components. We formally introduce the GSPN definition in the next section. In the general case, GSPNs are hard to study because of the generation of their reachability sets which is an NP-hard problem. Product-form GSPNs are studied in (Coleman et al. 1996, Balbo et al. 2002) and they define the stationary state probability as the product of functions depending on the marking of each place. However, these GSPNs still do not overcome the problem of deciding whether a marking is reachable from the model initial state or not. As a consequence this limits the applicability of the algorithms defined for product-form GSPNs (Sereno and Balbo 1997, Coleman et al. 1996). Hence the solution of such models can be derived by simulation in many practical complex cases. In this paper we present an algorithm that transforms a multiple class BCMP queueing network into a GSPN model. The task is not obvious for at least two reasons: 1Multiple class QNs cannot be simply associated to state machines as for single class QNs. In fact theoretical results (Baskett et al. 1975, Chandy et al. 1977) have shown that the queueing discipline influences the performance measures in multiple class QNs. Hence the GSPN that models a specific multiple class queueing station depends on the correspondent queueing discipline, and the equivalence between the QN and GSPN models must be proved. 2BCMP QNs allow some classes of stations to serve the customers with Coxian distributed service time. Moreover the service time distribution can be class dependent. Also in this case state machines cannot be used, and a different GSPN model has to be defined. The proposed algorithm is based on some equivalence results presented in previous papers (Balbo et al. 2003, Balsamo and Marin 4-6 June 2007; October 23-25, 2007) that we briefly recall in the section on related works. The proposed technique has the following properties: the modelling approach is modular and hierarchical. Indeed we define some GSPN blocks corresponding to queueing stations that can be combined into more complex systems preserving the product-form property. the GSPN models correspondent to the BCMP service stations can be combined with other no-BCMP productform models. We have proved in (Balsamo and Marin 4-6 June 2007; October 23-25, 2007) that any system that holds M ⇒ M Muntz (1972) property can be combined with our BCMP-equivalent models preserving the product-form solution. The algorithm is based on the following idea: For each BCMP queueing center type it defines a correspondent equivalent GSPN model. The GSPN models obtained from the queueing center translations are connected by arcs and immediate transitions according to the QN routing probability matrix. Theoretical results proved in (Balbo et al. 2003, Balsamo and Marin 4-6 June 2007; October 23-25, 2007) define the equivalence property between the GSPN and QN models, in terms of stationary state probability and average performance indices. Even if translating a QN into a GSPN increases the model complexity and reduces its readability, there are also some relevant advantages. First, the class of GSPN models is very expressive and its semantic if formally defined at a very low level. This means that given a model definition, the state representation can be obtained automatically. In general this does not happen when studying QNs where the system state has to be derived from a high level description of the queueing discipline. Therefore the class of GSPN models represents a suitable candidate for being the base model for a hybrid modelling tool. A second reason to use GSPN is related to the former one. We observe that product-form BCMP QNs exact analyzers or simulators usually do not allow the modeler to define new queuing centers with specific discipline. For example, to the best of our knowledge, modelling tools based on QNs do not allow the modeler to introduce in the network an MSCCC service center type defined in (Le Boudec 1986) that extends the BCMP theorem. Using GSPNs, one can define a model representing MSCCC discipline and then can embed it into the net. Therefore the performance indices can be derived, possibly by the product-form analysis. MODELS IN PRODUCT FORM In this section we introduce the formalisms that we use in the following and we briefly review the main definitions of the QNs and the GSPNs. We will limit our description to BCMP queueing networks. BCMP Queueing networks A queueing network consists of a set C = {c1, . . . , cN} of N service centers or stations. Each service center has a scheduling discipline. BCMP QNs allow the following service disciplines: First Come First Served (FCFS), Last Come First Served with preemptive resume (LCFSPR), Processor Sharing (PS) and Infinite Servers (IS). In a QN, the customer enters a service center, waits in the queue for the service, gets the service, and finally it either exits the network or enters another service center. Customers moves among the service centers according to routing probabilities. At a given time, every customer belongs to a class, but there can be class switchings, i.e., a customer can change its class after being served at a station. The class of the customer influences the routing probabilities and the service time at the stations. We denote by R the number of classes. The classes are labeled by 1, . . . , r, . . . R and can be partitioned into chains. A chain permanently characterizes a customer. In order to simplify the notation we consider BCMP QNs with multiple chains but only one class for chain. Hence in this context the terms class and chain becomes synonymous. In the section on supported extension of the algorithm, we show how it is possible to model class switching. We use the following notation for QNs: p ij with 1 ≤ i, j ≤ N and 1 ≤ c ≤ R is the probability that a chain c customer goes to station j after being served by station i p i0 with 1 ≤ i ≤ N and 1 ≤ c ≤ R is the probability that a chain c customer exits the network after being served by station i. Then the normalizing condition holds, i.e., ∑N j=0 p (c) ij = 1, for 1 ≤ i ≤ N and 1 ≤ c ≤ R μ i with 1 ≤ i ≤ N and 1 ≤ c ≤ R is the mean service rate for a chain c customer at station i. If the service time is Coxian distributed and L i is the number of stages for chain c customers at station i, then μ (c) `i , 1 ≤ ` ≤ L (c) i denotes the mean service rate for a chain c customer at stage of service ` of station i. If chain c is open, i.e., external arrivals and departures from the system are allowed, then λ > 0 denotes the external arrival rate for class c customers. The arrival probability at node i and chain c is denoted by p 0i . It is defined such that ∑N i=1 p (c) 0i = 1, for 1 ≤ c ≤ R. Then λp (c) 0i is the external arrival rate of chain c customers to station i. If chain c is closed then p 0i = 0. BCMP theorem considers four types of scheduling disciplines with some constraints. FCFS stations must have exponentially distributed service time. The service time must be chain-independent, i.e., μ = μ for 1 ≤ c ≤ R. LCFSPR, PS and IS station types have less restrictive conditions. The service time can be Coxian distributed and the mean service rate can depend on the customer being served. Let n = (n1, . . . ,nN) be a vector whose components are R-dimension vector of vectors and where component n i represents the number of class r customers at station i. Then BCMP theorem (Baskett et al. 1975) states that, under stability conditions, the stationary probability distribution is given by: