Adequate Models for Multiprocessor Stochastic Systems

In this paper, two models are presented for Multiprocessor Stochastic Systems (MSS). The first one is the Generalized Semi-Markov Process (GSMP), the second is the Generalized Stochastic Petri Net (GSPN). These models are used to study performance arbitrary parameters of (MSS). Our main result is by constructing an isomorphism mapping between (MSS) model and (GSPN) model. This isomorphism captures the essential dynamical structure of (MSS).


INTRODUCTION
Markov modulated processes are very often used to model different arrival processes of Multiprocessor Stochastic System (MSS). The major problem of each Markov model is the size of the state space which grows exponentially in the system complexity. Only some algorithms could by applied to that model. The other important mathematical model used to analyze Multiprocessor Stochastic System is the Generalized Stochastic Petri Net (GSPN) model which is an abstract formal model of information flow and is used to represent concurrency, synchronization, and communication in complex system. [9] [13] [11] [6].
In this paper we divided the work as follows: In section 2 we define and analyze the Multiprocessor Stochastic System with an example, while in section 3 we define and discuss the (GSPN) model also with an example. Finally, in section 4 we construct an isomorphism from a mathematical framework of (GSMP) model for (MSS) into a mathematical framework of (GSPN) model for (MSS), and in the end of this section we give conclusions.

ANALYSIS OF MULTIPROCESSOR STOCHASTIC SYSTEMS
Now we will present a multiprocessor stochastic system with a common storage, N different processors and only one server [4]. The different job arrivals and the characterization of the system are presented using (MSS) model and Markov modulated processes. A simple system specification looks as follows: Each of the N processors, which are connected with one common storage, sends requests to the common memory using the server  If for example in a (MSS) the request generation time is assumed to be exponentially distributed as well as the server allocation time. Also, all random variables are assumed to be stochastically independent and each processor is characterized by its own arrival and service rate. In this case some type of Markov processes models are used to analyzed the stochastic behavior of the system [5]. Since no computationally treble analytical results are available for calculating the exact value of mean value of the waiting time for single or Multiprocessor Stochastic System , simulation is a natural recourse [2]. Recently in [11] they present a method to obtain performance parameters from (GSPN) model to be able to analyze the stochastic behavior of (UML) system (which is a special case of (MSS) system). They derived Continuous Time Markov Chain (CTMC) from (GSPN) and they used Markov chain theory to obtain the performance parameters. Now, we can present a mathematical framework of Markov processes that may be used to analyze the stochastic behavior of (MSS). Theorem 2.1: [10] [8] Let {X (t): t ≥ 0} be a sequence of random variables with state space S. Assume that : 1-There exist a sequence {T k : k≥ 0} of stopping times such that {T k+1 -T k : k ≥ 0} are independent and identically distributed. 2-for every sequence of times 0 < t 1 < t 2 … < t m (m >1) and k ≥ 0, the random vectors {X(t 1 ) , … , X(t m )} and {X(T k +t 1 ), … , X(T k +t m )} have the same distribution and the processes {X(t): t ≤ T k } and A Generalized Semi-Markov Process (GSMP) were introduced by Schassberger in [14]. A GSMP moves from state to state with the destination and duration of each transition depending on which of several possible events associated with the occupied state occurs first. We give a formula of the following theorem from [14] , [2] and [4].

APPLICATIONS OF GSPN MODEL
Generalized Stochastic Petri Nets (GSPN) have recently emerged as principal performance modeling tool for distributed systems such as multiprocessor, local area networks, and automated manufacturing systems. In fact, there are many types of PN models [1]. The most important useful nowadays is the (GSPN) model. A (GSPN) have been used to analyze the performance of hardware and software systems. Note that the hardware and software systems that have been modeled successfully with GSPN , s include communication protocols, parallel programs, multiprocessor memory caches and distributed databases. Also GSPN , s are a popular graphical modeling formalism for investigating the qualitative and qualitative properties of concurrent systems [16]. Now, we can present the mathematical framework of a GSPN that may be also used to analyze the stochastic behavior of MSS.
GSPN are extensions of Place -Transition nets, which are untimed Petri nets with no transition firing delays. A Place -Transition net is formally defined as [3]: The reachability graph of a GSPN contains two types of markings. A vanishing marking is one in which an immediate transition is enabled. The sojourn time in such markings is zero. A tangible marking is one which enables only timed transitions. The sojourn time in such markings is exponentially distributed. We denote the set of reachable vanishing markings by and the set of reachable tangible markings by .
We define p ij to be the probability that j is the next marking entered after marking i, µ i 1  i M to be the man sojourn time in marking i and, for i , q ij = µ i p ij ;i.e. q ij is the instantaneous transition rate into marking j from marking i.
The following example is an application model for MSS for more details see [12].

Representation Models for MSS.
Our main result in this section is that for any MSS there exists a GSPN model that isomorphic to GSMP model as shown in the following theorem: for any state s in X(t) where  , µ are measurable mappings defined as follows:  (s) = s * (number of tokens per place) µ(E(s)) = T(s * )  T 1  T 2 for any time t and state s in X(t) since all mappings are assume to be onto, so that its clearly that X(t) is isomorphic to Y(t).

Example 4.2 (Producer/Consumer Model)
In the following example, we illustrate the use of the GSPN framework for formal specification of a MSS. In the graphical representation of a GSPN, places are drawn as circles, immediate transitions as thin bars, and timed transitions as thick bars, directed arcs connect transitions to output places and normal input places to transitions. Tokens are drawn as black dots Consider a system consisting of two producers, two consumers, two buffers, and a single shared channel for transmission of items from buffer to consumers; [7]. Producer i creates items for consumer i one at a time and items created, but not yet consumed, are placed in buffer i, i=1,2. Set The transitions have the following interpretation: e j = "creation of item by producer 1", e 2 = "start of transmission from buffer 1", e 3 = "end of transmission from buffer 1", e 4 = "creation of item by producer 2", e 5 = "start of transmission from buffer 2", and e 6 = "end of transmission from buffer 2". The interpretation of the places is as follows.
There is always one token in place d 1

CONCLUSION
1-Since GSPN and GSMP models captures the dynamic structure of MSS (Theorem 4.1) allows us to study the stochastic behavior of MSS. 2-Since no analytical results are available for calculating the parameter selected to be study (For example the waiting time parameter) in the MSS, simulation is a natural recourse and the choosing of the adequate model (GSPN or GSMP) play important role because depend in the nature of the system and parameters selected to be study. 3-Since the steadystate in GSMP is give by abstract form while in the GSPN model the steadystate is given by graphical form therefore this property in GSPN is better than GSMP in some real Multiprocessor Stochastic System. 4-The isomorphism in (theorem 4.1) can be applied by using semimarkov or markov processes by putting E(s)={e} or E(s)= respectively.