A comment on Boucherie product-form results

Transcription

1 A comment on Boucherie product-form results Andrea Marin Dipartimento di Informatica Università Ca Foscari di Venezia Via Torino 155, Venezia Mestre, Italy Abstract. This is a sketch about Boucherie paper on product form. It is quite informal, for a deeper view we suggest to study the interesting paper [2]. Our aim is just to point out the main main conditions for this product form. 1 The Boucherie result on the product form The starting CTMCs In this section we summarize the results presented in [2]. Let us consider K stable and regular CTMCs. The state spaces S 1,... S K of the CTMCs are finite or countable. Let q k (n k, n k ) be the transition rate from state n k to n k of k-th CTMC. Let π k : S k [0, 1] denote the normalized stationary probability function, i.e., n k S k π k (n k ) = 1. Let n = (n 1,..., n K ) with n k S k and k = 1,..., K be a state of the composed process. Clearly if the CTMCs are independent we have product form, i.e., π(n) = K k=1 π k(n k ). The competition Let use introduce an index set I. Intuitively I is a set which labels the resources. In each state a CTMC uses one and only one resource. So we can partition a state space S k in I mutually exclusive sets A ki such that: i I A ki = S k A ki If CTMCs k 1 and k 2 compete over resources i then this means that the combined process does not reach the state n where components n k1 A k1 i and n k2 A k2 i. We denote with C ki the set of CTMCs which compete with CTMC k on the resource i I. The product form condition Let us study the process with state space S = S 1... S k and let us define the following transition matrix for the CTMC on S: q(n, n ) = K k=1 [ q k (n k, n k) K r=1,r k Note that the definition of q is such that: ] 1(n r = n r)1(n r A ri = k / C ri ). (1)

2 It is impossible to reach a state where two CTMCs use a resource i I if they are in conflict on i. This means that the state space of the composed process can be reduced excluding the no-reachable states. Given two adjacent states n and n then just a single CTMC k changes its state, i.e., state n and state n differ just by the component k which changes from n k to n k, with k = 1,..., K. The transition rate from n to n is determined only by the transition rate between states n k and n k of the k th Markov chain. Suppose that CTMCs k, k conflict on resource i. Then when one of the two CTMCs is in a state belonging to the set A ki or A k i the other Markov chain cannot change its state at all. This is probably the main limit of this approach to the product form. We shall comment this condition in the conclusions. Boucherie gives the following important result on the stationary distribution of the composed process: K π(n) = B π k (n k ), (2) k=1 where n is a reachable state and B a normalizing constant. 2 An example Let us consider two identical CTMCs as showed in Figure 1. Fig. 1. Example CTMC Clearly the Markov chain is ergodic and its stationary probabilities are π(n) = 1 3 for n = a, b, c. Let us now define the competition. Suppose I = {1, 2}

3 and A k1 = {a, c}, A k2 = {b} and C 12 = {2}, C 22 = {1}, C k1 =, with k = 1, 2. In Figure 1 we use orange to denote the states which use resource 1 and blue for the state which uses resource 2. Note that competition is just over resource 2. The composed process on S = S 1 S 2 with the transition rates defined by formula 1 has 8 reachable states. By the result expressed by equation 2 we can conclude that the stationary probability of each state is the same, i.e Figure 2 shows the composed process S. Fig. 2. Composed process S. Read the picture as dotted lines are not present. All the transition rates are equal to v. It is easy to see that π(n) = 1 8 CTMC on space S. for all n satisfies the set of GBEs for the 3 Conclusions Boucherie product form condition is very interesting for various reasons. First of all it gives conditions on the Markov Chains so the result can be used as framework for studying the product form conditions for different stochastic model

4 formalisms, such as stochastic process algebra, stochastic Petri nets (SPN), and so on. If we apply Boucherie approach to SPNs, as pointed out in the original work [2], we note a second strength of the result. In fact if we consider Coleman et al. product form on SPNs [6, 3] the authors aim to give a stationary solution which is expressed in terms of product of functions of the number of tokens in each place. This means that, given a SPN, the authors give a set of sufficient conditions for the product form. Boucherie framework, on the other hand, is more suitable for hierarchical modelling. In fact, the idea is that given a set of SPNs (let us call them agent), each one with its associated CTMC, a set of sufficient conditions for the product form solution is defined on the interaction of the agents. However given an SPN it does not appear an easy task identifying algorithmically Boucherie product form: this would require to identify the agents and checking the interaction, but for the low level expressive power of SPNs this would be quite hard. It is worthwhile exploring the consequences of Boucherie conditions for product form. In the following we comment them one by one: The conditions on the single CTMC. These are very general, indeed it is just asked them to be ergodic and to have a unique stationary distribution. The conditions on the composed process state space. The exclusion mechanism appears to be very flexible. Defining a partition on the state space is a really general approach. In the original paper the author gives an example where he shows how the exclusion mechanism can model the availability of multiple resources. The conditions on the composed process transition rates. We think that it is important to understand well the meaning of this key-condition for the product form. Suppose that CTMCs k, k conflict on resource i. Then when one of the two CTMCs is in a state belonging to the set A ki or A k i the other Markov chain cannot change its state at all. In practice suppose you have two identical process which do this task: think }{{} print }{{} a b input data think }{{} c The printing state requires to use of the shared resource printer so there is a competition on that resource. This is basically the CTMC of the example of Section 2 where it is assumed that all the tasks has an exponential distributed duration with the same mean. The point is that Boucherie framework requires that when a process is printing the other one is stopped, independently on the state it is. There are no reasons, from a modelling point of view, for stopping a process which is performing an input action when the other process is printing. In Figure 2, by adding the dotted arcs, one obtains the process how it should be from a modelling point of view. But the product form is lost. It does not appear clear the relation between the class of product form SPNs (PFSPNs). As far as we know we can identify three classes of PFSPNs:

5 1. Lazar an Robertazzi product form [7, 8], 2. Coleman, Henderson, Taylor product form [6, 3] which includes also the product form defined in [5], 3. Boucherie product form [2]. As Boucherie points out in his paper, product form 3 generalizes product form 1, but the relations with 2 should be explored. In fact, even if some conditions are equivalent (e.g. each transition input vector has a correspondent transition output vector [4]) the product form model class is different. For example in [6, 3] the product form condition depends on the transition rate (this problem is overcome in [5] but the model class is smaller) and this has not an equivalent condition on product form 1 and 3. Finally we think that the relation between this conditions on product form and Muntz s M M [9] property should be explored. In [1] are showed some GSPN, i.e. they allow immediate transitions, with an associated Markov chain. It is proved that their composition is in product form, even if they do not fit any of the classes of product form cited here.

6 References 1. Balsamo, S., and Marin, A. On representing multiclass M/M/k queues by generalized stochastic Petri nets. In Proc. of ECMS/ASMTA-2007 conference. To appear. 2. Boucherie, R. J. A characterisation of independence for competing Markov chains with applications to stochastic Petri nets. Tech. Rep. RR Coleman, J. L., Henderson, W., and Taylor, P. G. Product form equilibrium distributions and a convolution algorithm for Stochastic Petri nets. Performance Evaluation 26 (1996), Donatelli, S., and Sereno, M. On the product form solution for stochastic petri nets. In Proceedings of the 13th International Conference on Application and Theory of Petri Nets (London, UK, 1992), Springer-Verlag, pp Haddad, S., Moreaux, P., Sereno, M., and Silva, M. Product-form and stochastic Petri nets: a structural approach. Perform. Eval. 59, 4 (2005), Henderson, W., Lucic, D., and Taylor, P. G. A net level performance analysis of Stochastic Petri Nets. J. Austral. Math. Soc. Ser. B 31 (1989), Lazar, A. A., and Robertazzi, T. G. Markovian Petri Net Protocols with Product Form Solution. In IEEE INFOCOM 87, The Conf. on Computer Communications, Proc., Sixth Annual Conference - Global Networks: Concept to Realization (Washington, DC, 1987), IEEE Computer Society Press, pp NewsletterInfo: Lazar, A. A., and Robertazzi, T. G. Markovian Petri Net Protocols with Product Form Solution. Performance Evaluation 12, 1 (jan 1991), NewsletterInfo: Muntz, R. R. Poisson departure processes and queueing networks. Tech. Rep. IBM Research Report RC4145, Yorktown Heights, New York, 1972.