F. Baccelli and J. Mairesse. Ergodic theorems for stochastic operators and discrete event networks. In Idempotency [16], pages 171208.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....a set of possible post transitions depending on the number of tokens previously arrived. This concept of routing arlready familiar in, e.g. queueing networks has been extended to Petri nets and used to investigate liveness properties and performance by several authors, see for example [3], 4] 5] 10] In routed nets, tokens arriving on a place p are routed towards some post transition(s) of p in the sense that they contribute to possible enablings only of those transitions and are not available for the others. The ring itself proceeds as usual. Routing thus re nes the usual ....

F. Baccelli and J. Mairesse. Ergodic theorems for stochastic operators and discrete event networks. In Idempotency [16], pages 171208.

....for nets with both choice free and unique choice places. The following theorem verifies that the average TSE of a transition pair satisfying Condition 1 converges for the Petri nets considered in this paper. The theorem generalizes the weak ergodicity result on the average cycle time in [CCS91, BM95] to deal with the TSEs of two arbitrary transitions rather than that of consecutive instances of a same transition. assumptions given in Section 2.2. For any two transitions s; t of Sigma for which Condition 1 holds, their average TSE with finite occurrent offset converges to a finite ....

F. Baccelli and J. Mairesse. Ergodic theorems for stochastic operators and discrete event networks. Technical Report No. 2641, INRIA, 1995.

....t with occurrence index offset is lim n 1 (s; t; Note that the average cycle time of a transition u is a special case of the average TSE defined above with s = t = u and = 1. It is well known that all the transitions in a live stochastic marked graph has the (finite) same cycle time [12, 27]. We shall show that the limit in the above definition in fact converges for every transition pair (s; t) and every finite . Theorem 1 states this result which generalizes the weak ergodicity property of the cycle time (of one transition) 12, 27] to that of the time separations of two arbitrary ....

....marked graph has the (finite) same cycle time [12, 27] We shall show that the limit in the above definition in fact converges for every transition pair (s; t) and every finite . Theorem 1 states this result which generalizes the weak ergodicity property of the cycle time (of one transition) [12, 27] to that of the time separations of two arbitrary transitions (see the Appendix for a proof of the theorem) Consequently, under fairly weak assumptions on delay models, we see that the average of the time separation sequence of arbitrary transitions converges almost surely to some constant. ....

F. Baccelli and J. Mairesse. Ergodic theorems for stochastic operators and discrete event networks. Technical Report No. 2641, INRIA, 1995.

....in [8] generalized this to systems involving operations min, max, plus and convex hull. For the broader class of systems involving random topical operators [11] which include the min max function as a special case, some sucient conditions on the existence of global cycle times were obtained in [2]. Further analysis of large deviations properties were carried out in [14] Analyticity of Lyapunov exponents in parameters for stochastic systems was analysed in [3] and [4] The results in [5] 7] or [2] can be used to determine the existence of global cycle times, provided that we have the ....

.... case, some sucient conditions on the existence of global cycle times were obtained in [2] Further analysis of large deviations properties were carried out in [14] Analyticity of Lyapunov exponents in parameters for stochastic systems was analysed in [3] and [4] The results in [5] 7] or [2] can be used to determine the existence of global cycle times, provided that we have the knowledge (value or distribution) of the parameters. However for the cases in which the parameters or their distributions are unknown, all results known can not be applied directly. In this paper, systems with ....

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F. Baccelli and J. Mairesse, \Ergodic Theorem for Stochastic Operators and Discrete Event Networks, " in J. Gunawardena Ed., Idempotency. Cambridge University Press, 1998.

....of m (i,j) Since x (i,j)t = m (i,j) y it y jt for all t, m (i,j) q y it q y jt q is a subgradient of x (i,j)t if F(q) is concave and differentiable. 5. ERGODICITY This section analyses the ergodic properties of y it and x t by using the results of [1]. For this purpose, let us rewrite equations (8) as follows: Y n 1 = W(n) Y n where the the (min, algebra is used, denotes the min operator and the operator, Y n = Y 1n , Y In ) T , and W(n) is II matrix with W(n) ij = A jn (t n 1 t n ) m (j,i) According to the ....

.... purpose, let us rewrite equations (8) as follows: Y n 1 = W(n) Y n where the the (min, algebra is used, denotes the min operator and the operator, Y n = Y 1n , Y In ) T , and W(n) is II matrix with W(n) ij = A jn (t n 1 t n ) m (j,i) According to the terminology of [1], W(n) for any n is a monotone homogeneous operator since (i) it is homogeneous, i.e. W(n) Y l1) l1 W(n)Y for all Y in IR I and l in IR, ii) it is monotone, i.e. W(n)X W(n)Y if X Y. Theorem 6: Assume that the event graph is strongly connected. If W(n) n IN is a stationary ergodic ....

F. Baccelli and J. Mairesse, "Ergodic theorems for stochastic operators and discrete event networks," Research Report #2641, INRIA, France, 1995.

....following theorem verifies that the running average of TSEs of a transition pair satisfying Condition 1 converges for the Petri nets considered in this paper. The theorem generalizes the weak ergodicity property of the cycle time sequence of the consecutive instances of a same transition [CCS91, BM95] to that of the TSE sequence of a pair of different transitions. Theorem 1 Let Sigma be a LS Petri net that has only free choices and unique choices and satisfies stochastic assumptions given in Section 2.2. For any transition pair (s; t) for which Condition 1 holds, its corresponding average ....

F. Baccelli and J. Mairesse. Ergodic theorems for stochastic operators and discrete event networks. Technical Report No. 2641, INRIA, 1995.

....index offset is lim n 1 1 n P n k=1 fl (k) s; t; Note that the average cycle time of a transition u is a special case of the average TSE defined above with s = t = u and = 1. It is well known that all the transitions in a live stochastic marked graph has the (finite) same cycle time [12, 27]. We shall show that the limit in the above definition in fact converges for every transition pair (s; t) and every finite . Theorem 1 states this result which generalizes the weak ergodicity property of the cycle time (of one transition) 12, 27] to that of the time separations of two arbitrary ....

....marked graph has the (finite) same cycle time [12, 27] We shall show that the limit in the above definition in fact converges for every transition pair (s; t) and every finite . Theorem 1 states this result which generalizes the weak ergodicity property of the cycle time (of one transition) [12, 27] to that of the time separations of two arbitrary transitions (see the Appendix for a proof of the theorem) Theorem 1 Let s; t be two arbitrary transitions of a LS stochastic timed MG (G; M 0 ) as defined in Section 2.2. Their average time separation (with occurrence offset ) converges to a ....

F. Baccelli and J. Mairesse. Ergodic theorems for stochastic operators and discrete event networks. Technical Report No. 2641, INRIA, 1995.

....The following theorem verifies that the running average of TSEs of a transition pair satisfying Condition 1 converges for the Petri nets considered in this paper. The theorem generalizes the weak ergodicity property of the cycle time sequence of the consecutive instances of a same transition [10, 2] to that of the TSE sequence of a pair of different transitions. Theorem 1 Let Sigma be a LS Petri net that has only freechoices and unique choices and satisfies stochastic assumptions given in Section 2.2. For any transition pair (s; t) for which Condition 1 holds, its corresponding average ....

F. Baccelli and J. Mairesse. Ergodic theorems for stochastic operators and discrete event networks. Technical Report No. 2641, INRIA, 1995.

....functions are homogeneous, F i (x 1 h; Delta Delta Delta ; x n h) F i (x 1 ; Delta Delta Delta ; x n ) h for all 1 i n, and nonexpansive in the 1 norm, kF ( x) Gamma F ( y)k k x Gamma yk. Functions with these properties have emerged recently in the work of several authors, [3, 23, 29, 34, 46]. We shall follow Gunawardena and Keane and call them topical functions. They include (possibly after suitable transformation) nonnegative matrices, Leontieff substitution systems and Bellman operators of games and of Markov decisions processes, 24] They also include examples less well known to ....

....cannot be sooner. These are intuitively very reasonable and are observed in some form in most discrete event systems. Recent work has suggested that semigroups of random topical functions are both mathematically tractable and capable of modelling a wide variety of discrete event systems, [3, 46]. What role do min max functions play within the class of topical functions In turns out to be an unexpectedly central one, as shown by the following observation of Gunawardena, Keane and Sparrow. Lemma 1.1. 24] Let T : R n R n be a topical function and let S R n be any finite set ....

F. Baccelli and J. Mairesse. Ergodic theorems for stochastic operators and discrete event systems. Appears in [22].

....of a class of operators for which these properties have been studied is that of topical operators, which includes the class of (min; max; functions. For the deterministic theory of such operators, see [10] and [6] for the random case, see [17] where an analogue of Result 2 can be found, and [3], which contains an analogue of Result 3. Theorem 3 Under (P1) P2) and (P3) fl f (p) is analytic at point 0; the radius of convergence is larger than or equal to 1 2cj and the coeOEcients of the analytic expansion are given by the RR n# 3427 18 Fran#ois BACCELLI and Dohy HONG following ....

Baccelli, F. and Mairesse, J. (1998) Ergodic theorems for stochastic operators and discrete event networks. In J. Gunawardena, editor, Idempotency. Cambridge University Press.

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