Browsing by Author "Baccelli, Francois"
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Item Dynamic, Transient and Stationary Behavior of the M/GI/1 Queue.(1988) Baccelli, Francois; Makowski, Armand M.; ISRAn exponential martingale is associated with the Markov chain of the number of customers in the M/GI/1 queue. This together with renewal theory are shown to provide a unified probabilistic framework for deriving several well-known generating functions for the M/GI/1 queue, including the Pollaczek-Khinchine formula, the transient generating function of the number of customers at departure epochs and the generating function of the number of customers served in a busy period.Item Multi-Dimensional Stochastic Ordering and Associated Random Variables.(1987) Baccelli, Francois; Makowski, Armand M.; ISRThis paper presents several relationships between the notion of associated random variables and notions of stochastic ordering which have appeared in the literature over the years. More concretely, the discussion centers around the following question: Under which conditions does the association of the R-valued RV's {X_1, ... ,X_n} imply a possible ordering in some stochastic sense between the R^n-valued RV X := (X_1, ... ,X_n) and its independent version X := ({X PRIME_1}, ... , {X PRIME_n})? Some of the results in that direction are as follows: (i) These R^n- valued RV's are comparable in either one of the orderings <=_st, <=_ci, and <=_cv iff they are identical in law, and (ii) if the RV's {X_1, ... ,X_n} are associated, certain comparison properties hold for the stochastic orderings <=_D, <=_K, and <=_L, defined in Stoyan [8, p.27]. Strengthening of result (i) leads to the following results on the stochastic ordering properties of R^n-valued RV's X and Y with identical mean: (j) The RV's X and Y are comparable for <=_st iff they are identical in law, and (jj) If X <=_D Y (resP. X <=_K Y), then X and Y are comparable for <=_ci, (resp. <=_cv) iff they are identical in law. These and related results are given direct applications to queueing theory and to the asymptotics of associated random variables.Item Stochastic Orders Associated with the Forward Recurrence Time of a Renewal Process(1992) Baccelli, Francois; Makowski, Armand M.; ISRWAITING FOR JAYA TO CREATE SYMBOLS FOR THE ABSTRACT OF THIS REPORT.