An inequality for Lp-norms with respect to the multivariate normal distribution

AbstractA partial converse of Jensen's inequality for integrals of norms on Rk is proved

Hitting times and the running maximum of Markovian growth collapse processes

We consider a Markovian growth collapse process on the state space e = [0;8) which evolves as follows. Between random downward jumps the process increases with slope one. Both the jump intensity and the jump sizes depend on the current state of the process. We are interested in the behavior of the first hitting time Ty = inf{t = 0¦Xt = y} as y becomes large and the growth of the maximum process Mt = sup{Xsj0= s = t} as t¿8. We consider the recursive sequence of equations Amn = mn-1, m0= = 1, where A is the extended generator of the MGCP, and show that the solution sequence (which is essentially unique and can be given in integral form) is related to the moments of Ty. The Laplace transform of Ty can be expressed in closed form (in terms of an integral involving a certain kernel) in a similar way. We derive asymptotic results for the running maximum: (i) if m1(y) is of rapid variation, we have Mt/m -1(t) d¿ 1; (ii) if m1(y) is of regular variation with index a ¿ 2 (0,8) and the MGCP is ergodic, then Mt/m-1(t) d¿ Za, where Za, has a Frechet distribution. We present several examples

Group Testing Models with Processing Times and Incomplete Identification

We consider the group testing problem for a finite population of possibly defective items with the objective of sampling a prespecified demanded number of nondefective items at minimum cost.Group testing means that items can be pooled and tested together; if the group comes out clean, all items in it are nondefective, while a "contaminated" group is scrapped.Every test takes a random amount of time and a given deadline has to be met.If the prescribed number of nondefective items is not reached, the demand has to be satisfied at a higher (penalty) cost.We derive explicit formulas for the distributions underlying the cost functionals of this model.It is shown in numerical examples that these results can be used to determine the optimal group size.testing;sampling

The M/G/1+G queue revisited

We consider an M/G/1 queue with the following form of customer impatience: an arriving customer balks or reneges when its virtual waiting time, i.e., the amount of work seen upon arrival, is larger than a certain random patience time. We consider the number of customers in the system, the maximum workload during a busy period, and the length of a busy period. We also briefl y treat the analogous model in which any customer enters the system and leaves at the end of his patience time or at the end of his virtual sojourn time, whichever occurs first

ML Characterization of the Multivariate Normal Distribution

AbstractIt is a well-known result (which can be traced back to Gauss) that the only translation family of probability densities on R for which the arithmetic mean is a maximum likelihood estimate of the translation parameter originates from the normal density. We generalize this characterization of the normal density to multivariate translation families

A compound Poisson EOQ model for perishable items with intermittent high and low demand periods

We consider a stochastic EOQ-type model, with demand operating in a two-state random environment. This environment alternates between exponentially distributed periods of high demand and generally distributed periods of low demand. The inventory level starts at some level q, and decreases according to different compound Poisson processes during the periods of high demand and of low demand. The inventory level is refilled to level q when level 0 is hit or when an expiration date is reached, whichever comes first. We determine various performance measures of interest, like the distribution of the time until refill, the expected amount of discarded material and of material held (inventory), and the expected values of various kinds of shortages. For a given cost/revenue structure, we can thus determine the long-run average profit

Group Testing Models with Processing Times and Incomplete Identification

We consider the group testing problem for a finite population of possibly defective items with the objective of sampling a prespecified demanded number of nondefective items at minimum cost.Group testing means that items can be pooled and tested together; if the group comes out clean, all items in it are nondefective, while a "contaminated" group is scrapped.Every test takes a random amount of time and a given deadline has to be met.If the prescribed number of nondefective items is not reached, the demand has to be satisfied at a higher (penalty) cost.We derive explicit formulas for the distributions underlying the cost functionals of this model.It is shown in numerical examples that these results can be used to determine the optimal group size.