We investigate the problem of minimum dispersion linear prediction of the components of a linear process driven by multivariate symmetric a-stable noise. The results generalize some of those obtained for univariate processes by Cline and Brockwell (1985) and Stuck (1978)

The distribution of earthquake seismic moment M is a Pareto (power-law) type with the exponent for shallow earthquakes. To ensure a finite total deformation energy, the distribution has to be modified for the largest earthquakes by introducing either a cutoff at or an exponential taper. Analysis of global seismicity suggests that both parameters do not change in a statistically significant way over continental areas and tectonic plate boundaries. Thus, very large earthquakes are possible in practically any urbanized areas. We present evidence that the statistical distribution of losses due to large earthquakes has a power-law (Pareto) tail with an exponent value less than 1.0: i.e., the distribution is heavy-tailed. If this statement is true, the earthquake average loss is controlled by the largest earthquakes. Neither the distribution of earthquake size at the maximum, nor the distribution of maximum losses is well known. This implies that insurance premiums can be determined only with significant uncertainty. We simulate the insurance ruin potential for three earthquake loss distributions: exponential, Pareto with exponent 3/2, and Pareto with exponent 2/3. The possibility of catastrophic losses due to great earthquakes suggests that the probabilities of the ruin of insurance companies are unacceptably high, unless the risk reserves are equal to or exceed the maximum possible loss. Although we do not give practical suggestions for earthquake insurance in the paper, these results point out the mathematical and geophysical constraints that insurance should operate within

We consider a non-preemptive head of the line multi-server priority model. A generalized Markovian arrival process with marked transitions is used as the arrival process. The service-time distribution is of phase-type and identical for the different priority classes. Applying matrix-geometric methods we compute the steady-state distribution of the corresponding single-class system where we do not distinguish between the different priority classes. Furthermore we compute the stationary distribution of the different priority classes immediately after arrival instants. We give an explicit representation for the actual waiting time distribution of the lowest priority customers. Matrix-geometric representations for the Laplace-Stieltjes-Transforms (LST) of the conditional waiting time distributions of different priority classes are derived. Using first passage time analysis and Little's formula, the mean actual waiting times and the mean queue lengths of different priority classes are calculated

The performance analysis of parallel systems is of great interest. It includes problems such as the comparison of different routing strategies in a communication network or the computation of the average waiting time of a process in a shared memory system. A common point to a large class of such systems is concurrency: the system is composed of a number of resources to which the access is exclusive. That is what limits parallelization: two different requests that require a common resource cannot be parallelized. In this paper we present a general model for such systems and we define a performance parameter that is roughly the throughput of the system. The evaluation of this parameter is known to be difficult. Under some probabilistic assumptions, we present an exact calculation for the case of a system composed of two resources and an upper bound for more general systems

Buffer overflow is important to communication and production systems. For the case where batches arrive according to a Poisson process and the service is in batches that are geometrically distributed, we describe a method to find the exact loss probability. We also give an asymptotic approximation for the loss probability. Numerical experiments indicate that the buffer sizes needed to keep the loss probabilities below a specified level increase when the expected service batch size increases while the offered load is kept constant

First-crossing times of linear boundaries by Poisson processes are considered. In particular, the moment generating functions of stopping times of the form are examined, where is a homogeneous Poisson counting process and . Exact and asymptotic (in B) expressions are obtained by analyzing certain delay-differential equations derived for this moment generating function by Dvoretsky, Kiefer and Wolfowitz. Further results are obtained for the case of a single, lower boundary by exploiting results of Zacks who gives a series representation for the stopping-time distribution in this case. These results are used to obtain a closed-form expression for the cumulants of the stopping time in this latter case

The explicit expressions for the mean and the second moment of the waiting time distribution for an M/G/l queue with random order of service can be found in the literature. However, higher-order moments have been unavailable, because their calculation is extremely laborious, albeit straightforward in principle. This paper presents the third through sixth order moments of the waiting time in symbolic form, which is obtained by a Mathematica program developed for formula manipulation. The results provide more information about the characteristic of the waiting time distribution. In particular, they are used to compare the skewness and kurtosis of the waiting time with those for the first-come first-served and last-come first-served systems

Keywords: M/G/1 type queuesMarkov chainsqueueing theoryserver vacationsstochastic service systemsworkload