Andreas Frey

Does Markov-Modulation Increase the Risk?

In this paper we compare ruin functions for two risk processes with respect to stochastic ordering, stop-loss ordering and ordering of adjustment coefficients. The risk processes are as follows: in the Markov-modulated environment and the associated averaged compound Poisson model. In the latter case the arrival rate is obtained by averaging over time the arrival rate in the Markov modulated model and the distribution of the claim size is obtained by averaging the ones over consecutive claim sizes.

A note on an M/GI/1/N queue with vacation time and exhaustive service discipline

We consider an M/GI/1/N queue with vacation time and exhaustive service discipline, the same queue as in Lee [Oper. Res. 32 (1984) 774-784]. Focusing only on the service completion epochs (as opposed to both service completion epochs and vacation completion epochs by Lee), we present a simple analysis for the queue length distribution at an arbitrary time as well as for the waiting time distribution.

ANMX/GI/1/N- QUEUE WITH CLOSE-DOWN AND VACATION TIMES

An MX/GI/1/N finite capacity queue with close-down time, vacation time and exhaustive service discipline is considered under the partial batch acceptance strategy as well as under the whole batch acceptance strategy. Applying the supplementary variable technique the queue length distribution at an arbitrary instant and at a departure epoch is obtained under both strategies, where no assumption on the batch size distribution is made. The loss probabilities and the Laplace-Stieltjes transforms of the waiting time distribution of the first customer and of an arbitrary customer of a batch are also given. Numerical examples give some insight into the behavior of the system.

Explicit solutions for the M/GI/1/N finite capacity queues with and without vacation time

We consider the M/GI/1/N finite capcaity queues with and without vacation time. We show that the embedded Markov chain formed by the service completion epochs makes the analysis simple and it enables us to find an explicit solution for the server-vacation queue. By taking the limit of our results as the vacation time length tends to zero, we also obtain an explicit solution for the ordinary queue (without vacation time).

Taylor-series expansion for multivariate characteristics of classical risk processes

Abstract For the continuous-time risk model with compound Poisson input, the (finite-horizont) joint probability P ( τ ≤ τ , X ≤ x , Y ≤ y ) of ruin time τ, surplus X just before ruin and deficit Y at ruin time τ is considered as a function of the arrival rate λ of claims. It is expanded into a Taylor series at λ = 0. A certain extension of a corresponding result for infinite-horizont joint probabilities, which previously has been derived in Gerber et al. (1987), is also given. For each n ≥ 1, the coefficient of λ n is determined by using a general representation formula for the derivatives of a wide class of functionals of independently marked Poisson processes.

Light-traffic approximations for Markov-modulated multi-server queues

A general concept is considered of expanding the expectation of a wide class of functional of marked point processes, which expresses this expectation by a sum of integrals over higher-order factorial moment measures of the underlying point process. The idea of factorial moment expansion is applied in order to derive approximation formulas for stationary characteristics of multi-server queues with Markov-modulated arrival process and with the first-come-first-served queueing discipline. Besides real-valued queueing characteristics like waiting time and total work load, we also give approximations for the Kiefer-Wolfowitz work-load vector. A boundedness condition on the service time distributions is given which ensures that the components of the expected stationary work-load vector are analytic functions of the arrival intensity in a neighborhood of zero. If the service times have phase-type distributions, the factorial moment expansion provides a useful computational technique for approximations of moment...

A single-server queueing system with modified service mechanism

We consider a single-server GI/G/1 queueing system with modified service mechanism. By modified service mechanism, we mean that the service time distribution (H0) for the customers arriving to find the system idle may be different from the service time distribution (H1) for the customers arriving to find the system busy. We present qualitative relationships among the performance measures in the system. Approximating the virtual waiting time process via the diffusion process and combining the qualitative relationships, we propose a new approximate formula for the mean performance measures. For special cases, our approximation is seen to be consistent with the previously-obtained exact results for the M/G/1 queueing system with modified service mechanism, and it is further seen to be consistent with the previously-proposed approximate results for the GI/GI/1 queueing system with standard ( H0 = H1 ) service mechanism.

Diffusion Approximation for a Web-Server System with Proxy Servers

It is an important and urgent Operations Research (OR) issue to evaluate the delay in a web-server system handling internet commerce real-time services. Usually, proxy servers in differently-located sites enable us to shorten the web-server access delay in order to guarantee the quality of real-time application services. However, there exists almost no literature on the queueing analyses for the web-server system with proxy servers. The goal of this paper is to provide a queueing analysis for the web-server system. We derive the statistics of the individual output processes from the proxy servers. Regarding the unfinished workload in the web-server system with input as a diffusion process, we derive a mean-delay explicit formula.

Approximate Formula of Delay-Time Variance in Renewal-Input General-Service-Time Single-Server Queueing System

Approximate formulas of the variance of the waiting-time (also called as delay-time variance) in a renewal-input general-service-time single-server (GI/GI/1) system play an important role in practical applications of the queueing theory. However, there exists almost no literature on the approximate formulas of the delay-time variance in the GI/GI/1 system. The goal of this paper is to present an approximate formula for the delay-time variance. Our approach is based on the combination of a higher-moment relationship between the unfinished work and the waiting time, and the diffusion process approximation for the unfinished work. To derive the former relationship, we apply Miyazawa’s rate conservation law for the stationary point process. Our approximate formula is shown to converge to the exact result for the Poisson-input system as traffic intensity goes to the unity. The accuracy of our approximation is validated by simulation results.