Wojciech M. Kempa

On Applying Evolutionary Computation Methods to Optimization of Vacation Cycle Costs in Finite-Buffer Queue

In this paper, problem of positioning and optimization of operation costs for finite-buffer queuing system with exponentially distributed server vacation is investigated. The problem is solved using evolutionary computation methods for independent 2-order hyper exponential input stream of packets and exponential service time distribution. Different scenarios of system operation are analyzed, i.e. different values of parameters of distribution functions describing evolution of the system.

A finite-buffer queue with a single vacation policy

Abstract In this paper, application of an evolutionary strategy to positioning a GI/M/1/N-type finite-buffer queueing system with exhaustive service and a single vacation policy is presented. The examined object is modeled by a conditional joint transform of the first busy period, the first idle time and the number of packets completely served during the first busy period. A mathematical model is defined recursively by means of input distributions. In the paper, an analytical study and numerical experiments are presented. A cost optimization problem is solved using an evolutionary strategy for a class of queueing systems described by exponential and Erlang distributions.

Genetic Cost Optimization of the GI/M/1/N Finite-Buffer Queue with a Single Vacation Policy

In the artice, problem of the cost optimization of the GI/M/1/N-type queue with finite buffer and a single vacation policy is analyzed. Basing on the explicit representation for the joint transform of the first busy period, first idle time and the number of packets transmitted during the first busy period and fixed values of unit costs of the server’s functioning an optimal set of system parameters is found for exponentially distributed vacation period and 2-Erlang distribution of inter arrival times. The problem of optimization is solved using genetic algorithm. Different variants of the load of the system are considered as well.

Some New Results for Departure Process in the M X /G/1 Queueing System with a Single Vacation and Exhaustive Service

A batch arrival queueing system with a single vacation between two successive busy periods and with exhaustive service is considered. The departure process h(t) is studied first on a single vacation cycle. The approach based on renewal theory is applied to obtain results in the general case. In particular, the explicit representation for the generating function of Laplace transform of the probability function of h(t) is derived. All formulae are written in terms of input parameters of the system and factors of a certain canonical factorization of Wiener–Hopf type. A numerical approach to results is discussed as well.

Application of the Superposition of Renewal Processes to the Study of Batch Arrival Queues

The GΘ/G/1-type batch arrival system is considered. We deal with non-steady-state characteristics of the system like the first busy period and the first idle time, the number of customers served on the first busy period. The study is based on a generalization of Korolyuks method which he developed for semi-Markov random walks.

Queue-Size Distribution in M/G/1-Type System with Bounded Capacity and Packet Dropping

A single-server queueing system of M/G/1-type with boun- ded total volume is considered. It is assumed that volumes of arriving packets are generally distributed random variables. The AQM-type mechanism is used to control the actual buffer state: each of arriving packets is dropped with probability depending on its volume and the occupied volume of the system at the pre-arrival epoch. The explicit formulae for the stationary queue-size distribution and loss probability are found.

GI/G/1/∞ batch arrival queueing system with a single exponential vacation

In the article the queueing system of GI/G/1 type with batch arrival of customers and a single exponentially distributed vacation period at the end of every busy period is considered. Basic characteristics of transient state of the system are investigated: the first busy period, the first vacation period and the number of customers served during the first busy period. New results for the Laplace transform of the joint distribution of these three variables are obtained in dependence on the initial conditions of the system.

The Virtual Waiting Time for the Batch Arrival Queueing Systems

Abstract In the article the G η I/G/1-type batch arrival system with infinite waiting-room is considered. The explicit formulae for the distribution of the virtual waiting time at any fixed moment t and as t → ∞ are obtained. The study is based on generalization of Korolyuks method for semi-markov random walks.

The transient analysis of the queue‐length distribution in the batch arrival system with N‐policy, multiple vacations and setup times

A batch arrival queueing system of the MX/G/1 type with unlimited queue is considered. After each busy period the server begins a multiple vacation period, consisting of independent single vacations, when the service process is blocked. The server begins successive single vacations as far as at the end of one of them the number of customers waiting in the queue equals at least N. The service of the first customer after the vacation period is preceded by a setup time.The analysis of the queue‐size distribution on the first vacation cycle is directed to the analysis of the same characteristic in the corresponding ”usual” system with unremovable server on its first busy period. The renewal‐theory approach is used to obtain results in the general case. As main result the explicit representation for the LT of queue‐size distribution is derived for the original system.

The virtual waiting time in a finite-buffer queue with a single vacation policy

A finite-buffer queueing system with Poisson arrivals and generally distributed service times is considered. Every time when the system empties, a single vacation is initialized, during which the service process is blocked. A system of integral equations for the transient distributions of the virtual waiting time v(t) at a fixed moment t, conditioned by the numbers of packets present in the system at the opening, is derived. A compact formula for the 2-fold Laplace transform of the conditional distribution of v(t) is found and written down using a special-type sequence called a potential. From this representation the stationary distribution of v(t) as t→∞ and its mean can be easily obtained. Theoretical results are illustrated by numerical examples as well.