Investigation of G-network with bypasses of queueing systems by positive customers at a non-stationary regime (original) (raw)
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Journal of Applied Mathematics and Computational Mechanics
In the first part of the article, an investigation of an open Markov queueing network with positive and negative customers (G-networks) has been carried out. The network receives two exponential arrivals of positive and negative customers. Negative customers do not receive service. The waiting time of customers of both types in each system is bounded by a random variable having an exponential distribution with different parameters. When the waiting time of a negative customer in the queue is over it reduces the number of positive customers per unit if the system has positive customers. The Kolmogorov system of difference-differential equations for non-stationary state probabilities has been derived. The method for finding state probabilities of an investigated network, based on the use of apparatus of multidimensional generating functions has been proposed. Expressions for finding the mean number of positive and negative customers in the network systems have also been found. In the second part the same network has been investigated, but with revenues. The case when revenues from the network transitions between states are random variables with given mean values has been considered. A method for finding expected revenues of the network systems has been proposed. Obtained results may be used for modeling of computer viruses in information systems and networks and also for forecasting of costs, considering the viruses penetration.
Journal of Applied Mathematics and Computational Mechanics, 2016
In the article a queueing network (QN) with positive customers and a random waiting time of negative customers has been investigated. Negative customers destroy positive customers on the expiration of a random time. Queueing systems (QS) operate under a heavy-traffic regime. The system of difference-differential equations (DDE) for state probabilities of such a network was obtained. The technique of solving this system and finding mean characteristics of the network, which is based on the use of multivariate generating functions was proposed.
A GI/Geo/1 queue with negative and positive customers
Applied Mathematical Modelling, 2010
The arrival of a negative customer to a queueing system causes one positive customer to be removed if any is present. Continuous-time queues with negative and positive customers have been thoroughly investigated over the last two decades. On the other hand, a discrete-time Geo/Geo/1 queue with negative and positive customers appeared only recently in the literature. We extend this Geo/Geo/1 queue to a corresponding GI/Geo/1 queue. We present both the stationary queue length distribution and the sojourn time distribution.
A MAP/G/1 Queue with Negative Customers
Queueing Systems, 2004
In this paper, we consider a M AP/G/1 queue with MAP arrivals of negative customers, where there are two types of service times and two classes of removal rules: the RCA and RCH, as introduced in Section 2. We provide an approach for analyzing the system. This approach is based on the classical supplementary variable method, combined with the matrix-analytic method and the censoring technique. By using this approach, we are able to relate the boundary conditions of the system of differential equations to a Markov chain of GI/G/1 type or a Markov renewal process of GI/G/1 type. This leads to a solution of the boundary equations, which is crucial for solving the system of differential equations. We also provide expressions for the distributions of stationary queue length and virtual sojourn time, and the Laplace transform of the busy period. Moreover, we provide an analysis for the asymptotics of the stationary queue length of the M AP/G/1 queues with and without negative customers.
Scientific Research of the Institute of Mathematics and Computer Science, 2012
The object of investigation is an open exponential network with a messages bypass of systems in transient behavior. The purpose of the research is to find stationary probabilities of states and the average characteristics of the network when the transition probabilities between the messages and bypass systems of the network, parameters of the incoming flow of messages and services are time-dependent. To find the state probabilities and the characteristics of a network is used the apparatus for the multivariate generating functions. The examples are calculated on a computer.
Analysis of the queuing network with messages bypass of systems in transient behavior
Scientific Research of the Institute of Mathematics and Computer Science, 2012
The article provides a survey of an open exponential network with a multiline queuing systems (QS) with a bypass service messages in the transient behavior. Messages with some probability join the QS, and with an additional probability to move immediately to another QS or leave the network. The paper describes the methodology finding the timedependent state probabilities of the network of its kind in the transient state. To find the state probabilities of the network, the method multivariate generating functions was applied. Examples are considered.
A Markov modulated multi-server queue with negative customers - The MM CPP/GE/c/L G-queue
Acta Informatica, 2001
We obtain the queue length probability distribution at equilibrium for a multi-server queue with generalised exponential service time distribution and either finite or infinite waiting room. This system is modulated by a continuous time Markov phase process. In each phase, the arrivals are a superposition of a positive and a negative arrival stream, each of which is a compound Poisson process with phase dependent parameters, i.e. a Poisson point process with bulk arrivals having geometrically distributed batch size. Such a queueing system is well suited to B-ISDN/ATM networks since it can account for both burstiness and correlation in traffic. The result is exact and is derived using the method of spectral expansion applied to the two dimensional (queue length by phase) Markov process that describes the dynamics of the system. Several variants of the system are considered, applicable to different modelling situations, such as server breakdowns, cell losses and load balancing. We also consider the departure process and derive its batch size distribution and the Laplace transform of the interdeparture time probability density function. From this, a recurrence formula is obtained for its moments. The analysis therefore provides the basis of a building block for modelling networks of switching nodes in terms of their internal arrival processes.
G-networks with multiple classes of negative and positive customers
Theoretical Computer Science, 1996
In recent years a new class of queueing networks with "negative and positive" customers was introduced by one of the authors [5], and shown to have a nonstandard product form. This model has undergone several generalizations to include triggers or signals which are special forms of customers whose role is to move other customers from some queue to another queue [9, 10, 6, 7-1. Positive customers are identical to the usual customers of a queueing network, while a negative customer which arrives to a queue simply destroys a positive customer. We call these generalized queueing networks G-networks. In this paper we extend the basic model of [5] to the case of multiple classes of positive customers, and multiple classes of negative customers. As in other multiple class queueing networks, a positive customer class is characterized by the routing probabilities and the service rate parameter at each service center while negative customers of different classes may have different "customer destruction" capabilities. In the present paper all service time distributions are exponential and the service centers can be of the following types: FIFO (first-in-first-out), LIFO/PR (last-in-first-out with preemption), PS (processor sharing), with class-dependent service rates.
Queueing Systems, 2004
The paper studies a closed queueing network containing two types of node. The first type (server station) is an infinite server queueing system, and the second type (client station) is a single server queueing system with autonomous service, i.e. every client station serves customers (units) only at random instants generated by strictly stationary and ergodic sequence of random variables. It is assumed that there are r server stations. At the initial time moment all units are distributed in the server stations, and the ith server station contains N i units, i = 1, 2,. .. , r, where all the values N i are large numbers of the same order. The total number of client stations is equal to k. The expected times between departures in the client stations are small values of the order O(N −1) (N = N 1 + N 2 + • • • + N r). After service completion in the ith server station a unit is transmitted to the j th client station with probability p i,j (j = 1, 2,. .. , k), and being served in the j th client station the unit returns to the ith server station. Under the assumption that only one of the client stations is a bottleneck node, i.e. the expected number of arrivals per time unit to the node is greater than the expected number of departures from that node, the paper derives the representation for non-stationary queue-length distributions in non-bottleneck client stations.
State-Dependent Signalling in Queueing Networks
Advances in Applied Probability, 1994
It has recently been shown that networks of queues with state-dependent movement of negative customers, and with state-independent triggering of customer movement have product-form equilibrium distributions. Triggers and negative customers are entities which, when arriving to a queue. force a single customer to be routed through the network or leave the network respectively. They are 'signals' which affect/control network behaviour. The provision of state-dependent intensities introduces queues other than single-server queues into the network.