A service model in which the server is required to search for customers

“…As observed by Neuts and Ramalhoto (1984), the number of arrivals during the waiting time of one customer in the MIGl1 queue has p.g.f.…”

Section: The MI G11 Queue Lengthmentioning

confidence: 91%

Section: -P+ajjw Z-q(z)mentioning

confidence: 99%

Asymptotic tail behaviour of Poisson mixtures by applications

Willmot

1990

“…Once the density {epv} has been evaluated, a large number of stationary queue-length densities are obtained by elementary numerical operations such as convolutions, mixing or shifts. We shall not review such algorithms in detail, but we draw attention to [33] for an unexpected application.…”

Section: The Discrete Pollaczek-khinchin Equationmentioning

confidence: 99%

“…The queue is stable if and only if Pt + pz < 1. Miller [23] has proved that the Laplace-Stieltjes transform wz(s) of the stationary waiting-time distribution of a customer of type 2 is given by (33) where hz(e) is the Laplace-Stieltjes transform of Hz(e). Yt(s) is the transform of the busy-period distribution G t (e) of an M / G /1 queue with arrival rate At and service-time distribution HI (e ).…”

Section: Priority Queuesmentioning

confidence: 99%

Generalizations of the Pollaczek-Khinchin integral equation in the theory of queues

Neuts

1986

Self Cite

A classical result in queueing theory states that in the stable M/G/1 queue, the stationary distribution W(x) of the waiting time of an arriving customer or of the virtual waiting time satisfies a linear Volterra integral equation of the second kind, of convolution type. For many variants of the M/G/1 queue, there are corresponding integral equations, which in most cases differ from the Pollaczek–Khinchin equation only in the form of the inhomogeneous term. This leads to interesting factorizations of the waiting-time distribution and to substantial algorithmic simplifications. In a number of priority queues, the waiting-time distributions satisfy Volterra integral equations whose kernel is a functional of the busy-period distribution in related M/G/1 queues. In other models, such as the M/G/1 queue with Bernoulli feedback or with limited admissions of customers per service, there is a more basic integral equation of Volterra type, which yields a probability distribution in terms of which the waiting-time distributions are conveniently expressed.For several complex queueing models with an embedded Markov renewal process of M/G/1 type, one obtains matrix Volterra integral equations for the waiting-time distributions or for related vectors of mass functions. Such models include the M/SM/1 and the N/G/1 queues, as well as the M/G/1 queue with some forms of bulk service.When the service-time distributions are of phase type, the numerical computation of waiting-time distributions may commonly be reduced to the solution of systems of linear differential equations with constant coefficients.

“…Other similar models can be found in the literature about multiclass retrial queues; see Falin et al [4] and its references. In related work, Neuts and Ramalhoto [8] analyzed a service model in which, at the end of the transmission, the channel is required to search the next unit to be processed.…”

Section: Introduction and Model Descriptionmentioning

confidence: 99%

Analysis of an M/G/1 queue with two types of impatient units

Martı́n

Artalejo

1995

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Applied Probability Trust is collaborating with JSTOR to digitize, preserve and extend access to Advances in Applied Probability. AbstractThis paper deals with a service system in which the processor must serve two types of impatient units. In the case of blocking, the first type units leave the system whereas the second type units enter a pool and wait to be processed later.We develop an exhaustive analysis of the system including embedded Markov chain, fundamental period and various classical stationary probability distributions. More specific performance measures, such as the number of lost customers and other quantities, are also considered. The mathematical analysis of the model is based on the theory of Markov renewal processes, in Markov chains of M/G/1 type and in expressions of 'Takacs' equation' type.Analysis of an M/G/1 queue be presented in a different way. Suppose that the pool permanently has information about the state of the channel and that the unit at the head of the pool could occupy the channel when this becomes free. Taking into account that a unit of the first type can be lost, the processor may consider the possibility of reserving a priority period of time for such a unit delaying the entrance at service for the units in the pool. In this case we can think of / as a 'priority parameter'.The model described above has applications in the study of the access of messages to a central processor in computer networks. The service time distribution, the units in the pool and the distribution of the successive retrials in the queueing terminology correspond to the transmission time, blocked terminals which can be stored in a buffer and retransmission policy. So our model has related applications in the study of packet switching networks and communications protocols arising in the modelling of local area networks. For a more detailed discussion of other similar protocols used in computer networks, see Hammond and O'Reilly [7]. Many queueing systems are closely related to our model. The particular case A1 = 0 was first introduced by Fayolle [6]. Farahmand [5] and Artalejo and Martin [1] examined the M/G/1 queue with similar retransmission control policies: the delays between successive attempts follow an exponential law with parameter ,u, but the pool discipline is not FCFS. Other similar models can be found in the literature about multiclass retrial queues; see Falin et al. [4] and its references. In related work, Neuts and Ramalhoto [8] analyzed a service model in which, at the end of the transmission, the channel is required to search the next unit to be processed.The rest of the paper is organized as follows. In Section 2, we consider a formula of 'Takacs' equati...