Analysis and optimization of vacation and polling models with retrials

We study a vacation-type queueing model, and a single-server multi-queue polling model, with the special feature of retrials. Just before the server arrives at a station there is some deterministic glue period. Customers (both new arrivals and retrials) arriving at the station during this glue period will be served during the visit of the server. Customers arriving in any other period leave immediately and will retry after an exponentially distributed time. Our main focus is on queue length analysis, both at embedded time points (beginnings of glue periods, visit periods and switch- or vacation periods) and at arbitrary time points.Comment: Keywords: vacation queue, polling model, retrials Submitted for review to Performance evaluation journal, as an extended version of 'Vacation and polling models with retrials', by Onno Boxma and Jacques Resin

EUROPEAN CONFERENCE ON QUEUEING THEORY 2016

International audienceThis booklet contains the proceedings of the second European Conference in Queueing Theory (ECQT) that was held from the 18th to the 20th of July 2016 at the engineering school ENSEEIHT, Toulouse, France. ECQT is a biannual event where scientists and technicians in queueing theory and related areas get together to promote research, encourage interaction and exchange ideas. The spirit of the conference is to be a queueing event organized from within Europe, but open to participants from all over the world. The technical program of the 2016 edition consisted of 112 presentations organized in 29 sessions covering all trends in queueing theory, including the development of the theory, methodology advances, computational aspects and applications. Another exciting feature of ECQT2016 was the institution of the Takács Award for outstanding PhD thesis on "Queueing Theory and its Applications"

Sojourn time in a single server queue with threshold service rate control

We study the sojourn time in a queueing system with a single exponential server, serving a Poisson stream of customers in order of arrival. Service is provided at low or high rate, which can be adapted at exponential inspection times. When the number of customers in the system is above a given threshold, the service rate is upgraded to the high rate, and otherwise, it is downgraded to the low rate. The state dependent changes in the service rate make the analysis of the sojourn time a challenging problem, since the sojourn time now also depends on future arrivals. We determine the Laplace transform of the stationary sojourn time and describe a procedure to compute all moments as well. First we analyze the special case of continuous inspection, where the service rate immediately changes once the threshold is crossed. Then we extend the analysis to random inspection times. This extension requires the development of a new methodological tool, that is "matrix generating functions". The power of this tool is that it can also be used to analyze generalizations to phase-type services and inspection times.Comment: 16 pages, 13 figure

The M/G/1 retrial queue: an information theoretic approach

In this paper, we give a survey of the use of information theoretic techniques for the estimation of the main performance characteristics of the M/G/1 retrial queue. We focus on the limiting distribution of the system state, the length of a busy period and the waiting time. Numerical examples are given to illustrate the accuracy of the maximum entropy estimations when they are compared versus the classical solutions.Peer Reviewe

Single server retrial queueing models.

Most retrial queueing research assumes that each retrial customer has its own orbit, and the retrial customers retry to enter service independently of each other. A small selection of papers assume that the retrial customers themselves form a queue, and only one customer from the retrial queue can attempt to enter at any given time. Retrial queues with exponential retrial times have been extensively studied, but little attention has been paid to retrial queues with general retrial times. In this thesis, we consider four retrial queueing models of the type in which the retrial customers form their own queue. Model I is a type of M/G/1 retrial queue with general retrial times and server subject to breakdowns and repairs. In addition, we allow the customer in service to leave the service position and keep retrying for service until the server has been repaired. After repair, the server is not allowed to begin service on other customers until the current customer (in service) returns from its temporary absence. We say that the server is in reserved mode, when the current customer is absent and the server has already been repaired. We define the server to be blocked if the server is busy, under repair or in reserved mode. In Model II, we consider a single unreliable server retrial queue with general retrial times and balking customers. If an arriving primary customer finds the server blocked, the customer either enters a retrial queue with probability p or leaves the system with probability 1 - p. An unsuccessful arriving customer from the retrial queue either returns to its position at the head of the retrial queue with probability q or leaves the system with the probability 1 - q. If the server fails, the customer in service either remains in service with probability r or enters a retrial service orbit with probability 1 - r and keeps returning until the server is repaired. We give a formal description for these two retrial queueing models, with examples. The stability of the system is analyzed by using an embedded Markov chain. We get a necessary and sufficient condition for the ergodicity of the embedded Markov chain. By employing the method of supplementary variables, we describe the state of the system at each point in time. A system of partial differential equations related to the models is derived from a stochastic analysis of the model. The steady state distribution of the system is obtained by means of probability generating functions. In steady state, some performance measures of the system are reported, the distribution of some important performance characteristics in the waiting process are investigated, and the busy period is discussed. In addition, some numerical results are given. Model III consists of a single-server retrial queue with two primary sources and both a retrial queue and retrial orbits. Some results are obtained using matrix analytic methods. Also simulation results are obtained. Model IV consists of a single server system in which the retrial customers form a queue. The service times are discrete. A stability condition and performance measures are presented.Dept. of Mathematics and Statistics. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis2005 .W87. Source: Dissertation Abstracts International, Volume: 67-07, Section: B, page: 3883. Thesis (Ph.D.)--University of Windsor (Canada), 2006

Performance analysis of polling systems with retrials and glue periods

We consider gated polling systems with two special features: (i) retrials, and (ii) glue or reservation periods. When a type-$i$ customer arrives, or retries, during a glue period of station $i$, it will be served in the next visit period of the server to that station. Customers arriving at station $i$ in any other period join the orbit of that station and retry after an exponentially distributed time. Such polling systems can be used to study the performance of certain switches in optical communication systems. For the case of exponentially distributed glue periods, we present an algorithm to obtain the moments of the number of customers in each station. For generally distributed glue periods, we consider the distribution of the total workload in the system, using it to derive a pseudo conservation law which in its turn is used to obtain accurate approximations of the individual mean waiting times. We also consider the problem of choosing the lengths of the glue periods, under a constraint on the total glue period per cycle, so as to minimize a weighted sum of the mean waiting times