Inward FDI in Ireland and its policy context

SIAM Journal on Applied Mathematics, 1992

Customers, in a Poisson stream at rate A, enter an infinite-server queue. Customer service times are independent and uniformly distributed on [0, s], s > 0. Gated service is performed in stages as follows. A stage begins with all customers transferred from the queue to the servers. The servers then begin serving these customers, all simultaneously. The stage ends when the service of all customers is complete. At this point, the next stage begins if the queue is nonempty. If the queue is empty, the servers just remain idle, awaiting the next arrival, at which time the next stage begins. This paper develops asymptotics for the equilibrium distribution of the number served in a stage in the light and heavy traffic regimes A --0 and A --oo, respectively. The results are obtained by the analysis of a Fredholm integral equation of the second kind. For computational purposes, the integral equation is transformed into an infinite system of linear algebraic equations. The effect of truncating the system to a finite size is then examined. Key words. gated service, infinite-server queue, Fredholm equation, asymptotics AMS(MOS) subject classifications. 41A60, 45B05, 60K25, 90B22 1. Introduction. Customers, in a Poisson stream at rate A, enter an infinite-server queue. Service is gated in that customers are served in stages, as follows. A stage begins with all customers transferred from the queue to the servers (the gate opens to admit waiting customers and then closes). The servers then begin serving these customers, all simultaneously. The stage ends when the service of all customers is complete. At this point, the next stage begins if the queue is nonempty. If the queue is empty, the servers just remain idle, awaiting the next arrival, at which time the next stage begins. Let k, denote the number of requests served during the nth stage. Customer service times are assumed to be independent, so {kI} is easily seen to be a Markov chain. This paper analyzes the behavior of {kJ} when service times are uniformly distributed on [0, s], s > 0. The normalization s = 1 is convenient and applies hereafter. By the analysis of a Fredholm integral equation of the second kind, asymptotics are developed for the mean, variance, and generating function of the equilibrium distribution in the light and heavy traffic regimes. In light traffic, when A is small, almost all stages serve only one customer. In heavy traffic, when A is large, stages serve many customers, so a stage duration (the maximum of the service times in the stage) is nearly one time unit. Then kn is approximately Poisson distributed with mean A. This paper derives asymptotic series that give higher-order corrections to these approximations. For other work on asymptotic solutions of Fredholm equations of the type considered here, see [5], [6], and [8]. A considerable literature exists on the analysis of gating disciplines restricted to single-server systems. Recent examples are [1] and. In [2] a gated infinite-server queue with vacations was studied. The infinite-server system "goes on vacation" whenever the server finds the queue empty on its return. While the model here is the special case without vacations and with uniformly distributed service times, the present analysis of {kn} is carried to greater depth. As noted in [2], there are many applications modeled by gated, parallel service. Among these are data transmission stations, in which servers are communication *

International journal of operations research, 2004

⎯We consider a queueing model with finite capacities. External arrivals follow a Coxian distribution. Due to the limitation of the capacity, arrivals may be lost if the buffer is full. Our goal is to study the probability of blocking. In order to obtain the steady-state probability distribution of this model, we construct an embedded Markov chain at the departure points. The solution is solved analytically and its analysis is extended to semi-Markovian representation of performance measures in queueing networks.

2016

This thesis includes three essays exploring some economic implications of queueing. A preliminary chapter introducing useful results from the literature which help contextualize the original research in the thesis is presented first. This introductory chapter starts by surveying queueing results from probability theory and operations research. Then it covers a few seminal papers on strategic queueing, mostly but not exclusively from the economics literature. These cover issues of individual and social welfare in the context of First Come First Served (FCFS) and Equitable Processor Sharing (EPS) queues, with one or multiple servers, as well as a discussion of strategic interactions surrounding queue cutting. Then an overview of some important papers on the impact of queueing on competitive behaviour, mostly Industrial Organization economists, is presented. The first original chapter presents a model for the endogenous determination of the number of queues in an M/M/2 system. Customer...

Proceedings of the 11th EAI International Conference on Performance Evaluation Methodologies and Tools

Routing jobs to parallel servers is a common and important task in today's computer systems. Join-the-shortest-queue (JSQ) routing minimizes the mean response time under rather general settings as long as the servers are identical and service times are independent and exponentially distributed. Apart from this, surprisingly few optimality results exist, mainly due to the complexities arising from the infinite state spaces. Indeed, it is difficult to analyze the performance of any given routing policy. In this paper, we consider an elementary job routing problem with heterogeneous servers and general cost structures. By a novel approximation, we reduce the state space to finite size, which enables us to estimate the mean performance, and to determine (practically) optimal routing policies, for a large class of cost structures. We demonstrate the approximation and its application to job routing policy optimization in numerical examples. CCS CONCEPTS • Mathematics of computing → Queueing theory; • Theory of computation → Markov decision processes; Routing and network design problems; • Information systems → Data centers;

Probability in the Engineering and Informational Sciences, 1992

We study a generalization of the GI/G/l queue in which the server is turned off at the end of each busy period and is reactivated only when the sum of the service times of all waiting customers exceeds a given threshold of size D. Using the concept of a “randomly selected” arriving customer, we obtain as our main result a relation that expresses the waiting-time distribution of customers in this model in terms of characteristics associated with a corresponding standard GI/G/1 queue, obtained by setting D = 0. If either the arrival process is Poisson or the service times are exponentially distributed, then this representation of the waiting-time distribution can be specialized to yield explicit, transform-free formulas; we also derive, in both of these cases, the expected customer waiting times. Our results are potentially useful, for example, for studying optimization models in which the threshold D can be controlled.

Probability in the Engineering and …, 1990

Queueing Systems, 2020

In recent years a number of models involving different compatibilities between jobs and servers in queueing systems, or between agents and resources in matching systems, have been studied, and, under Markov assumptions and appropriate stability conditions, the stationary distributions have been shown to have product forms. We survey these results and show how, under an appropriate detailed description of the state, many are corollaries of similar results for the Order Independent Queue. We also discuss how to use the product form results to determine distributions for steady-state response times.

Queueing Systems, 2008

We consider an infinite-server queue, where the arrival and service rates are both governed by a semi-Markov process that is independent of all other aspects of the queue. In particular, we derive a system of equations that are satisfied by various "parts" of the generating function of the steady-state queue-length, while assuming that all arrivals bring an amount of work to the system that is either Erlang or hyperexponentially distributed. These equations are then used to show how to derive all moments of the steady-state queue-length. We then conclude by showing how these results can be slightly extended, and used, along with a transient version of Little's law, to generate rigorous approximations of the steady-state queue-length in the case that the amount of work brought by a given arrival is of an arbitrary distribution.

Performance Evaluation, 2017

We consider a discrete-time queueing system having two distinct servers: one server, the "regular" server, is permanently available, while the second server, referred to as the "extra" server, is only allocated to the system intermittently. Apart from their availability, the two servers are identical, in the sense that the customers have deterministic service times equal to 1 fixed-length time slot each, regardless of the server that processes them. In this paper, we assume that the extra server is available during random "up-periods", whereas it is unavailable during random "down-periods". Up-periods and down-periods occur alternately on the time axis. The up-periods have geometrically distributed lengths (expressed in time slots), whereas the distribution of the lengths of the down-periods is general, at least in the first instance. Customers enter the system according to a general independent arrival process, i.e., the numbers of arrivals during consecutive time slots are i.i.d. random variables with arbitrary distribution. For this queueing model, we are able to derive closed-form expressions for the steadystate probability generating functions (pgfs) and the expected values of the numbers of customers in the system at various observation epochs, such as the start of an up-period, the start of a down-period and the beginning of an arbitrary time slot. At first sight, these formulas, however, appear to contain an infinite number of unknown constants. One major issue of the mathematical analysis turns out to be the determination of these constants. In the paper, we show that restricting the pgf of the down-periods to be a rational function of its argument, brings about the crucial simplification that the original infinite number of unknown constants appearing in the formulas can be expressed in terms of a finite number of independent unknowns. The latter can then be adequately determined based on the bounded nature of pgfs inside the complex unit disk, and an extensive use of properties of polynomials. Various special cases, both from the perspective of the arrival distribution and the down-period distribution, are discussed. The results are also illustrated by means of relevant numerical examples. Possible applications of this type of queueing model are numerous: the extra server could be the regular server of another similar queue, helping whenever an idle period occurs in its own queue; a geometric distribution for these idle times is then a very natural modelling assumption. A typical example would be the situation at the check-in counter at a gate in an airport: the regular server serves customers with a low-fare ticket, while the extra server gives priority to the business-class and first-class customers, but helps checking regular customers, whenever the priority line is empty.

Performance Evaluation, 1990

A discrete-time single server FIFO queue with Bernoulli arrivals and deterministic service times is studied. Both the finite capacity queue and the infinite capacity queue are investigated. The state of the system is observed at entry time and at the end of an arbitrary time slot. The closed-form expressions for the steady-state distributions of the queue length and of the unfinished work in system ( i.e. waiting time ) are obtained. The method of analysis, using Lindley's equation, naturally arises in case of deterministic service times