Results 1  10
of
69
Useful martingales for stochastic storage processes with Lévy input
 J. Appl. Probab
, 1992
"... In this paper we generalize the martingale of Kella and Whitt to the setting of Lévytype processes and show that under some quite minimal conditions the local martingales are actually L 2 martingales which upon dividing by the time index converge to zero a.s. and in L 2. We apply these results to g ..."
Abstract

Cited by 49 (14 self)
 Add to MetaCart
(Show Context)
In this paper we generalize the martingale of Kella and Whitt to the setting of Lévytype processes and show that under some quite minimal conditions the local martingales are actually L 2 martingales which upon dividing by the time index converge to zero a.s. and in L 2. We apply these results to generalize known decomposition results for Lévy queues with secondary jump inputs and queues with server vacations or service interruptions. Special cases are polling systems with either compound Poisson or more general Lévy inputs. Keywords: Lévytype processes, Lévy storage systems, KellaWhitt martingale, decomposition results, queues with server vacations
Polling Systems with Synchronization Constraints
, 1992
"... We introduce a new service discipline, called the synchronized gated discipline, for polling systems. It arises when there are precedence (or synchronization) constraints between the order that jobs in different queues should be served. These constraints are described as follows: There are N station ..."
Abstract

Cited by 17 (10 self)
 Add to MetaCart
(Show Context)
We introduce a new service discipline, called the synchronized gated discipline, for polling systems. It arises when there are precedence (or synchronization) constraints between the order that jobs in different queues should be served. These constraints are described as follows: There are N stations which are "fathers" of (zero or more) synchronized stations ("children"). Jobs that arrive at synchronized stations have to be processed only after jobs that arrived prior to them at their corresponding "father" station have been processed. We analyze the performance of the synchronized gated discipline and obtain expressions for the first two moments and the LaplaceStieltjes transform (LST) of the waiting times in different stations, and expressions for the moments and LST of other quantities of interest, such as cycle duration and generalized station times. We also obtain a "pseudo" conservation law for the synchronized gated discipline, and determine the optimal network topology that minimizes the weighted sum of the mean waiting times, as defined in the "pseudo" conservation law. Numerical examples are given for illustrating the dependence of the performance of the synchronized gated discipline on different parameters of the network.
Queueing in Space: design of Message Ferry Routes in static adhoc networks
"... Abstract—We study the concept of Ferry based Wireless Local Area Network (FWLAN), in which a number of isolated nodes are scattered over some area and where communication between a node and the outer world, or communication between the nodes, are made possible via a message ferry. The Ferry has a pr ..."
Abstract

Cited by 15 (7 self)
 Add to MetaCart
(Show Context)
Abstract—We study the concept of Ferry based Wireless Local Area Network (FWLAN), in which a number of isolated nodes are scattered over some area and where communication between a node and the outer world, or communication between the nodes, are made possible via a message ferry. The Ferry has a predetermined cyclic path which collects packets from a node and delivers packets to it when it is in the vicinity of the node. We use the mathematical theory of polling systems to study the performance of the FWLAN. We consider three different architectures and each of them is mapped to a polling model. The polling disciplines that are needed for modeling the FWLAN involve nonstandard variants of gating disciplines. Our goal is to design the routes of the Ferry as well as the points where it should stop to distribute and collect packets. This mathematical modeling brings another dimension to the classical related vehicle routing problem due to the radio channel: the cyclic path of the ferry need not touch every node. The distance between the node and the fairy at the point when communication occurs determines the transmission rate and hence the service time and thus the system’s capacity. I.
Singleserver queues with spatially distributed arrivals
 University of Twente/University of Ulm
, 1993
"... Consider a queueing system where customers arrive at a circle according to a homogeneous Poisson process. After choosing their positions on the circle, according to a uniform distribution, they wait for a single server who travels on the circle. The server’s movement is modelled by a Brownian motion ..."
Abstract

Cited by 14 (1 self)
 Add to MetaCart
(Show Context)
Consider a queueing system where customers arrive at a circle according to a homogeneous Poisson process. After choosing their positions on the circle, according to a uniform distribution, they wait for a single server who travels on the circle. The server’s movement is modelled by a Brownian motion with drift. Whenever the server encounters a customer, he stops and serves this customer. The service times are independent, but arbitrarily distributed. The model generalizes the continuous cyclic polling system (the diffusion coefficient of the Brownian motion is zero in this case) and can be interpreted as a continuous version of a Markov polling system. Using Tweedie’s lemma for positive recurrence of Markov chains with general state space, we show that the system is stable if and only if the traffic intensity is less than one. Moreover, we derive a stochastic decomposition result which leads to equilibrium equations for the stationary configuration of customers on the circle. Steadystate performance characteristics are determined, in particular the expected number of customers in the system as seen by a travelling server and at an arbitrary point in time.
Dynamic Server Assignment in a TwoQueue Model
 European Journal of Operational Research
, 1997
"... We consider a polling model of two M=G=1 queues, served by a single server. The service policy for this polling model is of threshold type. Service at queue 1 is exhaustive. Service at queue 2 is exhaustive unless the size of queue 1 reaches some level T during a service at queue 2; in the latte ..."
Abstract

Cited by 12 (2 self)
 Add to MetaCart
(Show Context)
We consider a polling model of two M=G=1 queues, served by a single server. The service policy for this polling model is of threshold type. Service at queue 1 is exhaustive. Service at queue 2 is exhaustive unless the size of queue 1 reaches some level T during a service at queue 2; in the latter case the server switches to queue 1 at the end of that service. Both zero and nonzero switchover times are considered. We derive exact expressions for the joint queue length distribution at customer departure epochs, and for the steadystate queuelength and sojourn time distributions. In addition, we supply a simple and very accurate approximation for the mean queue lengths, which is suitable for optimization purposes. AMS Subject Classification (1991): Primary: 60K25, Secondary: 90B22 Keywords & Phrases: Queueing, polling, ATM, threshold service, queue length distribution. 1 Introduction In this paper we consider a model of two M=G=1 queues, which are served by a single serve...
A Polling Model with Multiple Priority Levels
 Performance Evaluation
"... Version unknown ..."
(Show Context)
Closedform waiting time approximations for polling systems
 Performance Evaluation
"... A typical polling system consists of a number of queues, attended by a single server in a fixed order. The vast majority of papers on polling systems focusses on Poisson arrivals, whereas very few results are available for general arrivals. The current study is the first one presenting simple closed ..."
Abstract

Cited by 7 (3 self)
 Add to MetaCart
(Show Context)
A typical polling system consists of a number of queues, attended by a single server in a fixed order. The vast majority of papers on polling systems focusses on Poisson arrivals, whereas very few results are available for general arrivals. The current study is the first one presenting simple closedform approximations for the mean waiting times in polling systems with renewal arrival processes, performing well for all workloads. The approximations are constructed using heavy traffic limits and newly developed light traffic limits. The closedform approximations may prove to be extremely useful for system design and optimisation in application areas as diverse as telecommunication, maintenance, manufacturing and transportation.
Analysis and Design of Message Ferry Routes in Sensor Networks using Polling Models
"... We consider a Ferry based Wireless Local Area Network (FWLAN), in which information is forwarded from a base station to sensors, or gathered from sensors to a base station using a moving Ferry. The sensors are scattered in a large area and do not have direct radio connectivity with the base station ..."
Abstract

Cited by 6 (6 self)
 Add to MetaCart
(Show Context)
We consider a Ferry based Wireless Local Area Network (FWLAN), in which information is forwarded from a base station to sensors, or gathered from sensors to a base station using a moving Ferry. The sensors are scattered in a large area and do not have direct radio connectivity with the base station. The ferry thus serves as a relay that enables communication between the sensors and the base station. Our goal in this paper is to design optimal routes of the Ferry moving along which it distributes/collects the messages. Our analysis and optimization results build heavily on the theory of polling systems which we extend here in order to handle the case of continuous location of the demand. We derive optimal trajectories for various scenarios: uplink, downlink and their combination. We extend some of these results to the case of several base stations and several ferries.
Continuous Polling with Rerouting and Applications to Ferry Assisted Wireless LANs
"... In almost all studied continuous polling systems, the user leaves the system after his service is completed. There are interesting applications, in which the users demand a second service (or more). For example, in a ferry assisted wireless network, for every local data transfer the ferry has to col ..."
Abstract

Cited by 5 (5 self)
 Add to MetaCart
(Show Context)
In almost all studied continuous polling systems, the user leaves the system after his service is completed. There are interesting applications, in which the users demand a second service (or more). For example, in a ferry assisted wireless network, for every local data transfer the ferry has to collect the data from the source and then deliver the same to the sink. This type of application can be modeled by polling systems with rerouting. In polling systems with arrivals on a continuum (on a circle), a moving server attends the users as and when it encounters one. When rerouting is supported, after the service is completed, the users can reroute to a different point in the same circle to await another service. We obtain the performance of such a system under quite general conditions, via discretization approach. The results are applied to study a ferry assisted wireless local area network. Our results rely heavily on fixed point analysis of infinite dimensional operators.
When Does Forced Idle Time Improve Performance in Polling Models?
 in Polling models? Management Science 44
, 1998
"... Sarkar and Zangwill (1991) showed by numerical examples that reduction in setup times can, surprisingly, actually increase work in process in some cyclic production systems (that is, reduction in switchover times can increase waiting times in some polling models). We present, for polling models with ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
(Show Context)
Sarkar and Zangwill (1991) showed by numerical examples that reduction in setup times can, surprisingly, actually increase work in process in some cyclic production systems (that is, reduction in switchover times can increase waiting times in some polling models). We present, for polling models with exhaustive and gated service disciplines, some explicit formulas that provide additional insight and characterization of this anomaly. More specifically, we show that, for both of these models, there exist simple formulas that define for each queue a critical value z * of the mean total setup time z per cycle such that, if z õ z*, then the expected waiting time at that queue will be minimized if the server is forced to idle for a constant length of time z * 0 z every cycle; also, for the symmetric polling model, we give a simple explicit formula for the expected waiting time and the critical value z * that minimizes it.