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54
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 ..."
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Cited by 38 (13 self)
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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
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 ..."
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Cited by 13 (6 self)
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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.
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 ..."
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Cited by 11 (8 self)
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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.
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 ..."
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Cited by 10 (1 self)
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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 ..."
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Cited by 7 (2 self)
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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...
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 ..."
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Cited by 5 (5 self)
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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.
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 ..."
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Cited by 4 (1 self)
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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.
Mean value analysis for polling systems in heavy traffic
 In Proceedings of ValueTools, Pisa Article
, 2006
"... In this paper we present a new approach to derive heavytraffic asymptotics for polling models. We consider the classical cyclic polling model with exhaustive service at each queue, and with general servicetime and switchover time distributions, and study its behavior when the load tends to one. F ..."
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Cited by 4 (2 self)
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In this paper we present a new approach to derive heavytraffic asymptotics for polling models. We consider the classical cyclic polling model with exhaustive service at each queue, and with general servicetime and switchover time distributions, and study its behavior when the load tends to one. For this model, we explore the recently proposed mean value analysis (MVA), which takes a new view on the dynamics of the system, and use this view to provide an alternative way to derive closedfrom expressions for the expected asymptotic delay; the expressions were derived earlier in [32], but in a different way. Moreover, the MVAbased approach enables us to derive closedform expressions for the heavytraffic limits of the covariances between the successive visit periods, which are key performance metrics in many application areas. These results, which have not been obtained before, reveal a number of insensitivity properties of the covariances with respect to the system parameters under heavytraffic assumptions, and moreover, lead to simple approximations for the covariances between the successive visit times for stable systems. Numerical examples demonstrate that the approximations are accurate when the load is close enough to one.
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 ..."
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Cited by 4 (4 self)
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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.
Continuous Polling Models and Application to Ferry Assisted WLAN
 ANNALS OF OPERATIONS RESEARCH
"... In this paper we focus on a class of polling systems encountered while modeling the ferry based wireless local area network (FWLAN). A moving ferry, while walking in a predetermined cyclic path, communicates with the static nodes (or users) of the network via a wireless link. The ferry is assumed t ..."
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Cited by 4 (2 self)
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In this paper we focus on a class of polling systems encountered while modeling the ferry based wireless local area network (FWLAN). A moving ferry, while walking in a predetermined cyclic path, communicates with the static nodes (or users) of the network via a wireless link. The ferry is assumed to stop and communicate with a node that has a packet to send or to receive, when it is closest to that node. The location distribution of the node to which or from which a packet arrives is assumed to have a support of positive Lebesgue measure. These features imply that polling models with finite number of queues cannot be used to model the system. We study in this paper the continuous polling systems with service disciplines that model the use of the FWLAN (and that are more complex than the classical exhaustive or gated services). Our approach is based on discretization of the continuous polling model. We propose a special way of discretizing the continuous system such that: 1) the known Pseudo conservation laws can be applied to obtain the stationary expected workload of the discrete systems; 2) the limit, of these ’discretized’ expected workloads, equals the stationary expected workload of the continuous system. Our results rely heavily on fixed point analysis of infinite dimensional operators.