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A Duality Approach to Queues with Service Restrictions and Storage Systems with State-Dependent Rates

Part of: Operations research and management science Special processes

Published online by Cambridge University Press:  30 January 2018

D. Perry ,
W. Stadje  and
S. Zacks
D. Perry*
Affiliation:
University of Haifa
W. Stadje*
Affiliation:
University of Osnabrück
S. Zacks*
Affiliation:
Binghampton University
*
∗ Postal address: Department of Statistics, University of Haifa, 31905 Haifa, Israel. Email address: dperry@stat.haifa.ac.il
∗∗ Postal address: Fachbereich Mathematik/Informatik, University of Osnabrück, 49069 Osnabrück, Germany. Email address: wolfgang@mathematik.uni-osnabrueck.de
∗∗∗ Postal address: Department of Mathematical Sciences, Binghampton University, Binghampton, NY 13902-6000, USA. Email address: shelly@math.binghampton.edu
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Abstract

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Based on pathwise duality constructions, several new results on truncated queues and storage systems of the G/M/1 type are derived by transforming the workload (content) processes into certain ‘dual’ M/G/1-type processes. We consider queueing systems in which (a) any service requirement that would increase the total workload beyond the capacity is truncated so as to keep the associated sojourn time below a certain constant, or (b) new arrivals do not enter the system if they have to wait more than one time unit in line. For these systems, we derive the steady-state distributions of the workload and the numbers of customers present in the systems as well as the distributions of the lengths of busy and idle periods. Moreover, we use the duality approach to study finite capacity storage systems with general state-dependent outflow rates. Here our duality leads to a Markovian finite storage system with state-dependent jump sizes whose content level process can be analyzed using level crossing techniques. We also derive a connection between the steady-state densities of the non-Markovian continuous-time content level process of the G/M/1 finite storage system with state-dependent outflow rule and the corresponding embedded sequence of peak points (local maxima).

Keywords

Queue with service restrictionsstorage systemstate-dependent rateM/G/1G/M/1steady statedualitylevel crossingpeak point

MSC classification

Primary: 60K25: Queueing theory
Secondary: 90B22: Queues and service
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 3 , September 2013 , pp. 612 - 631
Copyright
© Applied Probability Trust 

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