Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-10T16:09:20.106Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Special processes Operations research and management science

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 

References

Asmussen, S. (2003). Applied Probability and Queues, 2nd edn. Springer, New York.Google Scholar
Bekker, R. (2005). Finite-buffer queues with workload-dependent service and arrival rates. Queueing Systems 50, 231–253.Google Scholar
Bekker, R. and Zwart, B. (2005). On an equivalence between loss rates and cycle maxima in queues and dams. Prob. Eng. Inf. Sci. 19, 241–255.Google Scholar
Bekker, R., Borst, S. C., Boxma, O. J. and Kella, O. (2004). Queues with workload-dependent arrival and service rates. Queueing Systems 46, 537–556.Google Scholar
Boxma, O., Perry, D., Stadje, W. and Zacks, S. (2009). The M/G/1 queue with quasi-restricted accessibility. Stoch. Models 25, 151–196.Google Scholar
Brandt, A. and Brandt, M. (2002). Asymptotic results and a Markovian approximation for the M(n)/M(n)/s + GI system. Queueing Systems 41, 73–94.Google Scholar
Brill, P. H. (2008). Level Crossing Methods in Stochastic Models. Springer, New York.Google Scholar
Chen, H. and Yao, D. D. (1992). A fluid model for systems with random disruptions. Operat. Res. S40, S239–S247.Google Scholar
Cohen, J. W. (1969). Single server queues with restricted accessibility. J. Eng. Math. 3, 265–285.Google Scholar
Cohen, J. W. (1982). The Single Server Queue, 2nd edn. North Holland, Amsterdam.Google Scholar
Conolly, B. W., Parthasarathy, P. R. and Selvaraju, N. (2002). Double-ended queues with impatience. Comput. Operat. Res. 29, 2053–2072.Google Scholar
Daley, D. J. (1964). Single-server queueing systems with uniformly limited queueing times. J. Austral. Math. Soc. 4, 489–505.Google Scholar
De Kok, A. G. and Tijms, H. C. (1985). A queueing system with impatient customers. J. Appl. Prob. 22, 688–696.Google Scholar
Gavish, B. and Schweitzer, P. J. (1977). The Markovian queue with bounded waiting time. Manag. Sci. 23, 1349–1357.Google Scholar
Harrison, J. M. and Resnick, S. I. (1976). The stationary distribution and first exit probabilities of a storage process with general release rule. Math. Operat. Res. 1, 347–358.CrossRefGoogle Scholar
Kaspi, H. and Perry, D. (1989). On a duality between a non-Markov storage/production process and a Markovian dam process with state dependent input and output. J. Appl. Prob. 26, 835–844.Google Scholar
Kaspi, H., Kella, O. and Perry, D. (1996). Dam processes with state dependent batch sizes and intermittent production processes with state dependent rates. Queueing Systems 24, 37–57.CrossRefGoogle Scholar
Kella, O. and Whitt, W. (1992). Useful martingales for stochastic storage processes with Lévy input. J. Appl. Prob. 29, 396–403.Google Scholar
Liu, L. and Kulkarni, V. G. (2006). Explicit solutions for the steady state distributions in M/PH/1 queues with workload dependent balking. Queueing Systems 52, 251–260.Google Scholar
Liu, L. and Kulkarni, V. G. (2008). Busy period analysis for M/PH/1 queues with workload dependent balking. Queueing Systems 59, 37–51.Google Scholar
Löpker, A. and Perry, D. (2010). The idle period of the finite G/M/1 queue with an interpretation in risk theory. Queueing Systems 64, 395–407.Google Scholar
Nahmias, S., Perry, D. and Stadje, W. (2004). Actuarial valuation of perishable inventory systems. Prob. Eng. Inf. Sci. 18, 219–232.Google Scholar
Perry, D. and Asmussen, S. (1995). Rejection rules in the M/G/1 type queue. Queueing Systems 19, 105–130.Google Scholar
Perry, D. and Stadje, W. (2003). Duality of dams via mountain processes. Operat. Res. Lett. 31, 451–458.Google Scholar
Perry, D., Stadje, W. and Zacks, S. (2000). Busy period analysis for M/G/1 and G/M/1 type queues with restricted accessibility. Operat. Res. Lett. 27, 163–174.Google Scholar
Stanford, R. E. (1990). On queues with impatience. Adv. Appl. Prob. 22, 768–769.Google Scholar
Zwart, A. P. (2006). A fluid queue with a finite buffer and subexponential input. Adv. Appl. Prob. 32, 221–243.Google Scholar