Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-28T12:42:35.818Z Has data issue: false hasContentIssue false
  • English
  • Français

Tollbooth tandem queues with infinite homogeneous servers

Part of: Operations research and management science Special processes

Published online by Cambridge University Press:  30 March 2016

Xiuli Chao ,
Qi-Ming He  and
Sheldon Ross
Xiuli Chao*
Affiliation:
University of Michigan
Qi-Ming He*
Affiliation:
University of Waterloo
Sheldon Ross*
Affiliation:
University of Southern California
*
∗Postal address: Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor, MI 48109, USA. Email address: xchao@umich.edu
∗∗Postal address: Department of Management Sciences, University of Waterloo, 200 University Avenue West, Waterloo, ON N2L 3G1, Canada. Email address: q7he@uwaterloo.ca
∗∗∗Postal address: Daniel J. Epstein Department of Industrial and Systems Engineering, University of Southern California, Los Angeles, CA 90089, USA. Email address: smross@usc.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.

In this paper we analyze a tollbooth tandem queueing problem with an infinite number of servers. A customer starts service immediately upon arrival but cannot leave the system before all customers who arrived before him/her have left, i.e. customers depart the system in the same order as they arrive. Distributions of the total number of customers in the system, the number of departure-delayed customers in the system, and the number of customers in service at time t are obtained in closed form. Distributions of the sojourn times and departure delays of customers are also obtained explicitly. Both transient and steady state solutions are derived first for Poisson arrivals, and then extended to cases with batch Poisson and nonstationary Poisson arrival processes. Finally, we report several stochastic ordering results on how system performance measures are affected by arrival and service processes.

Keywords

Tollbooth tandem queuedeparture delaydeparture-delayed customer

MSC classification

Primary: 60K25: Queueing theory
Secondary: 90B22: Queues and service
Type
Research Papers
Information
Journal of Applied Probability , Volume 52 , Issue 4 , December 2015 , pp. 941 - 961
Copyright
Copyright © Applied Probability Trust 2015 

References

Daskin, M. S., Shladover, S. and Sobel, K. (1976). An analysis of service station queues under gasoline shortage conditions. Comput. Operat. Res. 3, 83–93.Google Scholar
Eick, S. G., Massey, W. A. and Whitt, W. (1993). M(t)/G/8 queues with sinusoidal arrival rates. Manag. Sci. 39, 241–252.CrossRefGoogle Scholar
Eliazar, I. (2007). The M/G/8 system revisited: finiteness, summability, long range dependence, and reverse engineering. Queueing Systems 55, 71–82.Google Scholar
Hall, R. W. and Daganzo, C. F. (1983). Tandem tollbooths for the Golden Gate Bridge. Transportation Res. Rec. 905, 7–14.Google Scholar
He, Q-M. and Chao, X. (2014). A tollbooth tandem queue with heterogeneous servers. Europ. J. Operat. Res. 236, 177–189.Google Scholar
Holman, D. F., Chaudhry, M. L. and Kashyap, B. R. K. (1983). On the service system MX/G/8. Europ. J. Operat. Res. 13, 142–145.Google Scholar
Hong, Y-C. et al. (2009). Modeling and simulation of tandem tollbooth operations with max-algebra approach. In Future Generation Information Technology , Springer, Berlin, pp. 138–150.Google Scholar
Massey, W. A. and Whitt, W. (1993). Networks of infinite-server queues with nonstationary Poisson input. Queueing Systems 13, 183–250.Google Scholar
Papadopoulos, H. T. and O'Kelly, M. E. J. (1993). Exact analysis of production lines with no intermediate buffers. Europ. J. Operat. Res. 65, 118–137.CrossRefGoogle Scholar
Ramaswami, V. and Neuts, M. F. (1980). Some explicit formulas and computational methods for infinite-server queues with phase-type arrivals. J. Appl. Prob. 17, 498–514.Google Scholar
Reuveni, S., Eliazar, I. and Yechiali, U. (2012). Asymmetric inclusion process as a showcase of complexity. Phys. Rev. Lett. 109, 020603.Google Scholar
Reuveni, S., Hirschberg, O., Eliazar, I. and Yechiali, U. (2014). Occupation probabilities and fluctuations in the asymmetric simple inclusion process. Phys. Rev. E 89, 042109.Google Scholar
Ross, S. M. (1996). Stochastic Processes , 2nd edn. John Wiley, New York.Google Scholar
Ross, S. M. (2010). Introduction to Probability Models , 10th edn. Academic Press, Boston, MA.Google Scholar
Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York.CrossRefGoogle Scholar
Shanbhag, D. N. (1966). On infinite server queues with batch arrivals. J. Appl. Prob. 3, 274–279.Google Scholar
Teimoury, E., Yazdi, M. M., Haddadi, M. and Fathi, M. (2011). Modelling and improvement of non-standard queuing systems: a gas station case study. Internat. J. Appl. Decision Sci. 4, 324–340.Google Scholar
Zazanis, M. A. (2004). Infinite server queues with synchronized departures driven by a single point process. Queueing Systems 48, 309–338.Google Scholar