Journal of Applied Probability

Tollbooth tandem queues with infinite homogeneous servers

Xiuli Chao, Qi-Ming He, and Sheldon Ross

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Abstract

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 non\-stationary Poisson arrival processes. Finally, we report several stochastic ordering results on how system performance measures are affected by arrival and service processes.

Article information

Source
J. Appl. Probab., Volume 52, Number 4 (2015), 941-961.

Dates
First available in Project Euclid: 22 December 2015

Permanent link to this document
https://projecteuclid.org/euclid.jap/1450802745

Digital Object Identifier
doi:10.1239/jap/1450802745

Mathematical Reviews number (MathSciNet)
MR3439164

Zentralblatt MATH identifier
1334.60192

Subjects
Primary: 60K25: Queueing theory [See also 68M20, 90B22]
Secondary: 90B22: Queues and service [See also 60K25, 68M20]

Keywords
Tollbooth tandem queue departure delay departure-delayed customer

Citation

Chao, Xiuli; He, Qi-Ming; Ross, Sheldon. Tollbooth tandem queues with infinite homogeneous servers. J. Appl. Probab. 52 (2015), no. 4, 941--961. doi:10.1239/jap/1450802745. https://projecteuclid.org/euclid.jap/1450802745


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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.
  • Eick, S. G., Massey, W. A. and Whitt, W. (1993). $M(t)/G/\infty$ queues with sinusoidal arrival rates. Manag. Sci. 39, 241–252.
  • Eliazar, I. (2007). The M/G/$\infty$ system revisited: finiteness, summability, long range dependence, and reverse engineering. Queueing Systems 55, 71–82.
  • Hall, R. W. and Daganzo, C. F. (1983). Tandem tollbooths for the Golden Gate Bridge. Transportation Res. Rec. 905, 7–14.
  • He, Q-M. and Chao, X. (2014). A tollbooth tandem queue with heterogeneous servers. Europ. J. Operat. Res. 236, 177–189.
  • Holman, D. F., Chaudhry, M. L. and Kashyap, B. R. K. (1983). On the service system $M^X/G/\infty$. Europ. J. Operat. Res. 13, 142-145.
  • 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.
  • Massey, W. A. and Whitt, W. (1993). Networks of infinite-server queues with nonstationary Poisson input. Queueing Systems 13, 183–250.
  • 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.
  • 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.
  • Reuveni, S., Eliazar, I. and Yechiali, U. (2012). Asymmetric inclusion process as a showcase of complexity. Phys. Rev. Lett. 109, 020603.
  • 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.
  • Ross, S. M. (1996). Stochastic Processes, 2nd edn. John Wiley, New York.
  • Ross, S. M. (2010). Introduction to Probability Models, 10th edn. Academic Press, Boston, MA.
  • Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York.
  • Shanbhag, D. N. (1966). On infinite server queues with batch arrivals. J. Appl. Prob. 3, 274–279.
  • 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.
  • Zazanis, M. A. (2004). Infinite server queues with synchronized departures driven by a single point process. Queueing Systems 48, 309–338. \endharvreferences