### Queues with advanced reservations: an infinite-server proxy for the bookings diary

#### Abstract

Queues with advanced reservations are endemic in the real world. In such a queue, the 'arrival' process is an incoming stream of customer 'booking requests', rather than actual customers requiring immediate service. We consider a model with a Poisson booking request process with rate λ. Associated with each request is a pair of independent random variables (Ri, Si) constituting a request for service over a period Si, starting at a time Ri into the future. Our interest is in the probability that a customer will be rejected due to capacity constraints. We present a simulation of a finite-capacity queue in which we record the proportion of rejected customers, and then move to an analysis of a queue with infinitely-many servers. Obviously no customers are rejected in the latter case. However, the event that the arrival of the extra customer will cause the number of customers in the queue to exceed C at some point during its service can be used as a proxy for the event that the customer would have been rejected in a system with finite capacity C. We start by calculating the transient and stationary distributions for some performance measures for the infinite-server queue. By observing that the stationary measure for the bookings diary (that is, the list of customers currently on hand, together with their start times and service times) is the same as the law for the entire sample path of an infinite server queue with a specified nonhomogenous Poisson input process, which we call the bookings queue, we are able to write down expressions for the abovementioned probability that, at some time during a requested service, the number of customers exceeds C. This measure serves as a bound for the probability that an incoming arrival would be refused admission in a system with C servers and, for a well-dimensioned system, it is to be hoped that it is a good approximation. We test the quality of this approximation by comparing our analytical results for the infinite-server case against simulation results for the finite-server case.

#### Article information

Source
Adv. in Appl. Probab., Volume 48, Number 1 (2016), 13-31.

Dates
First available in Project Euclid: 8 March 2016

https://projecteuclid.org/euclid.aap/1457466153

Mathematical Reviews number (MathSciNet)
MR3473565

Zentralblatt MATH identifier
1337.60237

#### Citation

Maillardet, R. J.; Taylor, P. G. Queues with advanced reservations: an infinite-server proxy for the bookings diary. Adv. in Appl. Probab. 48 (2016), no. 1, 13--31. https://projecteuclid.org/euclid.aap/1457466153

#### References

• Abate, J. and Whitt, W. (1995). Numerical inversion of Laplace transforms of probability distributions. ORSA J. Comput. 7, 36–43.
• Abate, J. and Whitt, W. (1998). Calculating transient characteristics of the Erlang loss model by numerical transform inversion. Commun. Statist. Stoch. Models 14, 663–680.
• Barakat, N. and Sargent, E. H. (2004). An accurate model for evaluating blocking probabilities in multi-class OBS systems. IEEE Commun. Lett. 8, 119–121.
• Barakat, N. and Sargent, E. H. (2005). Analytical modelling of offset-induced priority in multiclass OBS networks. IEEE Trans. Commun. 53, 1343–1352.
• Borovkov, K. (2003). Elements of Stochastic Modelling. World Scientific, River Edge, NJ.
• Chihara, T. S. (1978). An Introduction to Orthogonal Polynomials. Gordon and Breach, New York.
• Coffman, E. G. Jr., Flatto, L. and Jelenković, P. (2000). Interval packing: the vacant interval distribution. Ann. Appl. Prob. 10, 240–257.
• Coffman, E. G. Jr., Jelenković, P. and Poonen, B. (1999). Reservation probabilities. Adv. Performance Anal. 2, 129–158.
• Coffman, E. G. Jr., Flatto, L., Jelenković, P. and Poonen, B. (1998). Packing random intervals on-line. Algorithmica 22, 448–476.
• Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes, Vol. 1, Elementary Theory and Methods. Springer, New York.
• Dolzer, K. and Gauger, C. (2001). On burst assembly in optical burst switching networks–-a performance evaluation of just-enough-time. In Proceedings of the 17th International Teletraffic Congress, pp. 149–160.
• Foley, R. D. (1982). The nonhomogeneous $M/G/ \infty$ queue. Opsearch 19, 40–48.
• Greenberg, A. G., Srikant, R. and Whitt, W. (1999). Resource sharing for book-ahead and instantaneous-request calls. IEEE/ACM Trans. Networking 7, 10–22.
• Kaheel, A., Alnuweiri, H. and Gebali, F. (2004). Analytical evaluation of blocking probability in optical burst switching networks. In Proc. IEEE Internat. Conf. Commun., Vol. 3, pp. 1548–1553.
• Kaheel, A. M., Alnuweiri, H. and Gebali, F. (2006). A new analytical model for computing blocking probability in optical burst switching networks. IEEE J. Selected Areas Commun. 24, 120–128.
• Levi, R. and Shi, C. (2014). Revenue management of reusable resources with advanced reservations. Submitted.
• Liang, Y., Liao, K., Roberts, J. W. and Simonian, A. (1988). Queueing models for reserved set up telecommunications services. In Proc. Teletraffic Science for New Cost-Effective Systems, Networks and Services, Session 4.4B, 1.1–1.7.
• Ramakrishnan, M., Sier, D. and Taylor, P. G. (2005). A two-time-scale model for hospital patient flow. IMA J. Manag. Math. 16, 197–215.
• Riordan, J. (1962). Stochastic Service Systems. John Wiley, New York.
• Takács, L. (1962). Introduction to the Theory of Queues. Oxford University Press.
• Van de Vrugt, M., Litvak, N. and Boucherie, R. J. (2014). Blocking probabilities in Erlang loss queues with advance reservation. Stoch. Models 30, 187–196.
• Virtamo, J. T. (1992). A model of reservation systems. IEEE Trans. Commun. 40, 109–118.
• Vu, H. L. and Zukerman, M. (2002). Blocking probability for priority classes in optical burst switching networks. IEEE Commun. Lett. 6, 214–216.
• Wolff, R. W. (1989). Stochastic Modeling and the Theory of Queues. Prentice-Hall, Englewood Cliffs, NJ.