Complutense University Library

The impact of self-generation of priorities on multi-server queues with finite capacity


Gómez-Corral, Antonio and Krishnamoorthy, A. and Narayanan, V.C. (2005) The impact of self-generation of priorities on multi-server queues with finite capacity. Stochastic Models, 21 . pp. 427-447. ISSN 1532-6349

[img] PDF
Restringido a Repository staff only hasta 2020.


Official URL:


This paper deals with multi-server queues with a finite buffer of size N in which units waiting for service generate into priority at a constant rate, independently of other units in the buffer. At the epoch of a unit's priority generation, the unit is immediately taken for service if there is one unit in service that did not generate into priority while waiting; otherwise such a unit leaves the system in search of immediate service elsewhere. The arrival stream Of units is a Markovian arrival process (MAP) and service requirements are of phase (PH) type. Our interest is in the continuous-time Markov chain describing the state of the queue at arbitrary times, which constitutes a finite quasi-birth-and-death (QBD) process. We give formulas well suited for numerical computation for a variety of performance measures, including the blocking probability, the departure process, and the stationary distributions of the system state at pre-arrival epochs, at post-departure epochs and at epochs at which arriving units are lost. Illustrative numerical examples show the effect of several parameters on certain probabilistic descriptors of the queue for various levels of congestion.

Item Type:Article
Uncontrolled Keywords:Finite-buffer; Markovian arrival process; Multi-server queue; Phase type distribution; Priority queue; Quasi-birth-and-death process.
Subjects:Sciences > Mathematics > Stochastic processes
ID Code:15624

Asmussen, S.; Koole, G. Marked point processes as limits of Markovian arrival streams. J. Appl.Probab. 1993, 30, 365–372.

Baccelli, F.; Hebuterne, G. On queues with impatient customers. In Performance’81; Kylstra, F.,Ed.; North-Holland: Amsterdam, 1981, 159–179.

Baccelli, F.; Boyer, P.; Hebuterne, G. Single-server queues with impatient customers. Adv. Appl.Probab. 1984, 16, 887–905.

Brahimi, M.; Worthington, D.J. Queueing models for out-patient appointment systems – A case study. J. Oper. Res. Soc. 1991, 42, 733–746.

Gail, H.R.; Hantler, S.L.; Taylor, B.A. Analysis of a non-preemptive priority multiserver queue.Adv. Appl. Probab. 1988, 20, 852–879.

Gaver, D.P.; Jacobs, P.A.; Latouche, G. Finite birth-and-death models in randomly changing environments. Adv. Appl. Probab. 1984, 16, 715–731.

Gómez-Corral, A. Analysis of a single-server retrial queue with quasi-random input and nonpreemptive priority. Comput. Math. Appl. 2002, 43, 767–782.

Graves, S.C. The application of queueing theory to continuous perishable inventory systems.Management Sci. 1982, 28, 400–406.

Gross, D.; Harris, C.M. Fundamentals of Queueing Theory, 3rd Ed.; John Wiley and Sons, Inc.:New York, 1998.

HE, Q.-M.; Li, H. Stability conditions of the MMAP[K]/G[K]/1/LCFS preemptive repeat queue.Queueing Syst. 2003, 44, 137–160.

Jaiswal, N.K. Priority Queues; Academic Press: New York, 1968.

Krieger, U.; Wagner, D. Analysis and application of an MMPP/PH/n/m multi-server model.In Performance of Distributed Systems and Integrated Communication Networks, IFIP Transaction C-5; Hasegawa, T., Takagi, H., Takahashi, Y. Eds.; North-Holland: Amsterdam, 1992, 323–342.

Langaris, C. Waiting time analysis of a two-stage queueing system with priorities. Queueing Syst.1993, 14, 457–473.

Langville, A.N.; Stewart, W.J. The Kronecker product and stochastic automata networks.J. Comput. Appl. Math. 2004, 167, 429–447.

Latouche, G.; Ramaswami, V. Introduction to Matrix Analytic Methods in Stochastic Modeling;ASA-SIAM: Philadelphia, 1999.

Leemans, H. Provable bounds for the mean queue lengths in a heterogenous priority queue.Queueing Syst. 2000, 36, 269–286.

Lucantoni, D.M.; Meier-Hellstern, K.S.; Neuts, M.F. A single-server queue with server vacations and a class of non-renewal arrival processes. Adv. in Appl. Probab. 1990, 22, 676–705.

Lucantoni, D.M. New results on the single server queue with a batch Markovian arrival process.Stochastic Models 1991, 7, 1–46.

Neuts, M.F. A versatile Markovian point process. J. Appl. Probab. 1979, 16, 764–779.

Neuts, M.F. Structured Stochastic Matrices of M/G/1 Type and Their Applications; Marcel Dekker, Inc.: New York, 1989.

Neuts, M.F. Matrix-Geometric Solutions in Stochastic Models. An Algorithmic Approach, 2nd Ed.; Dover Publications, Inc.: New York, 1994.

Perry, D. Analysis of a sampling control scheme for a perishable inventory system. Oper. Res.1999, 47, 966–973.

Stanford, D.A. Waiting and interdeparture times in priority queues with Poisson- and generalarrival streams. Oper. Res. 1997, 45, 725–735.

Takagi, H. Queueing Analysis. Volume 1: Vacations and Priority Systems; North-Holland: Amsterdam,1991.

Takine, T. The nonpreemptive priority MAP/G/1 queue. Oper. Res. 1999, 47, 917–927.

Taylor, I.D.S.; Templeton, J.G.C. Waiting time in a multi-server cutoff-priority queue, and its application to an urban ambulance service. Oper. Res. 1980, 28, 1168–1188.

Wagner, D. Analysis of a multi-server model with non-preemptive priorities and non-renewal input. In The Fundamental Role of Teletraffic in the Evolution of Telecommunication Networks, ITC-14,Labetoulle, J., Roberts, J.W. Eds.; Elsevier Science B.V.: Amsterdam, 1994; 757–766.

Wagner, D. Analysis of a finite capacity multiserver model with nonpreemptive priorities and nonrenewal input. In Matrix-Analytic Methods in Stochastic Models, Chakravarthy, S.R., Alfa, A.S.,Eds.; Lecture Notes in Pure and Applied Mathematics, Vol. 183; Marcel Dekker, Inc.: New York,1997; 67–86.

Wang, Q. Modeling and analysis of high risk patient queues. European J. Oper. Res. 2004, 155,502–515.

Zhao, Y.Q.; Alfa, A.S. Performance analysis of a telephone system with both patient and impatient customers. Telecommunication Systems 1995, 4, 201–215.

Zohar, E.; Mandelbaum, A.; Shimkin, N. Adaptive behavior of impatient customers in telequeues: Theory and empirical support. Management Sci. 2002, 48, 566–583.

Deposited On:14 Jun 2012 09:36
Last Modified:08 Mar 2016 16:06

Repository Staff Only: item control page