Large Closed Queueing Networks in SemiMarkov Environment and Their Application
 Vyacheslav M. Abramov
 … show all 1 hide
Purchase on Springer.com
$39.95 / €34.95 / £29.95*
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Abstract
The paper studies closed queueing networks containing a server station and k client stations. The server station is an infinite server queueing system, and client stations are singleserver queueing systems with autonomous service, i.e. every client station serves customers (units) only at random instants generated by a strictly stationary and ergodic sequence of random variables. The total number of units in the network is N. The expected times between departures in client stations are (N μ _{ j })^{−1}. After a service completion in the server station, a unit is transmitted to the jth client station with probability p _{ j } (j=1,2,…,k), and being processed in the jth client station, the unit returns to the server station. The network is assumed to be in a semiMarkov environment. A semiMarkov environment is defined by a finite or countable infinite Markov chain and by sequences of independent and identically distributed random variables. Then the routing probabilities p _{ j } (j=1,2,…,k) and transmission rates (which are expressed via parameters of the network) depend on a Markov state of the environment. The paper studies the queuelength processes in client stations of this network and is aimed to the analysis of performance measures associated with this network. The questions risen in this paper have immediate relation to quality control of complex telecommunication networks, and the obtained results are expected to lead to the solutions to many practical problems of this area of research.
 Abramov, V.M.: A large closed queueing network with autonomous service and bottleneck. Queueing Syst. 35, 23–54 (2000) CrossRef
 Abramov, V.M.: Some results for large closed queueing networks with and without bottleneck: Up and downcrossings approach. Queueing Syst. 38, 149–184 (2001) CrossRef
 Abramov, V.M.: A large closed queueing network containing two types of node and multiple customers classes: One bottleneck station. Queueing Syst. 48, 45–73 (2004) CrossRef
 Abramov, V.M.: The stability of jointheshortestqueue models with general input and output processes. arXiv: math/PR 0505040 (2005)
 Abramov, V.M.: The effective bandwidth problem revisited. arXiv: math/PR 0604182 (2006)
 Abramov, V.M.: Confidence intervals associated with performance analysis of symmetric large closed client/server computer networks. Reliab. Theory Appl. 2(2), 35–42 (2007)
 Abramov, V.M.: Further analysis of confidence intervals for large client/server computer networks. Reliab. Theory Appl. (2007, to appear)
 Anulova, S.V., Liptser, R.S.: Diffusion approximation for processes with normal reflection. Theory Probab. Appl. 35, 413–423 (1990)
 Baccelli, F., Makovsky, A.M.: Stability and bounds for singleserver queue in a random environment. Stoch. Models 2, 281–292 (1986) CrossRef
 Berger, A., Bregman, L., Kogan, Y.: Bottleneck analysis in multiclass closed queueing networks and its application. Queueing Syst. 31, 217–237 (1999) CrossRef
 Borovkov, A.A.: Stochastic Processes in Queueing Theory. Springer, Berlin (1976)
 Borovkov, A.A.: Asymptotic Methods in Queueing Theory. Wiley, New York (1984)
 Boxma, O.J., Kurkova, I.A.: The M/M/1 queue in heavytailed random environment. Stat. Neerl. 54, 221–236 (2000) CrossRef
 Dellacherie, C.: Capacités et Processus Stochastiques. Springer, Berlin (1972)
 D’Auria, B.: M/M/∞ queues in quasiMarkovian random environment. arXiv: math/PR 0701842 (2007)
 Fricker, C.: Etude d’une file GI/G/1 á service autonome (avec vacances du serveur). Adv. Appl. Probab. 18, 283–286 (1986) CrossRef
 Fricker, C.: Note sur un modele de file GI/G/1 á service autonomé (avec vacances du serveur). Adv. Appl. Probab. 19, 289–291 (1987) CrossRef
 Gelenbe, E., Iasnogorodski, R.: A queue with server of walking type (autonomous service). Ann. Inst. H. Poincare 16, 63–73 (1980)
 Helm, W.E., Waldmann, K.H.: Optimal control of arrivals to multiserver queues in a random environment. J. Appl. Probab. 21, 602–615 (1984) CrossRef
 Kalmykov, G.I.: On the partial ordering of onedimensional Markov processes. Theory Probab. Appl. 7, 456–459 (1962) CrossRef
 Kogan, Y.: Another approach to asymptotic expansions for large closed queueing networks. Oper. Res. Lett. 11, 317–321 (1992) CrossRef
 Kogan, Y., Liptser, R.S.: Limit nonstationary behavior of large closed queueing networks with bottlenecks. Queueing Syst. 14, 33–55 (1993) CrossRef
 Kogan, Y., Liptser, R.S., Smorodinskii, A.V.: Gaussian diffusion approximation of a closed Markov model of computer networks. Prob. Inf. Transm. 22, 38–51 (1986)
 Krichagina, E.V., Liptser, R.S., Puhalskii, A.A.: Diffusion approximation for a system that an arrival stream depends on queue and with arbitrary service. Theory Probab. Appl. 33, 114–124 (1988) CrossRef
 Krichagina, E.V., Puhalskii, A.A.: A heavytraffic analysis of closed queueing system with GI/∞ server. Queueing Syst. 25, 235–280 (1997) CrossRef
 Krieger, U., Klimenok, V.I., Kazmirsky, A.V., Breuer, L., Dudin, A.N.: A BMAP/PH/1 queue with feedback operating in a random environment. Math. Comput. Model. 41, 867–882 (2005) CrossRef
 Liptser, R.S., Shiryayev, A.N.: Theory of Martingales. Kluwer, Dordrecht (1989)
 McKenna, J., Mitra, D.: Integral representation and asymptotic expansions for closed Markovian queueing networks. Normal usage. Bell Syst. Tech. J. 61, 661–683 (1982)
 O’Cinneide, C., Purdue, P.: The M/M/∞ queue in a random environment. J. App. Probab. 23, 175–184 (1986) CrossRef
 Pittel, B.: Closed exponential networks of queues with saturation: The Jackson type stationary distribution and its asymptotic analysis. Math. Oper. Res. 6, 357–378 (1979)
 Purdue, P.R.: The M/M/1 queue in a random environment. Oper. Res. 22, 562–569 (1974) CrossRef
 Ramanan, K.: Reflected diffusions defined via extended Skorokhod map. Electron. J. Probab. 11, 934–992 (2006)
 Skorokhod, A.V.: Stochastic equations for diffusion processes in a bounded region. Theory Probab. Appl. 6, 264–274 (1961) CrossRef
 Tanaka, H.: Stochastic differential equations with reflected boundary conditions in convex regions. Hiroshima Math. J. 9, 163–177 (1979)
 Whitt, W.: Open and closed models for networks of queues. AT&T Bell Lab. Tech. J. 63, 1911–1979 (1984)
 Title
 Large Closed Queueing Networks in SemiMarkov Environment and Their Application
 Journal

Acta Applicandae Mathematicae
Volume 100, Issue 3 , pp 201226
 Cover Date
 20080201
 DOI
 10.1007/s1044000791804
 Print ISSN
 01678019
 Online ISSN
 15729036
 Publisher
 Springer Netherlands
 Additional Links
 Topics
 Keywords

 Closed queueing network
 Random environment
 Martingales and semimartingales
 Skorokhod reflection principle
 60K25
 60K30
 60H30
 60H35
 Authors

 Vyacheslav M. Abramov ^{(1)}
 Author Affiliations

 1. School of Mathematical Sciences, Monash University, Building 28M, Wellington road, Clayton, VIC, 3800, Australia