Abstract
The paper provides the up- and down-crossing method to study the asymptotic behavior of queue-length and waiting time in closed Jackson-type queueing networks. These queueing networks consist of central node (hub) and k single-server satellite stations. The case of infinite server hub with exponentially distributed service times is considered in the first section to demonstrate the up- and down-crossing approach to such kind of problems and help to understand the readers the main idea of the method. The main results of the paper are related to the case of single-server hub with generally distributed service times depending on queue-length. Assuming that the first k−1 satellite nodes operate in light usage regime, we consider three cases concerning the kth satellite node. They are the light usage regime and limiting cases for the moderate usage regime and heavy usage regime. The results related to light usage regime show that, as the number of customers in network increases to infinity, the network is decomposed to independent single-server queueing systems. In the limiting cases of moderate usage regime, the diffusion approximations of queue-length and waiting time processes are obtained. In the case of heavy usage regime it is shown that the joint limiting non-stationary queue-lengths distribution at the first k−1 satellite nodes is represented in the product form and coincides with the product of stationary GI/M/1 queue-length distributions with parameters depending on time.
Similar content being viewed by others
References
V.M. Abramov, Investigation of a Queueing System with Service Depending on Queue-Length (Donish, Dushanbe, 1991) (in Russian).
V.M. Abramov, On the asymptotic distribution of the maximum number of infectives in epidemic models with immigration, J. Appl. Probab. 31 (1994) 606-613.
V.M. Abramov, Queueing system with autonomous service: Up-and down-crossings approach, submitted for publication (2000).
V.M. Abramov, A large closed queueing network with autonomous service and bottleneck, Queueing Systems Theory Appl. 35 (2000) 23-54.
V.M. Abramov, Inequalities for the GI/M/1/n loss system, J. Appl. Probab. 38 (2001) to appear.
S.V. Anulova and R.Sh. Liptser, Diffusion approximation for the processes with normal reflection, Theory Probab. Appl. 35 (1990) 413-423.
A.T. Bharucha-Reid, Elements of the Theory of Markov Processes and Their Applications (McGraw-Hill, New York, 1960).
A.A. Borovkov, Stochastic Processes in Queueing Theory (Springer, New York, 1976).
A.A. Borovkov, Limit theorems for networks of queues I, II, Theory Probab. Appl. 31 (1986) 413-427; 32 (1987) 257-272.
P.H. Brill and M.J.M. Posner, Level crossing in point processes applied to queues: Single-server case, Oper. Res. 25 (1977) 662-674.
P.H. Brill and M.J.M. Posner, The system point method in exponential queues: A level crossing approach, Math. Oper. Res. 6 (1981) 31-49.
H. Chen and A. Mandelbaum, Discrete flow networks: Bottleneck analysis and fluid approximations, Math. Oper. Res. 16 (1991) 408-446.
H. Chen and A. Mandelbaum, Discrete flow networks: Diffusion approximations and bottlenecks, Ann. Probab. 19 (1991) 1463-1519.
J.W. Cohen, On up-and down-crossings, J. Appl. Probab. 14 (1977) 405-410.
J.G. Dai, On positive Harris recurrence of multiclass queueing networks: A unified approach via fluid limit models, Ann. Appl. Probab. 5 (1994) 49-77.
C. Dellacherie, Capacites et Processus Stochastiques (Springer, Berlin, 1972).
B. Gnedenko and I.N. Kovalenko, An Introduction to the Queueing Theory (Israel Program for Scientific Translations, Jerusalem, 1968).
D.L. Iglehart and W. Whitt, Multi-chanel queues in heavy traffic, I, Adv. Appl. Probab. 2 (1970) 150-177.
D.L. Iglehart and W. Whitt, Multi-chanel queues in heavy traffic, II, Adv. Appl. Probab. 2 (1970) 355-369.
G.I. Ivchenko, V.A. Kashtanov and I.N. Kovalenko, Theory of Queues (Vysshaya Shkola, Moscow, 1982) (in Russian).
J. Jacod and A.N. Shiryayev, Limit Theorems for Stochastic Processes (Springer, Berlin, 1987).
G.I. Kalmykov, On the partial ordering of one-dimensional Markov processes, Theory Probab. Appl. 7 (1962) 456-459.
S. Karlin and H.G. Taylor, A First Course in Stochastic Processes, 2nd edn. (Academic Press, New York, 1975).
H. Kaspi and A. Mandelbaum, Regenerative closed queueing networks, Stochastic Stochastic Rep. 39 (1992) 239-258.
H. Kaspi and A. Mandelbaum, On Harris recurrence in continuous time, Math. Oper. Res. 19 (1994) 211-222.
J. Keilson, Markov Chain Models–Rarity and Exponentiality (Springer, Heidelberg, 1979).
G.P. Klimov, Stochastical Service Systems (Nauka, Moscow, 1966) (in Russian).
C. Knessl and C. Tier, Asymptotic expansions for large closed queueing networks, J. Assoc. Comput. Mach. 37 (1990) 144-174.
Y. Kogan, Another approach to asymptotic expansions for large closed queueing networks, Oper. Res. Lett. 11 (1992) 317-321.
Y. Kogan and A. Birman, Asymptotic analysis of closed queueing networks with bottlenecks, in: Proc. Internat. Conf. on Perfomance of Distributed Systems and Integrated Communication Networks, eds. T. Hasegawa, H. Takagi and Y. Takahashi (Kyoto, 1991) pp. 237-252.
Y. Kogan and R.Sh. Liptser, Limit non-stationary behavior of large closed queueing networks with bottlenecks, Queueing Systems Theory Appl. 14 (1993) 33-55.
Y. Kogan, R.Sh. Liptser and M. Shenfild, State dependent Benes buffer model with fast loading and output rates, Ann. Appl. Probab. 5 (1995) 97-120.
Y. Kogan, R.Sh. Liptser and A.V. Smorodinskii, Gaussian diffusion approximation of closed Markov model of computer networks, Problems Inform. Transmission 22 (1986) 38-51.
E.V. Krichagina, Asymptotic analysis of queueing networks (martingale approach), Stochastic Stochastic Rep. 40 (1992) 43-76.
E.V. Krichagina, R.Sh. Liptser and A.A. Puhalskii, Diffusion approximation for system with arrivals depending on queue and with general service, Theory Probab. Appl. 33 (1988) 114-124.
A.J. Lemoine, Networks of queues–a survey of weak convergence results, Manag. Sci. 24 (1978) 1175-1193.
R.Sh. Liptser and A.N. Shiryayev, Statistics of Random Processes, Vols. I, II (Springer, Berlin, 1977, 1978).
R.Sh. Liptser and A.N. Shiryayev, Theory of Martingales (Kluwer, Dordrecht, 1989).
A. Mandelbaum and W. Massey, Strong approximations for time-dependent queues, Math. Oper. Res. 20 (1995) 33-64.
A. Mandelbaum, W.A. Massey and M.I. Reiman, Strong approximations for Markovian service networks, Queueing Systems Theory Appl. 30 (1998) 149-201.
A. Mandelbaum and G. Pats, State dependent stochastic networks. Part I. Approximations and applications with continuous diffusion limits, Ann. Appl. Probab. 8 (1998) 569-646.
J. McKenna, D. Mitra and K.G. Ramakrishnan, A class of closed Markovian queueing networks: Integral representation, asymptotic expansions and generalizations, Bell Syst. Technical J. 60 (1981) 599-641.
B. Pittel, Closed exponential networks of queues with saturation: the Jackson-type stationary distribution and its asymptotical analysis, Math. Oper. Res. 6 (1979) 357-378.
Y.V. Prohorov, Transient phenomena in queueing processes, Litovsk. Math. Sb. 3 (1963) 199-205 (in Russian).
M.I. Reiman, Open queueing networks in heavy traffic, Math. Oper. Res. 9 (1984) 441-458.
M.I. Reiman and B. Simon, A network of priority queues in heavy traffic: One bottleneck station, Queueing Systems Theory Appl. 6 (1990) 33-58.
M. Shaked and J.G. Shanthikumar, Stochastic Orders and Their Applications (Academic Press, Boston, 1994).
J.G. Shanthikumar, Some analysis of the control of queues using level crossing of regenerative processes, J. Appl. Probab. 17 (1980) 814-821.
J.G. Shanthikumar, Level crossing analysis of priority queues and a conservation identity for vacation models, Naval Res. Logist. Quat. 36 (1989) 797-806.
J.G. Shanthikumar and M.J. Chandra, Application of level crossing analysis to discrete state processes in queueing systems, Naval Res. Logist. Quat. 29 (1982) 593-608.
A.N. Shiryayev, Probability (Springer, Berlin, 1984).
A.V. Skorohod, Stochastic equations for diffusion processes in a bounded region, Theory Probab. Appl. 6 (1961) 264-274.
D. Stoyan, Comparison Methods for Queues and Other Stochastic Models (Wiley, Chichester, 1983).
L. Takacs, Introduction to the Theory of Queues (Oxford Univ. Press, New York, 1962).
H. Tanaka, Stochastic differential equations with reflecting boundary condition in convex regions, Hiroshima Math. J. 9 (1979) 163-177.
H. Thorisson, From coupling to shift-coupling, Theory Probab. Appl. 37 (1992) 105-112.
W. Whitt, Open and closed models for networks of queues, AT&T Bell Lab. Technical J. 63 (1984) 1911-1979.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Abramov, V.M. Some Results for Large Closed Queueing Networks with and without Bottleneck: Up- and Down-Crossings Approach. Queueing Systems 38, 149–184 (2001). https://doi.org/10.1023/A:1010954214228
Issue Date:
DOI: https://doi.org/10.1023/A:1010954214228