A Large Closed Queueing Network Containing Two Types of Node and Multiple Customer Classes: One Bottleneck Station Article

DOI :
10.1023/B:QUES.0000039887.30622.ef

Cite this article as: Abramov, V.M. Queueing Systems (2004) 48: 45. doi:10.1023/B:QUES.0000039887.30622.ef
Abstract The paper studies a closed queueing network containing two types of node. The first type (server station) is an infinite server queueing system, and the second type (client station) is a single server queueing system with autonomous service, i.e. every client station serves customers (units) only at random instants generated by strictly stationary and ergodic sequence of random variables. It is assumed that there are r server stations. At the initial time moment all units are distributed in the server stations, and the i th server station contains N _{i} units, i =1,2,...,r , where all the values N _{i} are large numbers of the same order. The total number of client stations is equal to k . The expected times between departures in the client stations are small values of the order O(N ^{−1} ) (N =N _{1} +N _{2} +...+N _{r} ). After service completion in the i th server station a unit is transmitted to the j th client station with probability p _{i,j} (j =1,2,...,k ), and being served in the j th client station the unit returns to the i th server station. Under the assumption that only one of the client stations is a bottleneck node, i.e. the expected number of arrivals per time unit to the node is greater than the expected number of departures from that node, the paper derives the representation for non-stationary queue-length distributions in non-bottleneck client stations.

closed queueing network autonomous service multiple customer classes bottleneck stochastic calculus martingales and semimartingales

References [1]

V.M. Abramov, A large closed queueing network with autonomous service and bottleneck, Queueing Systems 35 (2000) 23-54.

Google Scholar [2]

V.M. Abramov, Some results for large closed queueing networks with and without bottleneck: Upand down-crossings approach, Queueing Systems 38 (2001) 149-184.

Google Scholar [3]

S.V. Anulova and R.Sh. Liptser, Diffusion approximation for processes with normal reflection, Theory Probab. Appl. 35 (1990) 413-423.

Google Scholar [4]

A.A. Borovkov,

Stochastic Processes in Queueing Theory (Springer, Berlin, 1976).

Google Scholar [5]

A.A. Borovkov,

Asymptotic Methods in Queueing Theory (Wiley, New York, 1984).

Google Scholar [6]

Y.-J. Chao, Weak convergence of a sequence of semimartingales to a diffusion with discontinuous drift and diffusion coefficients, Queueing Systems 42 (2002) 153-188.

Google Scholar [7]

H. Chen and A. Mandelbaum, Discrete flow networks: Bottleneck analysis and fluid approximations, Math. Oper. Res. 16 (1991) 408-446.

Google Scholar [8]

H. Chen and A. Mandelbaum, Discrete flow networks: Diffusion approximations and bottlenecks, Ann. Probab. 19 (1991) 1463-1519.

Google Scholar [9]

C. Dellacherie,

Capacités et Processus Stochastiques (Springer, Berlin, 1972).

Google Scholar [10]

C. Knessl, B.J. Matkovsky, Z. Schuss and C. Tier, Asymptotic analysis of a state dependent

M/G/ 1 queueing system, SIAM J. Appl. Math. 46(3) (1986) 483-505.

Google Scholar [11]

C. Knessl, B.J. Matkovsky, Z. Schuss and C. Tier, The two repairmen problem: A finite source

M/G/ 2 queue, SIAM J. Appl. Math. 47(2) (1987) 367-397.

Google Scholar [12]

C. Knessl, B.J. Matkovsky, Z. Schuss and C. Tier, A Markov modulated

M/G/ 1 queue I. Stationary distribution, Queueing Systems 1 (1987) 355-374.

Google Scholar [13]

C. Knessl, B.J.Matkovsky, Z. Schuss and C. Tier, A Markov modulated

M/G/ 1 queue II. Busy period and time for buffer overflow, Queueing Systems 1 (1987) 375-399.

Google Scholar [14]

C. Knessl, B.J. Matkovsky, Z. Schuss and C. Tier, Busy period distribution in state dependent queues, Queueing Systems 2 (1987) 285-305.

Google Scholar [15]

C. Knessl and C. Tier, Asymptotic expansions for large closed queueing networks, J. Assoc. Comput. Mach. 37 (1990) 144-174.

Google Scholar [16]

Y. Kogan, and R.Sh. Liptser, Limit non-stationary behavior of large closed queueing networks with bottlenecks, Queueing Systems 14 (1993) 33-55.

Google Scholar [17]

Y. Kogan, R.Sh. Liptser and M. Shenfild, State dependent Benês buffer model with fast loading and output rates, Ann. Appl. Probab. 5 (1995) 97-120.

Google Scholar [18]

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.

Google Scholar [19]

T. Konstantopoulos, S.N. Papadakis and J. Walrand, Functional approximation theorems for controlled queueing systems, J. Appl. Probab. 31 (1994) 765-776.

Google Scholar [20]

E.V. Krichagina, R.Sh. Liptser and A.A. Puhalskii, Diffusion approximation for system with arrivals depending on queue and arbitrary service distribution, Theory Probab. Appl. 33 (1988) 114-124.

Google Scholar [21]

E.V. Krichagina and A.A. Puhalskii, A heavy traffic analysis of closed queueing system with

GI/ 8 service center, Queueing Systems 25 (1997) 235-280.

CrossRef Google Scholar [22]

N.V. Krylov and R. Liptser, On diffusion approximation with discontinuous coefficients, Stochastic Process. Appl. 102 (2002) 235-264.

Google Scholar [23]

R.Sh. Liptser, A large deviation problem for simple queueing model, Queueing Systems 14 (1993) 1-32.

Google Scholar [24]

R.Sh. Liptser and A.N. Shiryayev, Statistics of Random Processes , Vols. I, II (Springer, Berlin, 1977/1978).

[25]

R.Sh. Liptser and A.N. Shiryayev,

Theory of Martingales (Kluwer, Dordrecht, 1989).

Google Scholar [26]

A. Mandelbaum and W.A. Massey, Strong approximations for time dependent queues, Math. Oper. Res. 20 (1995) 33-64.

Google Scholar [27]

A. Mandelbaum, W.A. Massey and M.I. Reiman, Strong approximations for Markovian service networks, Queueing Systems 30 (1998) 149-201.

CrossRef Google Scholar [28]

A. Mandelbaum and G. Pats, State-dependent queues: approximations and applications, in:

IMA Volumes in Mathematics and Its Applications , eds. F.P. Kelly and R.J. Williams, Vol. 71 (Springer, Berlin, 1995) pp. 239-282.

Google Scholar [29]

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.

Google Scholar [30]

G. Pólya and G. Szegö,

Aufgaben und Lehrsatze aus der Analysis, Erster Band: Reihen, Integralrechnung, Functionentheorie (Springer, Berlin, 1964).

Google Scholar [31]

M.I. Reiman and B. Simon, A network of priority queues in heavy traffic: One bottleneck station, Queueing Systems 6 (1990) 33-58.

Google Scholar [32]

A.V. Skorokhod, Stochastic equations for diffusion processes in a bounded region, Theory Probab. Appl. 6 (1961) 264-274.

Google Scholar [33]

H. Tanaka, Stochastic differential equations with reflected boundary condition in convex regions, Hiroshima Math. J. 9 (1979) 163-177.

Google Scholar [34]

W. Whitt, Open and closed models for networks of queues, AT&T Bell. Lab. Tech. J. 63 (1984) 1911-1979.

Google Scholar [35]

R.J. Williams, On approximation of queueing networks in heavy traffic, in:

Stochastic Networks. Theory and Application , eds. F.P. Kelly, S. Zachary and I. Ziedins (Oxford Univ. Press, Oxford, 1996) pp. 35-56.

Google Scholar [36]

R.J. Williams, An invariance principle for semimartingale reflecting Brownian motion in orthant, Queueing Systems 30 (1998) 5-25.

CrossRef Google Scholar [37]

R.J. Williams, Diffusion approximation for open multiclass queueing networks: Sufficient conditions involving state space collapse, Queueing Systems 30 (1998) 27-88.

Google Scholar © Kluwer Academic Publishers 2004

Authors and Affiliations 1. The Sackler Faculty of Exact Sciences, School of Mathematical Sciences Tel Aviv University Tel Aviv Israel