Abstract
We consider a two-stage queueing system where the first (upstream) queue serves many flows, of which a fixed set of flows arrive to the second (downstream) queue. We show that as the capacity and the number of flows aggregated at the upstream queue increases, the overflow probability at the downstream queue converges to that of a simplified single queue obtained by removing the upstream queue from the original two-stage queueing system. Earlier work shows such convergence for fluid traffic, by exploiting the large deviation result that the workload goes to zero almost surely, as the number of flows and capacity is scaled. However, the analysis is quite different and more difficult for the point process traffic considered in this paper. The reason is that for point process traffic the large deviation rate function need not be strictly positive (i.e., I(0)=0), hence the workload at the upstream queue may not go to zero even though the number of flows and capacity go to infinity. The results in this paper thus make it possible to decompose the original two-stage queueing system into a simple single-stage queueing system.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
D.D. Botvich and N. Duffield, Large deviations, the shape of the loss curve, and economies of scale in large multiplexers, Queueing Systems 20 (1995) 293-320.
J. Cao and K. Ramanan, A Poisson limit for the unfinished work of superposed point processes, Bell Labs Technical Report (2001).
J. Choe and N.B. Shroff, Use of supremum distribution of Gaussian processes in queueing analysis with long-range dependence and self-similarity, Stochastic Models 16(2) (2000).
C. Courcoubetis and R.Weber, Buffer overflow asymptotics for a buffer handling many traffic sources, J. Appl. Probab. 33 (1996) 886-903.
D.J. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes(Springer, Berlin, 1988).
R. Durrett, Probability: Theory and Examples, 2nd ed. (Duxbury Press, Belmont, CA, 1996).
D.Y. Eun and N.B. Shroff, Network decomposition in the many-sources regime, Adv. in Appl. Probab. 36(3) (2004) to appear.
D.Y. Eun and N.B. Shroff, Simplification of network analysis in large-bandwidth systems, in: Proc. of IEEE INFOCOM, San Francisco, CA (2003).
N. Likhanov and R. Mazumdar, Cell loss asymptotics for buffers fed with a large number of independent stationary sources, J. Appl. Probab. 36(1) (1999) 86-96.
M. Mandjes and S. Borst, Overflow behavior in queues with many long-tailed inputs, Adv. in Appl. Probab. 32 (2000) 1150-1167.
O. Ozturk, R.Mazumdar and N. Likhanov, Many sources asymptotics for a feedforward network with small buffers, in: Proc. of the Allerton Conference, Monticello, IL (2002).
O. Ozturk, R. Mazumdar and N. Likhanov, Many sources asymptotics in networks with small buffers (2002) submitted.
J. Roberts, U. Mocci and J. Virtamo, Broadband Network Teletraffic, Final Report of Action COST 242(Springer, New York, 1996).
D. Wischik, The output of a switch, or, effective bandwidths for networks, Queueing Systems 32 (1999) 383-396.
D.Wischik, Sample path large deviations for queues with many inputs, Ann. Appl. Probab. 11 (2000) 379-404.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Eun, D.Y., Shroff, N.B. Analyzing a Two-Stage Queueing System with Many Point Process Arrivals at Upstream Queue. Queueing Systems 48, 23–43 (2004). https://doi.org/10.1023/B:QUES.0000039886.54866.c5
Issue Date:
DOI: https://doi.org/10.1023/B:QUES.0000039886.54866.c5