doi:10.1016/j.peva.2008.02.003
Crown copyright © 2008 Published by Elsevier B.V.
Merging and splitting autocorrelated arrival processes and impact on queueing performance
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Barış Balcıog˜lua, 
, 
, David L. Jagermanb, and Tayfur Altıokc,
aUniversity of Toronto, Department of Mechanical and Industrial Engineering, 5 King’s College Rd., Toronto, ON M5S 3G8, Canada
bRutgers University, RUTCOR, 640 Bartholomew Rd., Piscataway, NJ 08854, USA
cRutgers University, Department of Industrial and Systems Engineering, 96 Frelinghuysen Rd., Piscataway, NJ 08854, USA
Received 4 November 2005;
revised 25 October 2007;
accepted 18 February 2008.
Available online 25 February 2008.
Abstract
We have proposed a three-parameter renewal approximation to analyze splitting and superposition of autocorrelated processes. We define the index of dispersion for counts of an ordinary process used in a new and more accurate technique to estimate the third parameter. Then, we express this newly defined index of dispersion for the superposition in terms of the ordinary as well as the stationary indices of dispersion of the originally autocorrelated component processes. Hence, even if the superposition data is not observable, as long as sufficient information exists on component processes, the parameters of the proposed renewal approximation can be estimated accurately. The accurate renewal approximation of a general process helps in sustaining accuracy if it is split, by-passing the need to sample from branched processes. We have tested the impact of our approximation on the accuracy of the mean waiting time, which compared favorably with simulation results of the original systems.
Keywords: Autocorrelation; Peakedness; Indices of dispersion; Superposition; Splitting; Queueing networks; Mean waiting time; MMPP
Fig. 1. Index of dispersion for counts (
) of processes MMPP A and superposition of eight MMPP A’s.
Fig. 2. Superposition of n autocorrelated streams is split into m branched processes.
Table 1.
Comparison of equilibrium mean waiting times in the G/G/1 queue with MMPP A as the arrival process with intensity λX=0.8, residue AE=21.155, decrement αE=0.143, and ρ=0.7
Table 2.
Comparison of equilibrium mean waiting times in the G/G/1 queue with the superposition of two MMPP A’s as the arrival process with intensity λX=1.6, residue AE=21.155, decrement αE=0.177, and ρ=0.6
Table 3.
Comparison of equilibrium mean waiting times in the G/G/1 queue with the superposition of four MMPP A’s as the arrival process with intensity λX=3.2, residue AE=21.155, decrement αE=0.21, and ρ=0.8
Table 4.
Comparison of equilibrium mean waiting times in the G/G/1 queue with the superposition of eight MMPP A’s as the arrival process with intensity λX=6.4, residue AE=21.155, decrement αE=0.24, and ρ=0.8
Table 5.
Comparison of equilibrium mean waiting times in the G/G/1 queue with the superposition of MMPP A, MMPP B, MMPP C, and MMPP D as the arrival process with intensity λX=3, residue AE=12.877, decrement αE=0.28, and ρ=0.6
Table 6.
Comparison of equilibrium mean waiting times in the G/G/1 queue with the superposition of two MMPP A, two MMPP B, two MMPP C, and two MMPP D’s as the arrival process with intensity λX=6, residue AE=12.877, decrement αE=0.3 and ρ=0.8
Table 7.
Comparison of equilibrium mean waiting times in the G/G/1 queue with the superposition of Renewal A, Renewal B, Renewal C, and Renewal D streams as the arrival process with intensity λX=2, residue AE=−0.18155, decrement αE=6.1 and ρ=0.6
Table 8.
Comparison of equilibrium mean waiting times in the G/G/1 queue with the superposition of ten Renewal C streams as the arrival process with intensity λX=1, residue AE=−0.25, decrement αE=4 and ρ=0.8
Table 9.
Comparison of equilibrium mean waiting times after MMPP A with intensity λX=0.8, residue AE=21.155 and decrement αE=0.143 is split with p=0.4
Table 10.
Comparison of equilibrium mean waiting times after the superposition of eight MMPP A’s with intensity λX=6.4, residue AE=21.155 and decrement αE=0.24 is split with p=0.5
Table 11.
Comparison of equilibrium mean waiting times after the superposition of two MMPP A, two MMPP B, two MMPP C, and two MMPP D’s with intensity λX=6, residue AE=12.877 and decrement αE=0.3 is split with p1=0.3,p2=p4=0.25,p3=0.2
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