Analysis of batch arrivals with multiple service for unbalanced queueing systems
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0222
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Department of Mathematics, Faculty of Science ,Obafemi Awolowo University ,Ile -Ife
Abstract
This study designed a prototype of an unbalanced Combined Input Crosspoint Queued (CICQ) queueing system, obtained the possible states at the moment just before a departure for some specific gridpoints and provided the system solution in terms of those grid points and analysed the performanceof the proposed system. These were with a view to establishing relevant flow balance equations and analysing the traffic at various levels of interactions of contending units for congestion, complexity and grid points. Considered is a two-stream toll gate system modified from the CICQ switch system of homogeneous and heterogeneous packets. The Markov Modulated Poisson Process (MMPP) were superimposed at the input port to obtain the defined states at (u, v,w)k for epoch or moments just after a departure of type k which identified admissible states and their transition probability matrix was obtained using various combinatorial reasonings. The work thereafter, formulated governing equations from relevant flow balance equations, which were of the form ΠP = Π and solved them for stationary probability vectors. Subsequently, the work obtained the mean recurrence times for each state and mean first passage times from one state to the other. Highlighted in particular, were the situations of the extreme cases of heavy traffic (HT) with 27 possible states when the system was saturated with contending units and two cases of restricted light traffic. The restricted traffic was such that traffic (u, v,w) at the moment just before a departure, was guided by the conditions u ≤ 2n, v ≤ 2n, w ≤ n and v+w ≤ 2n where u, v,w and n were integers connoting number of different units in the queues. The light traffic case I (LTS I) connoted n = 1 with 17 admissible states and the light traffic case II (LTS II) connoted n = 2 with 86 admissible states for a 27 grid point system. For a combination of service times τ1, τ2, τ3, the resulting combination were evaluated for cohersion, stability and efficiency using measures of the minimum mean first passage times Tmin(ijk), Concentration Index (C) and the Euclidean Distance Measure (dI/II ). The result from the mean first passage time for all three systems provided a measure of the sensitivity for the stationary probability vectors as defined in their respective underlying states. The mean first passage times tijk was found to be a stochastic process of its own with respect to each of the models, matrices and values, in correspondence to the gridpoints as defined by the parameters of the
light traffic case I (LTSI), light traffic case II (LTSII) and the heavy traffic situations (HTS). Thetendency to concentrate (or dominate) on any state is lower for the heavy traffic, though steady, but cannot be sustained when compared with the light traffic cases. This tendency for concentration was found to be closer for both light traffic cases but the LTS II case was uniformly less concentrated than that of the LTS I case. The order, therefore, of the concentration Index C for the three systems is HTS < LTS II < LTS I by considering all the grid points. The value of the concentration index C for the heavy traffic ranges from 0.047 to 0.05, that of the LTS II ranges from 0.07 to 0.35 while the one for LTS I ranges from 0.08 to 0.37. The minimum Euclidean distance (dI/II ) compared to the restrictive light traffic cases, in relation to matching similar states, is wide. The width is in the multiple of 104. The least active destination states service type III has the lowest value of 4.34, while the most active destination state for service type III has a value of 6.42. It was noted that the minimum Euclidean distance dI/II appeared progressive as the destination states increases. However, the order of spryness for the three types of service is service type I > service type II > service type III. The study concluded that the Combined Input Crosspoint Queued switch model is able to predict relative performances of traffic systems based on the objective matrix for different levels of complexity and congestion.
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xxvi, 386p
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Babalola,A.R. (2022).Analysis of batch arrivals with multiple service for unbalanced queueing systems, Department of Mathematics, Faculty of Science, Obafemi Awolowo University ,Ile -Ife.