A BMAP/SM/1 queue with service times depending on the arrival process |
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Authors: | Machihara Fumiaki |
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Institution: | (1) Department of Information Sciences, Tokyo Denki University, Hatoyama‐machi, Hikigun, Saitama 350‐0394, Japan |
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Abstract: | We study a BMAP/>SM/1 queue with batch Markov arrival process input and semi‐Markov service. Service times may depend on arrival
phase states, that is, there are many types of arrivals which have different service time distributions. The service process
is a heterogeneous Markov renewal process, and so our model necessarily includes known models. At first, we consider the first
passage time from level {κ+1} (the set of the states that the number of customers in the system is κ+1) to level {κ} when a batch arrival occurs at time 0 and then a customer service included in that batch simultaneously starts. The service
descipline is considered as a LIFO (Last‐In First‐Out) with preemption. This discipline has the fundamental role for the analysis
of the first passage time. Using this first passage time distribution, the busy period length distribution can be obtained.
The busy period remains unaltered in any service disciplines if they are work‐conserving. Next, we analyze the stationary
workload distribution (the stationary virtual waiting time distribution). The workload as well as the busy period remain unaltered
in any service disciplines if they are work‐conserving. Based on this fact, we derive the Laplace–Stieltjes transform for
the stationary distribution of the actual waiting time under a FIFO discipline. In addition, we refer to the Laplace–Stieltjes
transforms for the distributions of the actual waiting times of the individual types of customers. Using the relationship
between the stationary waiting time distribution and the stationary distribution of the number of customers in the system
at departure epochs, we derive the generating function for the stationary joint distribution of the numbers of different types
of customers at departures.
This revised version was published online in June 2006 with corrections to the Cover Date. |
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Keywords: | BMAP heterogeneous input Markov renewal service first passage time waiting time queue length |
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