Block-Structured Double-Ended Queues and Bilateral QBD Processes
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This paper studies a block-structured double-ended queue, whose block structure comes from two independent Markovian arrival processes (MAPs), and its stability is guaranteed by customers' impatient behaviors. We show that such a queue can be expressed as a new bilateral quasi birth-and-death (QBD) process. For this purpose, we provide a detailed analysis for the bilateral QBD process, including the system stability, the stationary probability vector, the sojourn time, and so forth. Furthermore, we develop three effective algorithms for computing the performance measures (i.e., the probabilities of stationary queue lengths, the average stationary queue lengths, and the average sojourn times) of the block-structured double-ended queue. Finally, numerical examples are employed to verify the correctness of our theoretical results, and illustrate how the performance measures of this queue are influenced by key system parameters. We believe that the methodology and results described in this paper can be applied to deal with general matching queues (e.g., bilateral Markov processes of GI/M/1 type and those of M/G/1 type) via developing their corresponding bilateral block-structured Markov processes, which are very useful in analyzing many practical issues, such as those encountered in sharing economy, organ transplantation, intelligent manufacturing, intelligent transportation, and so on.
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