Algorithms for Queueing Systems with Reneging and Priorities Modeled as Quasi-Birth-Death Processes

Amir Rastpour (), Armann Ingolfsson () and Burhaneddin Sandıkçı ()
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Amir Rastpour: Faculty of Business and Information Technology, Ontario Tech University, Oshawa, Ontario L1H 7K5, Canada
Armann Ingolfsson: Alberta School of Business, University of Alberta, Edmonton, Alberta T6G 2R6, Canada
Burhaneddin Sandıkçı: Department of Industrial Engineering, Istanbul Technical University, 34367 Istanbul, Turkey

INFORMS Journal on Computing, 2022, vol. 34, issue 3, 1693-1710

Abstract: We consider a Markovian multiserver queueing system with two customer classes, preemptive priorities, and reneging. We formulate this system as an infinite level-dependent quasi-birth-death process (LDQBD). We introduce an algorithm that endogenously truncates the level and calculates lower and upper bounds on stationary probabilities of this LDQBD such that the gap between the bounds can be any desired amount. Our algorithm can be applied to any LDQBD for which the rate matrices become elementwise nonincreasing above some level. This appears to be the first algorithm that provides bounds on stationary probabilities for an infinite-level LDQBD. To obtain these bounds, the algorithm first obtains lower and upper bounds on the rate matrices of the LDQBD using a novel method, which can be applied to any LDQBD. We then extend this algorithm to approximate performance measures of the system of interest and calculate exact lower and upper bounds for those that can be expressed as probabilities, such as the probability that an incoming low-priority customer will wait. We generate a wide range of instances with up to 100 servers and compare the solution times and accuracy of our algorithm with two existing algorithms. These numerical experiments indicate that our algorithm is faster than the other two algorithms for a given accuracy requirement. We investigate the impact of changing service rates on the proportion of low-priority customers served and their wait time, and we demonstrate how ignoring one of these measures can possibly mislead decision makers. Summary of Contribution: We contribute to operations research by modeling a practically important queueing system and developing an algorithm to accurately compute performance measures for that system. We also contribute to computer science by providing error and complexity analysis for the algorithm to solve a broad class of two-dimensional Markov chains with infinite state space.

Keywords: service management; priority Erlang A; level-dependent QBD; bounding (search for similar items in EconPapers)
Date: 2022
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Citations: View citations in EconPapers (2)

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Persistent link: https://EconPapers.repec.org/RePEc:inm:orijoc:v:34:y:2022:i:3:p:1693-1710

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