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

Rastpour A., Ingolfsson A., Sandıkçı B.

INFORMS JOURNAL ON COMPUTING, vol.34, pp.1693-1710, 2022 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 34
  • Publication Date: 2022
  • Doi Number: 10.1287/ijoc.2021.1141
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.1693-1710
  • Keywords: service management, priority Erlang A, level-dependent QBD, bounding, COMPUTATION
  • Istanbul Technical University Affiliated: Yes


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.