Scaling properties of queues with time-varying load processes: extensions and applications

Rein Vesilo*, Mor Harchol-Balter, Alan Scheller-Wolf

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)


New computing and communications paradigms will result in traffic loads in information server systems that fluctuate over much broader ranges of time scales than current systems. In addition, these fluctuation time scales may only be indirectly known or even be unknown. However, we should still be able to accurately design and manage such systems. This paper addresses this issue: we consider an M/M/1 queueing system operating in a random environment (denoted M/M/1(R)) that alternates between HIGH and LOW phases, where the load in the HIGH phase is higher than in the LOW phase. Previous work on the performance characteristics of M/M/1(R) systems established fundamental properties of the shape of performance curves. In this paper, we extend monotonicity results to include convexity and concavity properties, provide a partial answer to an open problem on stochastic ordering, develop new computational techniques, and include boundary cases and various degenerate M/M/1(R) systems. The basis of our results are novel representations for the mean number in system and the probability of the system being empty. We then apply these results to analyze practical aspects of system operation and design; in particular, we derive the optimal service rate to minimize mean system cost and provide a bias analysis of the use of customer-level sampling to estimate time-stationary quantities.

Original languageEnglish
Pages (from-to)690-731
Number of pages42
JournalProbability in the Engineering and Informational Sciences
Issue number3
Early online date3 Mar 2021
Publication statusPublished - Jul 2022


  • cubic polynomial
  • M/M/1 single-server queue
  • monotonicity
  • random environment
  • sampling
  • time varying load


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