An Upper Bound on the M/M/k Queue With Deterministic Setup Times

2025-12-18

AI Summary

In multi-server queueing systems with frequent server activation/deactivation for energy savings, existing studies commonly assume exponential startup times; however, real-world startup processes are highly deterministic, leading to substantial inaccuracies in performance evaluation. Method: This paper introduces MIST (Markovian Intervention with Startup Time), a novel analytical framework integrating stochastic process analysis, time-integral estimation, and deterministic delay modeling, to derive tight upper and lower bounds on the mean waiting time for the M/M/k queue with deterministic startup times—differing by only a constant factor. Contributions: (i) The first closed-form characterization of waiting time for finite-capacity M/M/k queues with deterministic startup; (ii) a high-accuracy, computationally efficient approximation formula; and (iii) significant correction of the systematic underestimation of system performance caused by the conventional exponential-startup assumption. These results provide a more reliable foundation for queueing modeling in green data centers and other energy-constrained systems.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Multiagent Systems: Multiagent Systems under Uncertainty
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

In many systems, servers do not turn on instantly; instead, a setup time must pass before a server can begin work. These"setup times"can wreak havoc on a system's queueing; this is especially true in modern systems, where servers are regularly turned on and off as a way to reduce operating costs (energy, labor, CO2, etc.). To design modern systems which are both efficient and performant, we need to understand how setup times affect queues. Unfortunately, despite successes in understanding setup in a single-server system, setup in a multiserver system remains poorly understood. To circumvent the main difficulty in analyzing multiserver setup, all existing results assume that setup times are memoryless, i.e. distributed Exponentially. However, in most practical settings, setup times are close to Deterministic, and the widely used Exponential-setup assumption leads to unrealistic model behavior and a dramatic underestimation of the true harm caused by setup times. This paper provides a comprehensive characterization of the average waiting time in a multiserver system with Deterministic setup times, the M/M/k/Setup-Deterministic. In particular, we derive upper and lower bounds on the average waiting time in this system, and show these bounds are within a multiplicative constant of each other. These bounds are the first closed-form characterization of waiting time in any finite-server system with setup times. Further, we demonstrate how to combine our upper and lower bounds to derive a simple and accurate approximation for the average waiting time. These results are all made possible via a new technique for analyzing random time integrals that we named the Method of Intervening Stopping Times, or MIST.

Problem

• Characterize waiting time in multiserver systems with deterministic setup
• Derive bounds and approximation for average waiting time in M/M/k/Setup-Deterministic
• Address gap in understanding setup times in multiserver queueing systems

Innovation

• Derived bounds for M/M/k queue with deterministic setup times
• Introduced Method of Intervening Stopping Times (MIST) technique
• Provided closed-form waiting time characterization for multiserver systems