Abstract
In many systems, in order to fulfill demand (computing or other services) that varies over time, service capacities often change accordingly. In this paper, we analyze a simple two dimensional Markov chain model of a queueing system in which multiple servers can arrive to increase service capacity, and depart if a server has been idle for too long. It is well known that multi-dimensional Markov chains are in general difficult to analyze. Our focus is on an approximation method of stationary performance of the system via the Stein method. For this purpose, innovative methods are developed to estimate the moments of the Markov chain, as well as the solution to the Poisson equation with a partial differential operator.
- A. Bhandari, A. Scheller-Wolf, and M. Harchol-Balter. An exact and efficient algorithm for the constrained dynamic operator staffing problem for call centers. Management Science, 54(2):339--353, 2008.Google ScholarDigital Library
- A. Braverman and J. G. Dai. Stein's method for steady-state diffusion approximations of m/Ph/n + m systems. Ann. Appl. Probab., 27(1):550--581, 02 2017.Google ScholarCross Ref
- M. J. Davis, Y. Lu, M. Sharma, M. Squillante, and B. Zhang. Stochastic optimization models for workforce planning, operations, and risk management. Service Science, 10(1):40--57, 2018.Google ScholarDigital Library
- G. Fayolle, R. Iasnogorodski, V. Malyshev, and V. Malyshev. Random Walks in the Quarter-Plane: Algebraic Methods, Boundary Value Problems and Applications. Applications of mathematics. Springer, 1999.Google Scholar
- I. Gurvich. Diffusion models and steady-state approximations for exponentially ergodic markovian queues. Ann. Appl. Probab., 24(6):2527--2559, 12 2014.Google ScholarCross Ref
- I. Gurvich. Validity of heavy-traffic steady-state approximations in multiclass queueing networks: The case of queue-ratio disciplines. Mathematics of Operations Research, 39(1):121--162, 2014.Google ScholarDigital Library
- S. Halfin and W. Whitt. Heavy-Traffic Limits for Queues with Many Exponential Servers. Operations Research, 29(3):567--588, June 1981.Google Scholar
- N. Krylov and A. M. Society. Lectures on Elliptic and Parabolic Equations in Holder Spaces. Graduate studies in mathematics. American Mathematical Society, 1996.Google Scholar
- M. Lin, A. Wierman, L. L. H. Andrew, and E. Thereska. Dynamic right-sizing for power-proportional data centers. In 2011 Proceedings IEEE INFOCOM, pages 1098--1106, 2011.Google ScholarCross Ref
- Y. Lu. On a two-dimensional markov chain model for performance analysis of systems with varying capacities. arXiv:2106.03145, 2021.Google Scholar
- Y. Lu, M. Sharma, M. S. Squillante, and B. Zhang. Stochastic optimal dynamic control of gim/gim/1n queues with time-varying workloads. Probability in the Engineering and Informational Sciences, 30(3):470--491, 2016.Google ScholarCross Ref
- V. Mazalov and A. Gurtov. Queueing system with on-demand number of servers. Mathematica Applicanda, 40(2):1--12, 2012.Google Scholar
- R. Schollmeier. A definition of peer-to-peer networking for the classification of peer-to-peer architectures and applications. In Proceedings First International Conference on Peer-to-Peer Computing, pages 101--102, 2001.Google Scholar
- C. Stein. Approximate computation of expectations. Number 7. Institute of Mathematical Statistics Lecture Notes, Monograph Series, 1986.Google Scholar
Index Terms
- Performance Analysis of A Queueing System with Server Arrival and Departure
Recommendations
A queueing system with single arrival bulk service and single departure
In this note, we study a Markovian queueing system with accessible batches for service, but units depart individually. This system generalizes the M/M/C, bulk service, and accessible batch queueing systems. We compute the system size probabilities in ...
Retrial queuing system with Markovian arrival flow and phase-type service time distribution
We consider a multi-server queuing system with retrial customers to model a call center. The flow of customers is described by a Markovian arrival process (MAP). The servers are identical and independent of each other. A customer's service time has a ...
A Single-Server Priority Queuing System with General Holding Times, Poisson Input, and Reverse-Order-of-Arrival Queuing Discipline
This paper gives an explicit formula for the waiting-time distribution in a single-server system with general holding times subject to a Poisson input from any number of priority classes. In the general case, where the holding-time distributions of the ...
Comments