On multi-class multi-server queueing and spare parts management

Aart van Harten, Andrei Sleptchenko

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Abstract

Multi-class multi-server queuing problems are a generalization of the wellknown M/M/k situation to arrival processes with clients of N types that require exponentially distributed service with different averaged service time. Problems of this sort arise naturally in various applications, such as spare parts management, for example. In this paper we give a procedure to construct exact solutions of the stationary state equations. Essential in this procedure is the reduction of the problem for n = the number of clients in the system > k to a backwards second order difference equation with constant coefficients for a vector in a linear space with dimension depending on Nand k, denoted by d(N,k). Precisely d(N,k) of its solutions have exponential decay for n 00. Next, using this as input, the equations for n ::; k can be solved by backwards recursion. It follows that the exact solution does not have a simple product structure as one might expect intuitively. Further, using the exact solution several interesting performance measures related to spare parts management can be computed and compared with heuristic approximations. This is illustrated with numerical results.
Original languageEnglish
Place of PublicationEnschede
PublisherUniversiteit Twente
Number of pages35
Publication statusPublished - 2000

Publication series

NameBETA working paper
PublisherUniversity of Twente, Department of Technology and Management
No.WP-49
VolumeWP-49

Fingerprint

Spare Parts
Multi-class
Exact Solution
Multi-server
Second-order Difference Equations
Queuing
State Equation
Stationary States
Exponential Decay
Linear Space
Recursion
Performance Measures
Sort
Heuristics
Numerical Results
Coefficient
Approximation
Multi-server Queues

Keywords

  • IR-95548
  • METIS-128027

Cite this

van Harten, A., & Sleptchenko, A. (2000). On multi-class multi-server queueing and spare parts management. (BETA working paper; Vol. WP-49, No. WP-49). Enschede: Universiteit Twente.
van Harten, Aart ; Sleptchenko, Andrei. / On multi-class multi-server queueing and spare parts management. Enschede : Universiteit Twente, 2000. 35 p. (BETA working paper; WP-49).
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van Harten, A & Sleptchenko, A 2000, On multi-class multi-server queueing and spare parts management. BETA working paper, no. WP-49, vol. WP-49, Universiteit Twente, Enschede.

On multi-class multi-server queueing and spare parts management. / van Harten, Aart; Sleptchenko, Andrei.

Enschede : Universiteit Twente, 2000. 35 p. (BETA working paper; Vol. WP-49, No. WP-49).

Research output: Book/ReportReportProfessional

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AB - Multi-class multi-server queuing problems are a generalization of the wellknown M/M/k situation to arrival processes with clients of N types that require exponentially distributed service with different averaged service time. Problems of this sort arise naturally in various applications, such as spare parts management, for example. In this paper we give a procedure to construct exact solutions of the stationary state equations. Essential in this procedure is the reduction of the problem for n = the number of clients in the system > k to a backwards second order difference equation with constant coefficients for a vector in a linear space with dimension depending on Nand k, denoted by d(N,k). Precisely d(N,k) of its solutions have exponential decay for n 00. Next, using this as input, the equations for n ::; k can be solved by backwards recursion. It follows that the exact solution does not have a simple product structure as one might expect intuitively. Further, using the exact solution several interesting performance measures related to spare parts management can be computed and compared with heuristic approximations. This is illustrated with numerical results.

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van Harten A, Sleptchenko A. On multi-class multi-server queueing and spare parts management. Enschede: Universiteit Twente, 2000. 35 p. (BETA working paper; WP-49).