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Transient GI/MSP/1/N Queue.

Andrzej Chydzinski1

  • 1Department of Computer Networks and Systems, Silesian University of Technology, Akademicka 16, 44-100 Gliwice, Poland.

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Summary
This summary is machine-generated.

This study analyzes queue length in systems with correlated service times using the Markovian service process (MSP). We provide theorems and examples for queue length distribution and mean queue length over time.

Keywords:
GI/MSP/1/N queueMarkovian service processcorrelated service timesmean queue lengthqueue length distributionsingle-server queuetransient analysis

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Area of Science:

  • Operations Research
  • Applied Probability

Background:

  • Real-world queueing systems often exhibit non-zero correlations in service times.
  • The Markovian service process (MSP) is a tractable model for correlated service times, offering strong fitting capabilities.

Purpose of the Study:

  • To perform a transient analysis of queue length in a queueing model with MSP services and general interarrival times.
  • To prove theorems concerning the queue length distribution and mean queue length at arbitrary time t.
  • To illustrate the dynamic behavior of queue length metrics under varying conditions.

Main Methods:

  • Development of queueing models incorporating Markovian service processes.
  • Analytical derivation of theorems for queue length distribution and mean queue length.
  • Numerical simulations to demonstrate transient queue length behavior.

Main Results:

  • Two theorems are established: one for the queue length distribution at time t, and another for the mean queue length at time t.
  • Numerical examples showcase the evolution of mean queue length, standard deviation, and complete distribution.
  • The impact of service correlation strength, initial conditions, and interarrival time variance on queue dynamics is illustrated.

Conclusions:

  • The study provides a theoretical framework and practical insights into queueing systems with correlated service times.
  • The findings are applicable to optimizing performance in various queueing scenarios with non-exponential service times.
  • The Markovian service process proves effective for analyzing transient queue length dynamics.