Transient GI/MSP/1/N Queue

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.