  In 1987, J.W. Cohen analyzed the so-called Serve the Longest Queue (SLQ)
queueing system, where a single server attends two non-symmetric $M/G/1$-type
queues, exercising a non-preemptive priority switching policy. Cohen further
analyzed in 1998 a non-symmetric 2-queue Markovian system, where newly arriving
customers follow the Join the Shortest Queue (JSQ) discipline. The current
paper generalizes and extends Cohen's works by studying a combined JSQ-SLQ
model, and by broadening the scope of analysis to a non-symmetric 3-queue
system, where arriving customers follow the JSQ strategy and a single server
exercises the preemptive priority SLQ discipline. The system states'
multi-dimensional probability distribution function is derived while applying a
non-conventional representation of the underlying process's state-space. The
analysis combines both Probability Generating Functions and Matrix Geometric
methodologies. It is shown that the joint JSQ-SLQ operating policy achieves
extremely well the goal of balancing between queue sizes. This is emphasized
when calculating the Gini Index associated with the differences between mean
queue sizes: the value of the coefficient is close to zero. Extensive numerical
results are presented.
