Steady-State Probabilities of Two Queues in Parallel

Background: This study empirically examines a system of two parallel queues, each served by an exponentially distributed server with service rates μ₁ and μ₂, respectively. The arrival process follows a Poisson distribution with rate λ for both queues. Customers are directed to the queue with the shortest length; in the case of equal queue lengths, customers randomly select a queue with equal probability. The main focus of this study is on analyzing the steady-state probabilities of the two parallel queuing systems. After determining the performance measures, an application was studied to compare a one-queue, one-server system with the two parallel queues system to identify which is more efficient.

Methods: Two scenarios were explored in this research: one in which customers can switch between queues (queue hopping) and another where no switching is allowed. The steady-state probabilities of the system were analyzed. Wolfram Mathematica software was utilized to perform the computations and to find the transition matrix of the system.

Results: The study presents the steady-state probability distributions for the system, analyzing two cases: with and without queue hopping. These probabilities reveal the impact of allowing or restricting customer movement between queues on overall system performance. The two systems were compared to determine which one is more efficient.

Conclusion/Implications: The findings highlight the effect of the number of queues and the service rate on performance measures. The study also discusses two different cases of the two parallel queues system: when jockeying between queues is allowed and when it is not permitted. A command in Wolfram Mathematica is provided to demonstrate how the truncated transition matrix of the system can be applied.