Stationary Performance Evaluation of an M/M/2 Queue with Constant Retrials and State-Dependent Service Rate

Abstract

This paper investigates the stationary analysis of an M/M/2 retrial queuing system with a state dependent service rate. Customers arrive according to a Poisson process, and if both servers are busy, they enter an orbit and retry after a random time with a constant repeated attempt rate. The model is formulated as a Markov process with a two-dimensional state descriptor. Steady state equations are derived, and conditions for ergodicity are established. Numerical illustrations demonstrate how probabilities change with orbit size, providing insights into system performance.

Introduction

1. Overview of Queuing and Retrial Systems

• Queuing theory analyzes systems where customers arrive randomly to receive service from limited resources.

• In retrial queues, blocked customers do not leave the system permanently but retry after a random delay.

• This model is relevant in telecom, computer systems, and customer service, where temporary unavailability often leads to repeated attempts (e.g., redials).

2. The Proposed Model

• System: M/M/2 retrial queue with state-dependent service rates.

• Arrivals: Customers arrive according to a Poisson process (rate λ).

• Service Mechanism:

  • 2 servers available.

  • If a server is free → customer gets immediate service.

  • If both are busy → customer joins the orbit and retries later.

• Retrial Behavior:

  • Retrial rate is μ per orbiting customer.

  • Total retrial rate when j customers are in orbit = jμ.

• State-dependent service rates:

  • If one server is busy → rate is v?.

  • If both servers are busy → rate is v?.

• Assumptions:

  • All processes (arrivals, retrials, service) are mutually independent.

  • The system follows a Markov process defined by state (C(t), N(t)):

    • C(t): number of busy servers (0, 1, 2)

    • N(t): number of customers in orbit (0, 1, 2, ...)

3. Steady-State Analysis

• The steady-state probabilities P?? = P{C(t)=i, N(t)=j} are derived using:

  • Balance equations

  • Birth-death process formulation for {P??}, with:

    • Birth rate: α? (retrial and arrival components)

    • Death rate: β? (service completions)

Ergodicity Condition (Theorem)

• The system is ergodic (stable) if and only if the total arrival + retrial rate < total service rate.

• Mathematically, this means α < β.

  • If α = β → system is null recurrent

  • If α > β → system is transient

4. Numerical Observations

• When no one is in orbit, system utilization is high:

  • Probability of both servers busy (P??) > idle (P??).

• As the number of orbiting customers (j) increases:

  • All state probabilities (P??, P??, P??) decrease.

  • But P?? remains dominant, showing increased congestion.

• Retrial traffic acts like a secondary input, increasing load and reducing idle time.

• State-dependent service rates help moderate, but do not eliminate congestion.

5. Key Takeaways

• Retrial queues reflect realistic behavior in blocked service environments.

• State-dependent service rates model real-world scenarios where system efficiency changes with load.

• Proper stability analysis is crucial to ensure the system can handle incoming and retrial traffic without becoming overloaded.

• This model helps in performance optimization of heavily loaded service systems like call centers or network servers.

Conclusion

Numerical results confirm that as the number of customers in orbit increases, the probability of idle servers decreases, while the likelihood of both servers being busy rises. This demonstrates the direct impact of retrial traffic on system congestion. Moreover, the incorporation of state-dependent service rates offers a more realistic representation of systems where service efficiency is influenced by workload.Overall, the model contributes to the broader understanding of retrial queues by presenting a tractable yet practical framework. Its applicability extends to telecommunication systems, call centers, and computer networks, where managing retrial traffic and varying service capacities is critical for maintaining stability and service quality.

References

[1] J. R. Artalejo, “Stationary analysis of the characteristics of the M/M/2 queue with constant repeated attempts,” Opsearch, vol. 33, no. 2, pp. 107–123, 1996. [2] R. L. Garg and P. Singh, “Queue dependent servers queueing system,” Microelectronics and Reliability, vol. 33, no. 15, pp. 2289–2295, 1993. [3] D. Gross and C. M. Harris, Fundamentals of Queueing Theory, 3rd ed. New York, NY, USA: John Wiley & Sons, 1985. [4] L. Kleinrock, Queueing Systems, Volume 1: Theory. New York, NY, USA: Wiley, 1975. [5] J. Medhi, Stochastic Models in Queueing Theory, 2nd ed. San Diego, CA, USA: Academic Press, 1991.

Copyright

Copyright © 2025 V. Nandhini, Dr. J. Shiama, E. Dhanalakshmi, N. Sujithra, A. Elcy, K. Vijaya. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.