This paper investigates a correlated queueing model with customer abandonment, a framework applicable to various stochastic systems such as emergency departments and telecommunication networks. We assume that customer interarrival and service times follow a bivariate exponential distribution, capturing potential dependencies between these variables. Impatient customers may abandon the system before completing service, adding complexity to the analysis.

First, we use a bivariate distribution construction for arbitrarily correlated interarrival and service times, following an approach similar to that of [1, 4, 5], which builds on the bi variate phase-type framework of [2] and [3]. Second, we derive the stationary joint distribution of age processes by analyzing a multi-layer Markov-modulated fluid flow (MMFF) process.Third, we derive key performance metrics of the queueing model using the distribution of age processes, including loss probabilities, sojourn times, and queue length distributions.

Finally, we examine the influence of the correlation coefficient on these performance measures, providing insights into how dependency structures impact system behavior. Our results offer valuable tools for optimizing systems where customer abandonment and correlated service dynamics are critical factors.

1. REFERENCES

[1] E. Badila, O. Boxma, and J. Resing. Queues and risk

processes with dependencies. Stochastic Models,

30(3):390–419, 2014.

[2] M. Bladt and B. F. Nielsen. On the construction of

bivariate exponential distributions with an arbitrary

correlation coefficient. Stochastic Models, 26(2):295–308,

2010.

[3] Q.-M. He, H. Zhang, and J. C. Vera. On some properties

of bivariate exponential distributions. Stochastic Models,

28(2):187–206, 2012.

[4] H. Wu, Q.-M. He, and L. Xia. The impact of the

correlation coefficient of interarrival and service times on

Permission to make digital or hard copies of part or all of this work for personal or

classroom use is granted without fee provided that copies are not made or distributed

for profit or commercial advantage and that copies bear this notice and the full citation

on the first page. Copyrights for third-party components of this work must be honored.

For all other uses, contact the owner/author(s).

MAM12 2025, Toronto, Canada © 2025 Copyright held by the owner/author(s).

queueing performance: The M/M/1 case. European

Journal of Operational Research, 319(2):625–637, 2024.

[5] H. Wu, L. Xia, and Q.-M. He. A uniform approach for

analyzing queues with correlated interarrival and service

times. Journal of the Indian Society for Probability and

Statistics, 2024.