M/M(a,b)/1 Model Of Interdependent Queueing With Controllable Arrival Rates

K H Rahim; M Thiagarajan

doi:10.17485/IJST/v16i37.1259

Article

M/M(a,b)/1 Model Of Interdependent Queueing With Controllable Arrival Rates

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Indian Journal of Science and Technology

DOI: 10.17485/IJST/v16i37.1259

Year: 2023, Volume: 16, Issue: 37, Pages: 3100-3109

Original Article

M/M(a,b)/1 Model Of Interdependent Queueing With Controllable Arrival Rates

K H Rahim^1*, M Thiagarajan¹

¹Department of Mathematics, St. Joseph’s College (Autonomous), Affiliated to Bharathidasan University, Tiruchirappalli, 620002, Tamil Nadu, India

*Corresponding Author
Email: [email protected]

Received Date:24 May 2023, Accepted Date:30 August 2023, Published Date:03 October 2023

This work is licensed under a Creative Commons Attribution 4.0 International License.

Abstract

Objectives: Instead of only providing individualized one-on-one assistance, some studies in the literature on queueing theory describe systems that provide services in batches. This study introduces controllable arrival rates and interdependency in such a system’s service and arrival processes and then obtains the queueing system’s probabilities and characteristics. It also verified the obtained results numerically. Methods: Controlling the arrival rates by faster and slower arrival rates are expected for the input, with Poisson (each time Poisson occurrence has one arrival) being the default assumption. The general bulk service rule dictates that the service be delivered in batches. Service begins only when the count of customers in the queue approaches or surpasses a and the capacity b (a 1). For brevity, a batch’s service time distribution is assumed to be exponential and is not dependent on the batch size. Then, all the steady-state probabilities are derived using a recursive approach. Findings: We used M/M(a,b)/1 as the notation. For this model, steady-state solutions & characteristics are derived and explored. All the probabilities are expressed in terms of P0;0(0). The expected count of customers and waiting time depends on the interdependency, service rate, faster arrival rate, and slower arrival rate. According to each parameter, all the results are verified. Novelty: There are works related to bulk service in queuing theory, but this is a new approach to give a bridge between bulk service and controllable arrival rates along with interdependency in the arrival and service process.

Keywords: M/M(a; b)/1 Queueing System; Bulk Service; Controllable Arrival Rates; Steady States; Interdependent Model; Stochastic Processes

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Copyright

© 2023 Rahim & Thiagarajan. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Published By Indian Society for Education and Environment (iSee)