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Priority retrial queueing model operating in random environment with varying number and reservation of servers

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  • Dudin, Alexander
  • Kim, Chesoong
  • Dudin, Sergey
  • Dudina, Olga

Abstract

We consider a multi-server queueing system with two types of customers operating in the Markovian random environment. Under the fixed state of the random environment, the arrival flow is described by the marked Markovian arrival process. Type 1 customers have preemptive priority over type 2 customers. Type 1 customer is lost only if, at its arrival moment, all servers are busy by type 1 customers. The service times of both types of customers have an exponential distribution with the rate depending on the type of a customer. Type 2 customer is accepted for service if the number of busy servers at its arrival epoch does not exceed a fixed threshold. Otherwise, the arriving customer makes a randomized choice to leave the system permanently (to balk) or to join so called orbit and try to obtain service later. The inter-retrial times have an exponential distribution. Customers in orbit can be impatient and may leave the system after an exponentially distributed amount of time. Due to preemptive priority of type 1 customers, service of type 2 customer can be interrupted. In this case, the interrupted customer makes a randomized choice to leave the system permanently or go into orbit. When the random environment changes its state, immediately the following parameters of the system change their value: the total number of servers, the number of servers available for type 2 customers, the matrices defining the arrival process, the rates of service of customers, the rate of retrials, the impatience intensity, the probability of balking the system at the arrival moment or the moment of termination of service of type 2 customer. Behavior of the system is described by the level dependent multi-dimensional Markov chain that belongs to the class of asymptotically quasi-Toeplitz Markov chains. This allows us to derive the ergodicity condition for this Markov chain and compute its stationary distribution. The main performance measures of the system are expressed via the stationary state probabilities. Numerical illustrations are presented.

Suggested Citation

  • Dudin, Alexander & Kim, Chesoong & Dudin, Sergey & Dudina, Olga, 2015. "Priority retrial queueing model operating in random environment with varying number and reservation of servers," Applied Mathematics and Computation, Elsevier, vol. 269(C), pages 674-690.
    • Handle: RePEc:eee:apmaco:v:269:y:2015:i:c:p:674-690
    DOI: 10.1016/j.amc.2015.08.005
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    References listed on IDEAS

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    1. Marcel F. Neuts & David M. Lucantoni, 1979. "A Markovian Queue with N Servers Subject to Breakdowns and Repairs," Management Science, INFORMS, vol. 25(9), pages 849-861, September.
    2. Jain, Madhu & Bhagat, Amita & Shekhar, Chandra, 2015. "Double orbit finite retrial queues with priority customers and service interruptions," Applied Mathematics and Computation, Elsevier, vol. 253(C), pages 324-344.
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    Cited by:

    1. Ashis Kumar Chakraborty & Ritwika Chattopadhyay & Inder Kaur & Sayaran Mittra, 2022. "Optimization of the number of maintenance crew in a manufacturing unit," OPSEARCH, Springer;Operational Research Society of India, vol. 59(1), pages 1-19, March.
    2. Kim, Chesoong & Klimenok, V.I. & Dudin, A.N., 2017. "Analysis of unreliable BMAP/PH/N type queue with Markovian flow of breakdowns," Applied Mathematics and Computation, Elsevier, vol. 314(C), pages 154-172.
    3. V. Vinitha & N. Anbazhagan & S. Amutha & K. Jeganathan & Bhanu Shrestha & Hyoung-Kyu Song & Gyanendra Prasad Joshi & Hyeonjoon Moon, 2022. "Analysis of a Stochastic Inventory Model on Random Environment with Two Classes of Suppliers and Impulse Customers," Mathematics, MDPI, vol. 10(13), pages 1-18, June.
    4. Alexander Dudin & Olga Dudina & Sergei Dudin & Konstantin Samouylov, 2021. "Analysis of Multi-Server Queue with Self-Sustained Servers," Mathematics, MDPI, vol. 9(17), pages 1-18, September.

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