Queueing System with Two Input Flows, Preemptive Priority, and Stochastic Dropping


We consider a single-line queuing system with an infinite buffer that receives two Poisson flows of customers with different intensities. Customers of the first type have preemptive priority over customers of the second type. In addition, at the time of the end of servicing, a high-priority customer with some probability can drop all low-priority customers in the queue. Serving both types of customers has an exponential distribution with different parameters. We show expressions for calculating stationary probabilities in this system, the probability of servicing a low-priority customer in terms of the generating function, and a formula for the average number of customers of the second type.

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  1. 1.

    At the same time, logical errors were made in (4) and (17).


  1. 1.

    Gelenbe, E., Glynn, P. & Sigman, K. Queues with Negative Arrivals 28(no. 1), 245–250 (1991).

    Google Scholar 

  2. 2.

    Do, T. V. Bibliography on G-Networks, Negative Customers and Applications. Math. Comput. Model. 53, 205–212 (2011).

    Article  Google Scholar 

  3. 3.

    Caglayan, M. G-Networks and Their Applications to Machine Learning, Energy Packet Networks and Routing: Introduction to the Special Issue. Prob. Eng. Inform. Sci. 31(no. 4), 381–395 (2017).

    MathSciNet  Article  Google Scholar 

  4. 4.

    Malinkovskii, Yu.V. and Borodin, N.N.Queueing Networks with Finite Number of Flows of Negative Customers and with Limited Sojourn Time, Probl. Fiz., Mat. Tekh., 2017, no. 1, pp. 64–68.

  5. 5.

    Malinkovskii, Yu. V. Stationary Probability Distribution for States of G-Networks with Constrained Sojourn Time. Autom. Remote Control 78(no. 10), 1857–1866 (2017).

    MathSciNet  Article  Google Scholar 

  6. 6.

    Dimitriou, I. A Mixed Priority Retrial Queue with Negative Arrivals, Unreliable Server and Multiple Vacations. Appl. Math. Model. 37, 1295–1309 (2013).

    MathSciNet  Article  Google Scholar 

  7. 7.

    Rajkumar, M. An (sS) Retrial Inventory System with Impatient and Negative Customers. Int. J. Math. Oper. Res. 6, 106–122 (2014).

    MathSciNet  Article  Google Scholar 

  8. 8.

    Farkhadov, M. and Fedorova, E.Asymptotic Analysis of Retrial Queue MM∣1 with Negative Calls Under Heavy Load Condition, in Proc. 20th Int. Conf. "Distributed Computer and Communication Networks: Control, Computation, Communications” (DCCN-2017), Moscow: TEKHNOSFERA, 2017, pp. 470–475.

  9. 9.

    Farkhadov, M. & Fedorova, E. Retrial Queue MM∣1 with Negative Calls Under Heavy Load Condition. Commun. Comput. Inform. Sci. 700, 406–416 (2017).

    Article  Google Scholar 

  10. 10.

    Zidani, N., Spiteri, P. & Djellab, N. Numerical Solution for the Performance Characteristics of the MMCK Retrial Queue with Negative Customers and Exponential Abandonments by Using Value Extrapolation Method. RAIRO-Oper. Res. 53, 767–786 (2019).

    MathSciNet  Article  Google Scholar 

  11. 11.

    Kyung, C. Chae, Hyun, M. Park & Won, S. Yang A GIGeo∣1 Queue with Negative and Positive Customers. Appl. Math. Model. 34, 1662–1671 (2010).

    MathSciNet  Article  Google Scholar 

  12. 12.

    Wang, J., Huang, Y. & Dai, Z. A Discrete-Time On-Off Source Queueing System with Negative Customers. Comput. Ind. Eng. 61, 1226–1232 (2011).

    Article  Google Scholar 

  13. 13.

    Gao, Sh, Wang, J. & Zhang, D. Discrete-Time GIXGeo∣1∣N Queue with Negative Customers and Multiple Working Vacations. J. Korean Stat. Soc. 42, 515–528 (2013).

    Article  Google Scholar 

  14. 14.

    Wang, J., Huang, Y. & Do, T. A Single-Server Discrete-Time Queue with Correlated Positive and Negative Customer Arrivals. Appl. Math. Model. 37, 6212–6224 (2013).

    MathSciNet  Article  Google Scholar 

  15. 15.

    Lee, D.H. and Kim, K.Analysis of Repairable GeoG∣1 Queues with Negative Customers, Appl. Math. Model., 2014, Article ID 350621.

  16. 16.

    Senthil, VadivuA., Arumuganathan, R. & Senthil, KumarM. Analysis of Discrete-Time Queues with Correlated Arrivals, Negative Customers and Server Interruption. RAIRO-Oper. Res. 50, 67–81 (2016).

    MathSciNet  Article  Google Scholar 

  17. 17.

    Klimenok, V. I. & Dudin, A. N. A BMAPPHN Queue with Negative Customers and Partial Protection of Service. Commun. Statist. Simulat. Comput. 41(no. 7), 1062–1082 (2012).

    MathSciNet  Article  Google Scholar 

  18. 18.

    Rajadurai, P., Chandrasekaran, M. & Saravanarajan, M. C. Steady State Analysis of Batch Arrival Feedback Retrial Queue with Two Phases of Service, Negative Customers, Bernoulli Vacation and Server Breakdown. Int. J. Math. Oper. Res. 7, 519–546 (2015).

    MathSciNet  Article  Google Scholar 

  19. 19.

    Singh, C.J., Jain, M., Kaur, S., and Meena, R.K.Retrial Bulk Queue with State Dependent Arrival and Negative Customers, Proc. Sixth Int. Conf. on Soft Computing for Problem Solving: SocProS 2016, vol. 2: Advances in Intelligent Systems and Computing, Patiala, India, Switzerland: Springer, 2017, vol. 547, pp. 290–301.

  20. 20.

    Ayyappan, G. and Thamizhselvi, P.Transient Analysis of \({M}^{[{X}_{1}]},{M}^{[{X}_{2}]}/{G}_{1},{G}_{2}/1\) Retrial Queueing System with Priority Services, Working Vacations and Vacation Interruption, Emergency Vacation, Negative Arrival and Delayed Repair, Int. J. Appl. Comput. Math., 2018, vol. 4, article number 77.

  21. 21.

    Matalytski, M. A. Forecasting Anticipated Incomes in the Markov Networks with Positive and Negative Customers. Autom. Remote Control 78(no. 5), 815–825 (2017).

    MathSciNet  Article  Google Scholar 

  22. 22.

    Lee, D.H.Optimal Pricing Strategies and Customers’ Equilibrium Behavior in an Unobservable MM∣1 Queueing System with Negative Customers and Repair, Math. Probl. Eng., 2017, article ID 8910819.

  23. 23.

    Matalytski, M. Finding Expected Revenues in G-network with Multiple Classes of Positive and Negative Customers Probability in the Engineering and Informational Sciences. Prob. Eng. Inform. Sci. 33(no. 1), 105–120 (2019).

    MathSciNet  Article  Google Scholar 

  24. 24.

    Matalytski, M. Analysis of the Network with Multiple Classes of Positive and Negative Customers at a Transient Regime. Prob. Eng. Inform. Sci. 33(no. 2), 172–185 (2019).

    MathSciNet  Article  Google Scholar 

  25. 25.

    Sun, K. & Wang, J. Equilibrium Joining Strategies in the Single Server Queues with Negative Customers. Int. J. Comput. Math. 96(no. 6), 1169–1191 (2019).

    MathSciNet  Article  Google Scholar 

  26. 26.

    Xiu-li, Xu, Xian-ying, Wang, Xiao-feng, Song & Xiao-qing, Li Fluid Model Modulated by an MM∣1 Working Vacation Queue with Negative Customer. Acta. Math. Appl. Sin.-E. 34(no. 2), 404–415 (2018).

    MathSciNet  Article  Google Scholar 

  27. 27.

    Chin, C.H., Koh, S.K., Tan, Y.F., Pooi, A.H., and Goh, Y.K.Stationary Queue Length Distribution of A Continuous-Time Queueing System with Negative Arrival, J. Phys. Conf., 2018, vol. 1132, article ID 012057.

  28. 28.

    Peng, Y. The MAP/G/1 G-queue with Unreliable Server and Multiple Vacations. Informatica 43(no. 4), 545–550 (2019).

    Article  Google Scholar 

  29. 29.

    Gupta, U., Kumar, N., and Barbhuiya, F.A Queueing System with Batch Renewal Input and Negative Arrivals, arXiv:2002.08209v1, 2020.

  30. 30.

    Kreinin, A. Queueing Systems with Renovation. J. Appl. Math. Stoch. Anal. 10(no. 4), 431–443 (1997).

    MathSciNet  Article  Google Scholar 

  31. 31.

    Bocharov, P. P. & Zaryadov, I. S. Queueing Systems with Renovation. Stationary Probability Distribution. Vest. Ross. Univ. Druzh. Narod., Ser. "Mat. Informat. Fiz.” no. 1-2, 14–23 (2007).

    Google Scholar 

  32. 32.

    Zaryadov, I. S. & Pechinkin, A. V. Stationary Time Characteristics of the GIMn System with Some Variants of the Generalized Renovation Discipline. Autom. Remote Control 70(no. 12), 2085–2097 (2009).

    MathSciNet  Article  Google Scholar 

  33. 33.

    Zaryadov, I. S. The GIMn Queuing System with Generalized Renovation. Autom. Remote Control 71(no. 4), 663–671 (2010).

    MathSciNet  Article  Google Scholar 

  34. 34.

    Grishunina, Y. B. Optimal Control of Queue in the MG∣1∣ System with Possibility of Customer Admission Restriction. Autom. Remote Control 76(no. 3), 433–445 (2015).

    MathSciNet  Article  Google Scholar 

  35. 35.

    Agalarov, Ya. M. & Shorgin, V. S. About the Problem of Profit Maximization in GM∣1 Queuing Systems with Threshold Control of the Queue. Informat. Primen. 11(no. 4), 55–64 (2017).

    Google Scholar 

  36. 36.

    Konovalov, M. G. & Razumchik, R. V. Comparison of Two Active Queue Management Schemes through the MD∣1∣N Queue. Inform. Appl. 12(no. 4), 9–15 (2018).

    Google Scholar 

  37. 37.

    Adams, R. Active Queue Management: A Survey. IEEE Commun. Surveys Tutorials 15(no. 3), 1425–1476 (2013).

    Article  Google Scholar 

  38. 38.

    Sh, Anup & Ashok, B. A Survey on Active Queue Management Techniques. Int. J. Eng. Comput. Sci. 5, 18993–18997 (2016).

    Google Scholar 

  39. 39.

    White, H. & Christie, LeeS. Queuing with Preemptive Priorities or with Breakdown. Oper. Res. 6(no. 1), 79–95 (1958).

    MathSciNet  Article  Google Scholar 

  40. 40.

    Zaryadov, I. S. & Gorbunova, A. V. Analysis of the Characteristics of a Queueing System with Two Input Flows, Non-Preemptive Priority, and Dropping. Sovr. Inform. Tekhnol. IT-Obrazovanie no. 10, 388–393 (2014).

    Google Scholar 

  41. 41.

    Zaryadov, I. S. & Gorbunova, A. V. The Analysis of Queueing System with Two Input Flows and Stochastic Drop Mechanism. Bull. Peopl. Friends. Univ. Russ., Ser. Math., Informat., Phys. no. 2, 33–37 (2015).

    Google Scholar 

  42. 42.

    Zaryadov, I. S. & Korol’kova, A. V. The Application of Model with General Renovation to the Analysis of Characteristics of Active Queue Management with Random Early Detection (RED). T-Comm–Telekom. Transport no. 7, 84–88 (2011).

    Google Scholar 

  43. 43.

    Chydzinski, A. and Mrozowski, P.Queues with Dropping Functions and General Arrival Processes, PLoS ONE, 2016, vol. 11, article ID 0150702.

  44. 44.

    Wang, Y., Lin, Ch, Li, Q.-L. & Fang, Y. A Queueing Analysis for the Denial of Service (DoS) Attacks in Computer Networks. Comput. Networks 51, 3564–3573 (2007).

    Article  Google Scholar 

  45. 45.

    Imamverdiyev, Y. & Nabiyev, B. Queuing Model for Information Security Monitoring Systems. PIT 07(no. 1), 28–32 (2016).

    Google Scholar 

  46. 46.

    Kammas, P., Komninos, T., and Stamatiou, Y.C.Queuing Theory Based Models for Studying Intrusion Evolution and Elimination in Computer Networks, Fourth Int. Conf. on Information Assurance and Security, Napoli, Italy, 2008, pp. 167–171.

  47. 47.

    Ariba, Y., Gouaisbaut, F., Rahme, S. & Labit, Y. Traffic Monitoring in Transmission Control Protocol/Active Queue Management Networks through a Time-Delay Observer. IET Control Theory A 6(no. 2), 506–517 (2012).

    MathSciNet  Article  Google Scholar 

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Gorbunova, A., Lebedev, A. Queueing System with Two Input Flows, Preemptive Priority, and Stochastic Dropping. Autom Remote Control 81, 2230–2243 (2020). https://doi.org/10.1134/S0005117920120073

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  • queuing system
  • preemptive priority
  • generalized renovation
  • stochastic dropping