Priority retrial queueing model operating in random environment with varying number and reservation of servers. (English) Zbl 1410.90050
Summary: 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.
MSC:
| 90B22 | Queues and service in operations research |
| 60K25 | Queueing theory (aspects of probability theory) |
Keywords:
random environment; multi-server queue; random number of servers; servers reservation; retrialsReferences:
| [1] | Kim, C. S.; Dudin, A. N.; Klimenok, V. I.; Khramova, V. V., Erlang loss queueing system with batch arrivals operating in a random environment, Comput. Operat. Res., 36, 674-967 (2009) · Zbl 1179.90077 |
| [2] | Kim, C. S.; Klimenok, V.; Mushko, V.; Dudin, A., The BMAP/PH/N retrial queueing system operating in markovian random environment, Comput. Operat. Res., 37, 1228-1237 (2010) · Zbl 1178.90097 |
| [3] | Wu, J.; Liu, Z.; Yang, G., Analysis of the finite source \(MAP / PH /N\) retrial g-queue operating in a random environment, Appl. Math. Model., 35, 1184-1193 (2011) · Zbl 1211.90053 |
| [4] | Cordeiro, J. D.; Kharoufeh, J. P., The unreliable \(M/M/1\) retrial queue in a random environment, Stochastic Models, 28, 29-48 (2012) · Zbl 1237.60070 |
| [5] | Yang, G.; Yao, L.-G.; Ouyang, Z.-S., The \(MAP / PH /N\) retrial queue in a random environment, Acta Math. Appl. Sin., 29, 725-738 (2013) · Zbl 1288.60124 |
| [6] | Mitrani, I. L.; Avi-Itzhak, B., A many-server queue with server interruptions, Operat. Res., 16, 628-638 (1968) · Zbl 0239.60094 |
| [7] | Neuts, M. F.; Lucantoni, D., A markovian queue with \(n\) servers subject to breakdowns and repair, Manag. Sci., 25, 849-861 (1979) |
| [8] | Al-Begain, K.; Dudin, A.; Klimenok, V.; Dudin, S., Generalised survivability analysis of systems with propagated failures, Comput. Math. Appl., 64, 3777-3791 (2012) · Zbl 1268.90014 |
| [9] | Klimenok, V. I.; Dudin, A. N., A \(BMAP / PH /N\) queue with negative customers and partial protection of service, Commun. Stat. - Simul. Comput., 41, 1062-1082 (2012) · Zbl 1267.60105 |
| [10] | Yang, X.; Alfa, A. F., A class of multi-server queueing systems with server failures, Comput. Ind. Eng., 56, 33-43 (2009) |
| [11] | Kuoa, C. C.; Sheub, S. H.; Ke, J. C.; Zhang, Z. G., Reliability-based measures for a retrial system with mixed standby components, Appl. Math. Model., 38, 4640-4651 (2014) · Zbl 1428.90054 |
| [12] | Hsu, Y. L.; Ke, J. C.; Liu, T. H.; Wu, C. H., Modeling of multi-server repair problem with switching failure and reboot delay and related profit analysis, Comput. Ind. Eng., 69, 21-28 (2014) |
| [13] | Wu, C. H.; Ke, J. C., Multi-server machine repair problems under a (v, r) synchronous single vacation policy, Appl. Math. Model., 38, 2180-2189 (2014) · Zbl 1427.90114 |
| [14] | Dimitriou, I., A batch arrival priority queue with recurrent repeated demands, admission control and hybrid failure recovery discipline, Appl. Math. Comput., 219, 11327-11340 (2013) · Zbl 1304.90065 |
| [15] | Jain, M.; Bhagat, A.; Shekhar, C., Double orbit finite retrial queues with priority customers and service interruptions, Appl. Math. Comput., 253, 324-344 (2015) · Zbl 1338.90130 |
| [16] | Akyildiz, I. F.; Lee, W. Y.; Vuran, M. C.; Mohanty, S., Next generation dynamic spectrum access cognitive radio wireless networks: a survey, Comput. Netw., 50, 2127-2159 (2006) · Zbl 1107.68018 |
| [17] | Chen, S.; Wyglinski, A.; Vuyyuru, R.; Altinas, O., Feasibility analysis of vehicular dynamic spectrum access via queueing theory model, IEEE Commun. Mag., 49, 156-163 (2011) |
| [18] | Konishi, Y.; Masuyama, H.; Kasara, S.; Takahashi, Y., Performance analysis of dynamic spectrum handoff scheme with variable bandwidth demand on secondary users for cognitive radio networks, Wireless Netw., 19, 607-617 (2013) |
| [19] | Mitola, J.; Maguire, G. Q., Cognitive radio: making software radios more personal, IEEE Personal Commun., 6, 13-18 (1999) |
| [20] | Zahed, S.; Awan, I.; Cullen, A., Analytical modeling for spectrum handoff decision in cognitive radio networks, Simul. Model. Pract. Theory, 38, 98-114 (2013) |
| [21] | Zhu, X. A.; Shen, L. A.; Yum, T.-S., Analysis of cognitive radio spectrum access with optimal channel reservation, IEEE Commun. Lett., 11, 304-306 (2007) |
| [22] | Sun, B.; Lee, M. H.; Dudin, A. N.; Dudin, S. A., Analysis of multiserver queueing system with opportunistic occupation and reservation of servers, Math. Probl. Eng., 2014, 1-13 (2014) · Zbl 1407.90120 |
| [23] | Graham, A., Kronecker Products and Matrix Calculus with Applications (1981), Ellis Horwood: Ellis Horwood Cichester · Zbl 0497.26005 |
| [24] | Klimenok, V. I.; Dudin, A. N., Multi-dimensional asymptotically quasi-toeplitz markov chains and their application in queueing theory, Queueing Syst., 54, 245-259 (2006) · Zbl 1107.60060 |
| [25] | Gantmakher, F. R., The Matrix Theory (1967), Science: Science Moscow |
| [26] | Dudina, O.; Kim, C.; Dudin, S., Retrial queuing system with markovian arrival flow and phase-type service time distribution, Comput. Ind. Eng., 66, 360-373 (2013) |
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