Control Systems and Computers, N6, 2016, Article 5
DOI: https://doi.org/10.15407/usim.2016.06.040
Upr. sist. maš., 2016, Issue 6 (266), pp. 40-47.
UDC 519.872
Bagirova Sevindj Alakber kyzy – doctoral student, Baku State University (Baku, Azerbaijan), E-mail: aseva@gmail.com
Analysis of Queuing–Inventory Systems with Instantaneous Service and Variable Size of Order
Introduction. In the scientific literature, the queuing-inventory systems with the different kinds of lead policy and service mechanisms are investigated. In this paper a model of the queuing-inventory systems with the variable order size and instantaneous service is considered. It is assumed that the customers can form orbit of the retrial customers. The exact and approximate methods to calculate the characteristics of the systems are developed.
Purpose. The following model of queuing-inventory systems with the finite size of stock is proposed. The input flow of customers is Poisson one and service time is zero. The backlog customers forms the orbit of the finite size and retrial calls from an orbit either lost the orbit or return to the orbit in order to repeat in the future. Sojourn time in orbit is random variable with exponential distribution function. The models of queuing-inventory systems with the finite and infinite size of orbit for the retrial customers are investigated. Lead policy is in class of variable size of order. The main characteristics of the system are average level of inventory, probability of loss of the initial and retrial customers and the average number of customers in the orbit. The effective method to calculate the indicated characteristics of the system is worked out.
Method. The functioning of the investigated model is described by two-dimensional Markov Chain (2-D MC). Infinitesimal matrix of the appropriate 2-D MC is developed. It is shown that stationary distribution of indicated 2-D MC has no analytical solution and the method to calculate the approximate values of characteristics of the system is demonstrated.
Results. The results of numerical experiments for both kinds of models with finite and infinite size of orbit for retrial customers are represented. Different schemas for changing the lead times are investigated. These results allow to perform the detailed analysis of the system behavior in the wide area of the changing parameters.
Conclusion. In this paper the simple algorithms to calculate the characteristics of queuing-inventory systems are developed. Complexities of the algorithms are very low and they allow to select an optimal size of order in the investigated systems. The last problems will be studied in the future works.
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Keywords: queuing–inventory systems, lead policy, variable order size.
1. Schwarz, M., Daduna, H., 2006. Queuing systems with inventory management with random lead times and with backordering. Mathematical Methods of Operations Research, 64 (3), pp. 383–414.
https://doi.org/10.1007/s00186-006-0085-1
2. M. Schwarz, C. Sauer, H. Daduna et al., 2006. M/M/1 queuing systems with inventory. Queuing systems. Theory and applications, 54 (1), pp. 55–78.
3. Ryzhikov, Yu.I., 2001. Teoriya ocheredey i upravleniye zapasami, SPb.: Piter, 384 p. (In Russian).
4. Melikov, A.Z., Ponomarenko, L.A., Bagirova, S.A., 2016. Analysis of queueing-inventory systems with impatience consume customers, J. of Autom. and Inform. Sci., 2016. 48 (1), pp. 53–68.
https://doi.org/10.1615/JAutomatInfScien.v48.i1.70
5. Melikov, A.Z., Ponomarenko, L.A., Bagirova, S.A., 2016. Models of queuing-inventory systems with randomized lead policy. J. of Autom. and Inform. Sci., 48 (10), pp. 53–68.
https://doi.org/10.1615/JAutomatInfScien.v48.i1.70
6. Demchenko, I.Yu., 2016. Statsionarnyye raspredeleniya v modelyakh upravleniya zapasami. Kibernetika i sistemnyy analiz, 2, pp. 163–166. (In Russian).
Received 09.12.2016
